Abstract

Thermal drifts in long fiber-optic delay lines are compensated based on chromatic dispersion. An arbitrary input radio-frequency (RF) waveform and a control RF sine wave modulate two different tunable laser sources and are coupled into the fiber delay line. The RF phase of the control tone at the output of the delay line is monitored and used to adjust the wavelengths of both sources, so that the effects of thermal drifts and dispersion cancel out. The input and control waveforms are separated in the optical domain, and no restrictions are imposed on their RF spectra. A figure of merit is proposed, in terms of the fiber delay, range of temperature changes that may be compensated for, and residual delay variations. An upper bound on performance is established in terms of the specifications of the tunable lasers. The principle is used in the stable distribution of sine waves and of broadband linear frequency-modulated (LFM) waveforms, which are commonly employed in radar systems. Lastly, the method is incorporated in stable interrogation of a localized hot-spot within a high-resolution, distributed Brillouin fiber sensing setup. The results demonstrate the applicability of the proposed protocol in the processing of arbitrary waveforms, as part of larger, more complex systems.

© 2015 Optical Society of America

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References

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2015 (1)

Y. London, Y. Antman, N. Levanon, and A. Zadok, “Brillouin analysis with 8.8 km range and 2 cm resolution,” Proc. SPIE 9634, 96340G (2015).

2014 (3)

2013 (1)

2012 (5)

L. Yaron and M. Tur, “RF nonlinearities in an analog optical link and their effect on radars carrying linear and nonlinear frequency modulated pulses,” J. Lightwave Technol. 30(22), 3475–3483 (2012).
[Crossref]

Y. Antman, N. Levanon, and A. Zadok, “Low-noise delays from dynamic Brillouin gratings based on perfect Golomb coding of pump waves,” Opt. Lett. 37(24), 5259–5261 (2012).
[Crossref] [PubMed]

J. Yao, ““A tutorial on microwave photonics, Part 2,” IEEE Photonics Soc. News. 26(2), 4–12 (2012).

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

2011 (2)

2007 (3)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

A. Zadok, O. Raz, A. Eyal, and M. Tur, “Optically controlled low distortion delay of GHz-wide RF signals using slow light in fibers,” IEEE Photonics Technol. Lett. 19(7), 462–464 (2007).
[Crossref]

A. Zadok, A. Eyal, and M. Tur, “GHz-wide optically reconfigurable filters using stimulated Brillouin scattering,” J. Lightwave Technol. 25(8), 2168–2174 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

R. Rotman, O. Raz, and M. Tur, “Small signal analysis for analogue optical links with arbitrary optical transfer function,” Electron. Lett. 40(8), 504–505 (2004).
[Crossref]

2000 (1)

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique -proposal, experiment and simulation,” IEICE Trans. Electron E83-C(3), 405–412 (2000).

1998 (2)

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

D. T. K. Tong and M. C. Wu, “Multiwavelength optically controlled phased array antennas,” IEEE Trans. Microw. Theory Tech. 46(1), 108–115 (1998).
[Crossref]

1997 (1)

1996 (1)

1992 (1)

J. Wang and K. Petermann, “Small signal analysis for dispersive optical fiber communication systems,” J. Lightwave Technol. 10(1), 96–100 (1992).
[Crossref]

1990 (2)

I. L. Newberg, C. M. Gee, G. D. Thurmond, and H. W. Yen, “Long microwave delay fiber optic link for radar testing,” IEEE Trans. Micro. Theory Tech. 38(5), 664–666 (1990).
[Crossref]

T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990).
[Crossref]

Antman, Y.

Y. London, Y. Antman, N. Levanon, and A. Zadok, “Brillouin analysis with 8.8 km range and 2 cm resolution,” Proc. SPIE 9634, 96340G (2015).

D. Elooz, Y. Antman, N. Levanon, and A. Zadok, “High-resolution long-reach distributed Brillouin sensing based on combined time-domain and correlation-domain analysis,” Opt. Express 22(6), 6453–6463 (2014).
[Crossref] [PubMed]

Y. Antman, N. Levanon, and A. Zadok, “Low-noise delays from dynamic Brillouin gratings based on perfect Golomb coding of pump waves,” Opt. Lett. 37(24), 5259–5261 (2012).
[Crossref] [PubMed]

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

Ben-Amram, A.

A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

Brito, R.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Byrd, J. M.

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Castro, J.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Chang, L.

Choi, H. S.

Cliché, J.-F.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Dai, Y.

Denisov, A.

A. Denisov, M. A. Soto, and L. Thévenaz, “1’000’000 resolved points along a Brillouin distributed fibre sensor,” Proc. SPIE 9157, 9157D2 (2014).

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

Dong, Y.

Doolittle, L.

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

Elooz, D.

Esman, R. D.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

Eyal, A.

A. Zadok, O. Raz, A. Eyal, and M. Tur, “Optically controlled low distortion delay of GHz-wide RF signals using slow light in fibers,” IEEE Photonics Technol. Lett. 19(7), 462–464 (2007).
[Crossref]

A. Zadok, A. Eyal, and M. Tur, “GHz-wide optically reconfigurable filters using stimulated Brillouin scattering,” J. Lightwave Technol. 25(8), 2168–2174 (2007).
[Crossref]

Gee, C. M.

I. L. Newberg, C. M. Gee, G. D. Thurmond, and H. W. Yen, “Long microwave delay fiber optic link for radar testing,” IEEE Trans. Micro. Theory Tech. 38(5), 664–666 (1990).
[Crossref]

Grammer, W.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Hasegawa, T.

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique -proposal, experiment and simulation,” IEICE Trans. Electron E83-C(3), 405–412 (2000).

He, H.

He, Z.

Horiguchi, T.

T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990).
[Crossref]

Hotate, K.

K. Y. Song, Z. He, and K. Hotate, “Distributed strain measurement with millimeter-order spatial resolution based on Brillouin optical correlation domain analysis,” Opt. Lett. 31(17), 2526–2528 (2006).
[Crossref] [PubMed]

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique -proposal, experiment and simulation,” IEICE Trans. Electron E83-C(3), 405–412 (2000).

Hou, D.

Hu, W.

Jacques, C.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Ji, Y.

Kurashima, T.

T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990).
[Crossref]

Lee, C. E.

Levanon, N.

Li, J.

Li, P.

Lin, J.

Liu, C.

Livingston, M.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

London, Y.

Y. London, Y. Antman, N. Levanon, and A. Zadok, “Brillouin analysis with 8.8 km range and 2 cm resolution,” Proc. SPIE 9634, 96340G (2015).

A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

Masui, Y.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Meadows, J.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Newberg, I. L.

I. L. Newberg, C. M. Gee, G. D. Thurmond, and H. W. Yen, “Long microwave delay fiber optic link for radar testing,” IEEE Trans. Micro. Theory Tech. 38(5), 664–666 (1990).
[Crossref]

Nichols, L. T.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

Niklès, M.

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Parent, M. G.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

Petermann, K.

J. Wang and K. Petermann, “Small signal analysis for dispersive optical fiber communication systems,” J. Lightwave Technol. 10(1), 96–100 (1992).
[Crossref]

Primerov, N.

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

Ratti, A.

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

Raz, O.

A. Zadok, O. Raz, A. Eyal, and M. Tur, “Optically controlled low distortion delay of GHz-wide RF signals using slow light in fibers,” IEEE Photonics Technol. Lett. 19(7), 462–464 (2007).
[Crossref]

R. Rotman, O. Raz, and M. Tur, “Analysis of a true time delay photonic beamformer for transmission of a linear frequency-modulated waveform,” J. Lightwave Technol. 23(12), 4026–4036 (2005).
[Crossref]

R. Rotman, O. Raz, and M. Tur, “Small signal analysis for analogue optical links with arbitrary optical transfer function,” Electron. Lett. 40(8), 504–505 (2004).
[Crossref]

Ren, T.

Robert, P. A.

Román, J. E.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

Rotman, R.

R. Rotman, O. Raz, and M. Tur, “Analysis of a true time delay photonic beamformer for transmission of a linear frequency-modulated waveform,” J. Lightwave Technol. 23(12), 4026–4036 (2005).
[Crossref]

R. Rotman, O. Raz, and M. Tur, “Small signal analysis for analogue optical links with arbitrary optical transfer function,” Electron. Lett. 40(8), 504–505 (2004).
[Crossref]

Sancho, J.

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

Shillue, W.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Song, K. Y.

Soto, M. A.

A. Denisov, M. A. Soto, and L. Thévenaz, “1’000’000 resolved points along a Brillouin distributed fibre sensor,” Proc. SPIE 9157, 9157D2 (2014).

Staples, J. W.

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

Stern, Y.

A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

Tang, G.

Tateda, M.

T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990).
[Crossref]

Tavik, G. C.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
[Crossref]

Taylor, H. F.

Thevenaz, L.

A. Zadok, Y. Antman, N. Primerov, A. Denisov, J. Sancho, and L. Thevenaz, “Random-access distributed fiber sensing,” Laser Photonics Rev. 6(5), L1–L5 (2012).
[Crossref]

Thévenaz, L.

A. Denisov, M. A. Soto, and L. Thévenaz, “1’000’000 resolved points along a Brillouin distributed fibre sensor,” Proc. SPIE 9157, 9157D2 (2014).

M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett. 21(10), 758–760 (1996).
[Crossref] [PubMed]

Thurmond, G. D.

I. L. Newberg, C. M. Gee, G. D. Thurmond, and H. W. Yen, “Long microwave delay fiber optic link for radar testing,” IEEE Trans. Micro. Theory Tech. 38(5), 664–666 (1990).
[Crossref]

Tong, D. T. K.

D. T. K. Tong and M. C. Wu, “Multiwavelength optically controlled phased array antennas,” IEEE Trans. Microw. Theory Tech. 46(1), 108–115 (1998).
[Crossref]

Treacy, R.

W. Shillue, W. Grammer, C. Jacques, R. Brito, J. Meadows, J. Castro, Y. Masui, R. Treacy, and J.-F. Cliché, “The ALMA photonic local oscillator system,” Proc. SPIE 8452, 845216 (2012).
[Crossref]

Tur, M.

Wang, J.

J. Wang and K. Petermann, “Small signal analysis for dispersive optical fiber communication systems,” J. Lightwave Technol. 10(1), 96–100 (1992).
[Crossref]

Wilcox, R.

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

Wiliams, K. J.

J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
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Y. London, Y. Antman, N. Levanon, and A. Zadok, “Brillouin analysis with 8.8 km range and 2 cm resolution,” Proc. SPIE 9634, 96340G (2015).

D. Elooz, Y. Antman, N. Levanon, and A. Zadok, “High-resolution long-reach distributed Brillouin sensing based on combined time-domain and correlation-domain analysis,” Opt. Express 22(6), 6453–6463 (2014).
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A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

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A. Zadok, O. Raz, A. Eyal, and M. Tur, “Optically controlled low distortion delay of GHz-wide RF signals using slow light in fibers,” IEEE Photonics Technol. Lett. 19(7), 462–464 (2007).
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I. L. Newberg, C. M. Gee, G. D. Thurmond, and H. W. Yen, “Long microwave delay fiber optic link for radar testing,” IEEE Trans. Micro. Theory Tech. 38(5), 664–666 (1990).
[Crossref]

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J. E. Román, L. T. Nichols, K. J. Wiliams, R. D. Esman, G. C. Tavik, M. Livingston, and M. G. Parent, “Fiber-optic remoting of an ultrahigh dynamic range radar,” IEEE Trans. Microw. Theory Tech. 46(12), 2317–2323 (1998).
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Other (6)

J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples, and R. Wilcox, “Timing distribution in accelerators via stabilized optical fiber links,” in Proceeding of LINAC (2006), pp. 577–579.

A. M. Koonen, M. G. Larrode, A. Ng’oma, K. Wang, H. Yang, Y. Zheng, and E. Tangdiongga, “Perspectives of radio-over-fiber technologies,” paper OThP3 in Proceedings of Optical Fiber Communication Conference (OFC 2008), Optical Society of America (2008).

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[Crossref]

A. Ben-Amram, Y. Stern, Y. London, Y. Antman, and A. Zadok, “Stabilized fiber-optic delay of arbitrary waveforms with application in distributed sensors,” in IEEE Conference on Microwave Photonics (MWP) (IEEE, 2015).

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Figures (9)

Fig. 1
Fig. 1 Experimental setup for the stabilized delay of arbitrary input RF waveforms based on chromatic dispersion, using two tunable laser sources. Blue, solid lines: fiber-optic paths. Green, dashed lines: RF paths of the control tone. Red, dashed lines: RF paths of the input waveform. Black, dashed-dotted lines, DC control signals. PC: polarization controller [22].
Fig. 2
Fig. 2 Persistence traces of a 3 GHz control sine wave, following two-way propagation along 9 km of fiber. Left – free-running delay. Right – closed-loop delay.
Fig. 3
Fig. 3 Left - output DC voltage obtained in the mixing of a delayed 3 GHz sine wave and a non-delayed replica, as a function of time. The delay line was free-running in the first ten minutes of the experiment, and the feedback loop was closed in the final ten minutes. Right - wavelength of the tunable laser source of control channel λ 1 as a function of time, during the same measurement. The wavelength was not modified in the first ten minutes of the measurements, and was continuously adjusted during closed-loop operation over the final ten minutes, to maintain a stable delay.
Fig. 4
Fig. 4 Persistence traces of compressed LFM waveforms, sampled at the output of a 9 km-long fiber-optic delay line every 5 s, over 2 minutes. The central frequency, bandwidth and duration of the waveforms were 500 MHz, 1.75 GHz and 5 µs, respectively. The main lobe of the compressed waveform is 2 ns wide. Left – free-running delay line. Right – closed-loop operation. Delay drifts are reduced from 200 ps to 20 ps in closed-loop operation.
Fig. 5
Fig. 5 Wavelengths of the tunable laser sources as a function of time: control channel λ 1 (blue trace, left-hand axis), and channel of the LFM waveform λ 2 (green trace, right-hand axis), during closed-loop operation.
Fig. 6
Fig. 6 Schematic illustration of a phase-coded Brillouin optical correlation-domain analysis setup, incorporating a stabilized fiber-optic delay line in the path of the Brillouin signal wave. Solid lines denote fiber paths, dashed lines represent RF cable paths, and dashed-dotted, black lines correspond to DC control signals. Blue color represents paths and components related to the sensing functionality, whereas green paths and components are part of the delay stabilization module. Details of the 'Pump wave processing' module are not shown, for better clarity (see [12]).
Fig. 7
Fig. 7 Top – Measured Brillouin gain as a function of frequency offset between pump and signal, and of time. The phase-coded B-OCDA setup was adjusted to monitor the Brillouin gain spectrum in a 2 cm-wide hot-spot. The delay of the Brillouin signal was free-running, without closed-loop feedback to the wavelength of the sensor laser. The Brillouin gain spectrum changed after 10 minutes, from that of the hot-spot to that of the fiber under test at room temperature. Bottom – Wavelength of the control channel λ 1 (left-hand axis), and the implied thermal drift in the path length of the 25 km-long delay line (right-hand axis), as a function of time. Thermal drift on the order of 10 cm is observed. SBS: stimulated Brillouin scattering.
Fig. 8
Fig. 8 Top – Measured Brillouin gain as a function of frequency offset between pump and signal, and of time. The phase-coded B-OCDA setup was adjusted to monitor the Brillouin gain spectrum in a 2 cm-wide hot-spot. Unlike Fig. 7, the delay of the Brillouin signal was stabilized through closed-loop feedback to the wavelength of the sensor laser. The Brillouin gain spectrum remained that of the hot-spot throughout the measurements. Bottom – Wavelength of the Brillouin sensor laser source λ 2 (left-hand axis), and the implied thermal drift in the path length of the 25 km-long delay line (right-hand axis), as a function of time. Thermal drift on the order of 9 cm is observed, which is several times larger than the extent of the hot-spot. SBS: stimulated Brillouin scattering.
Fig. 9
Fig. 9 Measured Brillouin frequency shifts as a function of time. The phase-coded B-OCDA setup was adjusted to monitor the location of a 2 cm-wide hot-spot. Thermal drift in a 25 km-long delay line incorporated in the setup leads to incorrect interrogation after 10 minutes of free-running operation (blue). Thermal drift is overcome with closed-loop, stabilized delay (red).

Equations (12)

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Δ τ T =( 1 n n T + 1 L L T ) n c LΔT=( 1 n n T + 1 L L T )τΔT α τ τΔT.
Δ τ D ( λ )=D( λ )LΔλ= c n D( λ )τΔλ.
Δλ( λ )= n cD( λ ) α τ ΔT.
Δ φ 1T =2πf2Δ τ T .
Δ φ 1D =Δ φ 1T Δ λ 1 = Δ τ T LD( λ 1 ) = Δ φ 1T 2πf2LD( λ 1 ) .
V= V 0 cos( Δ φ 1B +Δ φ 1T +Δ φ 1D ).
V= V 0 sin( Δ φ 1T +Δ φ 1D ) V 0 ( Δ φ 1T +Δ φ 1D ).
Δ λ 1 = 1 2πf2LD( λ 1 ) V V 0 α λ1 V.
Δ λ 2 = 1 2πf2LD( λ 2 ) V V 0 = D( λ 1 ) D( λ 2 ) α λ1 V α λ2 V.
FoM τ δτ Δ T max = τ LDδλ cD n α τ Δ λ max = 1 α τ Δ λ max δλ 1 α τ N λ .
f 3dB = 1 2π π 3| β 2 |L = 1 2λ 2n 3| D |τ 1 2λ | D |τ .
A( t )= A 0 cos( 2π f 0 t+π B T LFM t 2 )rect( t T LFM ).

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