Abstract

A general analytic solution for Brillouin distributed sensing in optical fibers with sub-meter spatial resolution is obtained by solving the acoustical-optical coupled wave equations by a perturbation method. The Brillouin interaction of a triad of square pump pulses with a continuous signal is described, covering a wide range of pumping schemes. The model predicts how the acoustic wave, the signal amplitude and the optical gain spectral profile depend upon the pumping scheme. Sub-meter spatial resolution is demonstrated for bright-, dark- and π-shifted interrogating pump pulses, together with disturbing echo effects, and the results compare favorably with experimental data. This analytic solution is an excellent tool not only for optimizing the pumping scheme but also for post-processing the measured data to remove resolution degrading features.

© 2011 OSA

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    [CrossRef] [PubMed]

2010

L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
[CrossRef]

S. Foaleng Mafang, M. Tur, J.-C. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[CrossRef]

2009

L. Thevenaz and J.-C. Beugnot, ““General analytical model for distributed Brillouin sensors with sub-meter spatial resolution”, Proceedings of 20th International Conference of Fiber Sensors, (SPIE, Edinburgh, United Kingdom),” Proc. SPIE 7503, 75036A, 75036A-4 (2009).
[CrossRef]

2008

2007

2006

2005

2004

R. Bernini, A. Minardo, and L. Zeni, “Accuracy Enhancement in Brillouin Distributed Fiber-Optic Temperature Sensors Using Signal Processing Techniques,” IEEE Photon. Technol. Lett. 16(4), 1143–1145 (2004).
[CrossRef]

2003

2002

I. Alasaarela, P. Karioja, and H. Kopola, “Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements,” Opt. Eng. 41(1), 181–189 (2002).
[CrossRef]

2000

1999

1992

R. Chu, M. Kanefsky, and J. Falk, “Numerical study of transient stimulated Brillouin scattering,” J. Appl. Phys. 71(10), 4653–4658 (1992).
[CrossRef]

Afshar, S.

Alasaarela, I.

I. Alasaarela, P. Karioja, and H. Kopola, “Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements,” Opt. Eng. 41(1), 181–189 (2002).
[CrossRef]

Bao, X.

F. Wang, X. Bao, L. Chen, Y. Li, J. Snoddy, and X. Zhang, “Using pulse with a dark base to achieve high spatial and frequency resolution for the distributed Brillouin sensor,” Opt. Lett. 33(22), 2707–2709 (2008), http://www.opticsinfobase.org/abstract.cfm?uri=ol-33-22-2707 .
[CrossRef] [PubMed]

W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express 16(26), 21616–21625 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-26-21616 .
[CrossRef] [PubMed]

W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express 16(26), 21616–21625 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-26-21616 .
[CrossRef] [PubMed]

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

V. P. Kalosha, E. A. Ponomarev, L. Chen, and X. Bao, “How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses,” Opt. Express 14(6), 2071–2078 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=OE-14-6-2071 .
[CrossRef] [PubMed]

Y. Wan, S. Afshar, L. Zou, L. Chen, and X. Bao, “Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors,” Opt. Lett. 30(10), 1099–1101 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-10-1099 .
[CrossRef] [PubMed]

S. Afshar, X. Bao, L. Zou, and L. Chen, “Brillouin spectral deconvolution method for centimeter spatial resolution and high-accuracy strain measurement in Brillouin sensors,” Opt. Lett. 30(7), 705–707 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-7-705 .
[CrossRef] [PubMed]

X. Bao, Y. Wan, L. Zou, and L. Chen, “Effect of optical phase on a distributed Brillouin sensor at centimeter spatial resolution,” Opt. Lett. 30(8), 827–829 (2005), http://www.opticsinfobase.org/abstract.cfm?&uri=ol-30-8-827 .
[CrossRef] [PubMed]

S. Afshar, G. A. Ferrier, X. Bao, and L. Chen, “Effect of the finite extinction ratio of an electro-optic modulator on the performance of distributed probe-pump Brillouin sensor systems,” Opt. Lett. 28(16), 1418–1420 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-16-1418 .
[CrossRef] [PubMed]

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Spatial Resolution Enhancement of Brillouin-Distributed Sensor Using a Novel Signal Processing Method,” J. Lightwave Technol. 17(7), 1179–1183 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=JLT-17-7-1179 .
[CrossRef]

X. Bao, A. Brown, M. Demerchant, and J. Smith, “Characterization of the Brillouin-loss spectrum of single-mode fibers by use of very short (<10-ns) pulses,” Opt. Lett. 24(8), 510–512 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-8-510 .
[CrossRef]

Bernini, R.

A. Minardo, R. Bernini, and L. Zeni, “Stimulated Brillouin scattering modeling for high-resolution, time-domain distributed sensing,” Opt. Express 15(16), 10397–10407 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-16-10397 .
[CrossRef] [PubMed]

A. Minardo, R. Bernini, and L. Zeni, “A Simple Technique for Reducing Pump Depletion in Long-Range Distributed Brillouin Fiber Sensors,” Meas. Sci. Technol. 16, 633–634 (2005).

R. Bernini, A. Minardo, and L. Zeni, “Accuracy Enhancement in Brillouin Distributed Fiber-Optic Temperature Sensors Using Signal Processing Techniques,” IEEE Photon. Technol. Lett. 16(4), 1143–1145 (2004).
[CrossRef]

Beugnot, J.-C.

S. Foaleng Mafang, M. Tur, J.-C. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[CrossRef]

L. Thevenaz and J.-C. Beugnot, ““General analytical model for distributed Brillouin sensors with sub-meter spatial resolution”, Proceedings of 20th International Conference of Fiber Sensors, (SPIE, Edinburgh, United Kingdom),” Proc. SPIE 7503, 75036A, 75036A-4 (2009).
[CrossRef]

Bremner, T. W.

Brown, A.

Brown, A. W.

Brown, K.

Cambon, P.

Chen, L.

F. Wang, X. Bao, L. Chen, Y. Li, J. Snoddy, and X. Zhang, “Using pulse with a dark base to achieve high spatial and frequency resolution for the distributed Brillouin sensor,” Opt. Lett. 33(22), 2707–2709 (2008), http://www.opticsinfobase.org/abstract.cfm?uri=ol-33-22-2707 .
[CrossRef] [PubMed]

W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express 16(26), 21616–21625 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-26-21616 .
[CrossRef] [PubMed]

W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express 16(26), 21616–21625 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-26-21616 .
[CrossRef] [PubMed]

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

V. P. Kalosha, E. A. Ponomarev, L. Chen, and X. Bao, “How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses,” Opt. Express 14(6), 2071–2078 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=OE-14-6-2071 .
[CrossRef] [PubMed]

Y. Wan, S. Afshar, L. Zou, L. Chen, and X. Bao, “Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors,” Opt. Lett. 30(10), 1099–1101 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-10-1099 .
[CrossRef] [PubMed]

S. Afshar, X. Bao, L. Zou, and L. Chen, “Brillouin spectral deconvolution method for centimeter spatial resolution and high-accuracy strain measurement in Brillouin sensors,” Opt. Lett. 30(7), 705–707 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-7-705 .
[CrossRef] [PubMed]

X. Bao, Y. Wan, L. Zou, and L. Chen, “Effect of optical phase on a distributed Brillouin sensor at centimeter spatial resolution,” Opt. Lett. 30(8), 827–829 (2005), http://www.opticsinfobase.org/abstract.cfm?&uri=ol-30-8-827 .
[CrossRef] [PubMed]

S. Afshar, G. A. Ferrier, X. Bao, and L. Chen, “Effect of the finite extinction ratio of an electro-optic modulator on the performance of distributed probe-pump Brillouin sensor systems,” Opt. Lett. 28(16), 1418–1420 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-16-1418 .
[CrossRef] [PubMed]

Chu, R.

R. Chu, M. Kanefsky, and J. Falk, “Numerical study of transient stimulated Brillouin scattering,” J. Appl. Phys. 71(10), 4653–4658 (1992).
[CrossRef]

Colpitts, B. G.

Demerchant, M.

DeMerchant, M. D.

Falk, J.

R. Chu, M. Kanefsky, and J. Falk, “Numerical study of transient stimulated Brillouin scattering,” J. Appl. Phys. 71(10), 4653–4658 (1992).
[CrossRef]

Ferrier, G. A.

Foaleng Mafang, S.

S. Foaleng Mafang, M. Tur, J.-C. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[CrossRef]

L. Thévenaz and S. Foaleng Mafang, ““Distributed fiber sensing using echoes”, Proceedings of 19th International Conference of Fiber Sensors, (SPIE, Perth, WA, Australia),” Proc. SPIE 7004, 70043N, 70043N-4 (2008).

Jackson, D. A.

Jenkins, R. B.

Joseph, R. I.

Kalosha, V. P.

Kanefsky, M.

R. Chu, M. Kanefsky, and J. Falk, “Numerical study of transient stimulated Brillouin scattering,” J. Appl. Phys. 71(10), 4653–4658 (1992).
[CrossRef]

Karioja, P.

I. Alasaarela, P. Karioja, and H. Kopola, “Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements,” Opt. Eng. 41(1), 181–189 (2002).
[CrossRef]

Kolosha, V. P.

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

Kopola, H.

I. Alasaarela, P. Karioja, and H. Kopola, “Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements,” Opt. Eng. 41(1), 181–189 (2002).
[CrossRef]

Le Floch, S.

Lecoeuche, V.

Li, W.

Li, Y.

Minardo, A.

A. Minardo, R. Bernini, and L. Zeni, “Stimulated Brillouin scattering modeling for high-resolution, time-domain distributed sensing,” Opt. Express 15(16), 10397–10407 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-16-10397 .
[CrossRef] [PubMed]

A. Minardo, R. Bernini, and L. Zeni, “A Simple Technique for Reducing Pump Depletion in Long-Range Distributed Brillouin Fiber Sensors,” Meas. Sci. Technol. 16, 633–634 (2005).

R. Bernini, A. Minardo, and L. Zeni, “Accuracy Enhancement in Brillouin Distributed Fiber-Optic Temperature Sensors Using Signal Processing Techniques,” IEEE Photon. Technol. Lett. 16(4), 1143–1145 (2004).
[CrossRef]

Pannell, C. N.

Ponomarev, E. A.

Ravet, F.

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

Smith, J.

Snoddy, J.

Sova, R.

Thevenaz, L.

L. Thevenaz and J.-C. Beugnot, ““General analytical model for distributed Brillouin sensors with sub-meter spatial resolution”, Proceedings of 20th International Conference of Fiber Sensors, (SPIE, Edinburgh, United Kingdom),” Proc. SPIE 7503, 75036A, 75036A-4 (2009).
[CrossRef]

Thévenaz, L.

L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
[CrossRef]

S. Foaleng Mafang, M. Tur, J.-C. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[CrossRef]

L. Thévenaz and S. Foaleng Mafang, ““Distributed fiber sensing using echoes”, Proceedings of 19th International Conference of Fiber Sensors, (SPIE, Perth, WA, Australia),” Proc. SPIE 7004, 70043N, 70043N-4 (2008).

Tur, M.

Wan, Y.

Wang, F.

Webb, D. J.

Yale, A.

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

Zeni, L.

A. Minardo, R. Bernini, and L. Zeni, “Stimulated Brillouin scattering modeling for high-resolution, time-domain distributed sensing,” Opt. Express 15(16), 10397–10407 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-16-10397 .
[CrossRef] [PubMed]

A. Minardo, R. Bernini, and L. Zeni, “A Simple Technique for Reducing Pump Depletion in Long-Range Distributed Brillouin Fiber Sensors,” Meas. Sci. Technol. 16, 633–634 (2005).

R. Bernini, A. Minardo, and L. Zeni, “Accuracy Enhancement in Brillouin Distributed Fiber-Optic Temperature Sensors Using Signal Processing Techniques,” IEEE Photon. Technol. Lett. 16(4), 1143–1145 (2004).
[CrossRef]

Zhang, X.

Zou, L.

Front. Optoelectron. China

L. Thévenaz, “Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives,” Front. Optoelectron. China 3(1), 13–21 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

A. W. Brown, B. G. Colpitts, and K. Brown, “Distributed Sensor Based on Dark-Pulse Brillouin Scattering,” IEEE Photon. Technol. Lett. 17(7), 1501–1503 (2005).
[CrossRef]

R. Bernini, A. Minardo, and L. Zeni, “Accuracy Enhancement in Brillouin Distributed Fiber-Optic Temperature Sensors Using Signal Processing Techniques,” IEEE Photon. Technol. Lett. 16(4), 1143–1145 (2004).
[CrossRef]

IEEE Sens. J.

F. Ravet, X. Bao, Y. Li, A. Yale, V. P. Kolosha, and L. Chen, “Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution,” IEEE Sens. J. 6, 3610–3618 (2007).

J. Appl. Phys.

R. Chu, M. Kanefsky, and J. Falk, “Numerical study of transient stimulated Brillouin scattering,” J. Appl. Phys. 71(10), 4653–4658 (1992).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

A. Minardo, R. Bernini, and L. Zeni, “A Simple Technique for Reducing Pump Depletion in Long-Range Distributed Brillouin Fiber Sensors,” Meas. Sci. Technol. 16, 633–634 (2005).

Opt. Eng.

I. Alasaarela, P. Karioja, and H. Kopola, “Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements,” Opt. Eng. 41(1), 181–189 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

F. Wang, X. Bao, L. Chen, Y. Li, J. Snoddy, and X. Zhang, “Using pulse with a dark base to achieve high spatial and frequency resolution for the distributed Brillouin sensor,” Opt. Lett. 33(22), 2707–2709 (2008), http://www.opticsinfobase.org/abstract.cfm?uri=ol-33-22-2707 .
[CrossRef] [PubMed]

V. Lecoeuche, D. J. Webb, C. N. Pannell, and D. A. Jackson, “Transient response in high-resolution Brillouin-based distributed sensing using probe pulses shorter than the acoustic relaxation time,” Opt. Lett. 25(3), 156–158 (2000), http://www.opticsinfobase.org/abstract.cfm?&uri=ol-25-3-156 .
[CrossRef]

X. Bao, A. Brown, M. Demerchant, and J. Smith, “Characterization of the Brillouin-loss spectrum of single-mode fibers by use of very short (<10-ns) pulses,” Opt. Lett. 24(8), 510–512 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-8-510 .
[CrossRef]

S. Afshar, G. A. Ferrier, X. Bao, and L. Chen, “Effect of the finite extinction ratio of an electro-optic modulator on the performance of distributed probe-pump Brillouin sensor systems,” Opt. Lett. 28(16), 1418–1420 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-16-1418 .
[CrossRef] [PubMed]

S. Afshar, X. Bao, L. Zou, and L. Chen, “Brillouin spectral deconvolution method for centimeter spatial resolution and high-accuracy strain measurement in Brillouin sensors,” Opt. Lett. 30(7), 705–707 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-7-705 .
[CrossRef] [PubMed]

X. Bao, Y. Wan, L. Zou, and L. Chen, “Effect of optical phase on a distributed Brillouin sensor at centimeter spatial resolution,” Opt. Lett. 30(8), 827–829 (2005), http://www.opticsinfobase.org/abstract.cfm?&uri=ol-30-8-827 .
[CrossRef] [PubMed]

Y. Wan, S. Afshar, L. Zou, L. Chen, and X. Bao, “Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors,” Opt. Lett. 30(10), 1099–1101 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-10-1099 .
[CrossRef] [PubMed]

Proc. SPIE

L. Thévenaz and S. Foaleng Mafang, ““Distributed fiber sensing using echoes”, Proceedings of 19th International Conference of Fiber Sensors, (SPIE, Perth, WA, Australia),” Proc. SPIE 7004, 70043N, 70043N-4 (2008).

L. Thevenaz and J.-C. Beugnot, ““General analytical model for distributed Brillouin sensors with sub-meter spatial resolution”, Proceedings of 20th International Conference of Fiber Sensors, (SPIE, Edinburgh, United Kingdom),” Proc. SPIE 7503, 75036A, 75036A-4 (2009).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics (Academic, NY, 2003) 3th ed.

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution”, Proceedings of 12th International Conference on Optical Fiber Sensors, WilliamsburgVA, OSA publications, Washington DC, 324–327 (1997). http://www.opticsinfobase.org/abstract.cfm?URI=OFS-1997-OWD3

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Figures (9)

Fig. 1
Fig. 1

(a) Brillouin amplifier configuration in a fiber segment of length Δz. (b) Pump coding waveform where α is purely real and β,γ are complex. The pump pulse duration T determines the spatial resolution.

Fig. 2
Fig. 2

Signal gain at resonance in time domain for different pump waveforms, as calculated by plotting Eq. (6) with 2z0/Vg = 20ns, T = 5ns. (a) Intensity pulse (α = γ = 0 and β = 1); (b) bright pulse with a CW component (α = γ = 0.5 and β = 1); (c) dark pulse (α = γ = 1 and β = 0); and (d) π−phase pulse (α = γ = 1 and β = −1).

Fig. 3
Fig. 3

(a) Calculated (Eq. (A12) in (3)) 3D signal gain for a 1ns intensity pulse configuration (α = γ = 0, β = 1) and for an interacting fiber segment Δz = 1m, arbitrarily starting at t = 10ns; (b) Calculated Brillouin gain spectrum in a bright pulse configuration (β = 1) for different CW backgrounds at the end of the pulse: t = 11ns. The red line corresponds to a vertical cut through the spectrum of (a) at t = 11ns.

Fig. 4
Fig. 4

(a) 3D gain signal diagram in a dark pulse configuration (α = γ = 1, β = 0) as calculated from the full analytic solution of Eq. (A12) for an interacting fiber segment Δz = 1m and T = 1ns. The pulse enters the fiber at t = 10ns. The background Brillouin signal, introduced by the nonzero α or γ has been removed in this 3D diagram by subtraction and the polarity of the spectrum was inverted for clarity. (b) Time-domain plots of the Brillouin gain for different frequency detunings, Eq. (A12) (Here, the background Brillouin signal, mentioned above was not removed and the true polarity of the signal is shown). The observed oscillation frequencies are equal to the detuning values.

Fig. 5
Fig. 5

(a) Calculated (Eq. (A12) in (3)) 3D gain signal diagram for a 1ns π-phase pulse as a function of distance and frequency from a 1m long fiber segment. The plot has been vertically inverted and CW background were subtracted. (b) Calculated Brillouin gain spectrum at the end of pulse for three different pump pulse configuration for T = 1ns. Black line, π-phase β pulse (α = γ = 1, β = −1), red line, dark pulse (α = γ = 1, β = 0) and blue line, intensity pulse (α = γ = 0, β = 1). The dark-pulse spectrum and π-phase spectrum have been inverted for clarity. The π-phase shift technique provides the highest Brillouin signal.

Fig. 6
Fig. 6

(a) Experimental gain distribution with 0.5ns π−phase pump pulse. (b) Calculated gain distribution for the three concatentated fiber segments used in the experiment, as obtained from the full analytic solution in Eqs. (A10), (A12).

Fig. 7
Fig. 7

A distance-domain plot of the Brillouin gain at 10.73GHz, as obtained from the full analytic solution (average was not subtracted), see text for details.

Fig. 8
Fig. 8

(a) Experimental signal gain at 10.73 GHz and (b) signal spectrum at 10.7m. The blue and green lines correspond to acquired and processed data (Sec. 6), respectively. The scale is linear.

Fig. 9
Fig. 9

(a) Experimental gain distribution after filtering by the impulse response of Eq. (8).The scale is linear. (b) Effect of filtering on the frequency distribution along the fiber. The green and blue line corresponds to the original measured data and processed data, respectively.

Equations (33)

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E ^ p ( z , t ) = 1 2 A P ( z , t ) e i ( k P z ω P t ) + c c ,
E ^ S ( z , t ) = 1 2 A S ( z , t ) e i ( k S z ω S t ) + c c ,
Q ^ ( z , t ) = 1 2 Q ( z , t ) e i ( q z Ω t ) + c c .
A p ( z , t ) z + 1 V g A p ( z , t ) t = i 1 2 g 2 ( z ) A s ( z , t ) Q ( z , t ) ,
A S ( z , t ) z 1 V g A S ( z , t ) t = i 1 2 g 2 ( z ) A P ( z , t ) Q * ( z , t ) ,
Q ( z , t ) t + Γ A ( z ) Q ( z , t ) = i g 1 ( z ) A P ( z , t ) A s * ( z , t ) .
P S ( z = 0 , Ω , t ) = | A S 0 + a S | 2 | A S 0 | 2 + 2 Re [ A S 0 * a S ( z = 0 , Ω , t ) ] .
a S ( z , t ) z 1 V g a S ( z , t ) t = i 1 2 g 2 A P ( z , t ) Q * ( z , t ) ,
Q ( z , t ) t + Γ A Q ( z , t ) = i A S 0 * g 1 ( z ) A P ( z , t ) .
A p ( z , t ) = A p 0 [ α u ( t z V g )     +     ( β α ) u ( t t 0 z V g )     +     ( γ β ) u ( t t 0 T z V g ) ] ,
a s S H O R T ( z z 0 , t | t 0 τ A , t t 0 ) = g ( z 0 ) I P 0 A S 0 2 Γ A * Δ z { α 2 [ u ( t 2 z 0 z V g ) u ( t t 0 2 z 0 z V g ] } + β [ β * ( β * α ) exp [ Γ A * ( t t 0 2 z 0 z V g ) ] ] [ u ( t t 0 2 z 0 z V g ) u ( t t 0 T 2 z 0 z V g ) ] + γ [ γ * [ β * ( 1 exp [ Γ A * T ] ) ( α γ * exp [ Γ A * T ] ) ] exp [ Γ A * ( t t 0 2 z 0 z V g ] u ( t t 0 T 2 z 0 z V g ) ] ,
Γ A = i Ω B 2 Ω 2 i Ω Γ B 2 Ω Γ B 2 + i ( Ω B Ω )     ( f o r     Ω Ω B ) ,
h ( t , z = 0 ) = g I P 0 A S 0 Γ B Δ z { [ 1 2 e Γ B 2 t ]     [ u ( t ) u ( t T ) ] + [ 1 + 2 e Γ B 2 t ( 1 e Γ B 2 T ) ] u ( t T ) } .
s Q ˜ ( z , s ) + Γ A Q ˜ ( z , s ) = i g 1 ( z ) A ˜ p ( z , s ) A s 0 * ,
A ˜ p ( z , s ) = A p 0 [ α s exp [ z V g s ] + β α s exp [ ( z V g + t 0 ) s ] + γ β s exp [ ( z V g + t 0 + T ) s ] ] ,
Q ( z , t ) = i g 1 ( z ) A p 0 A s 0 * Γ A { α ( 1 exp [ Γ A ( t z V g ) ] )     u ( t z V g ) + ( β α ) ( 1 exp [ Γ A ( t t 0 z V g ) ] ) u ( t t 0 z V g ) + ( γ β ) ( 1 exp [ Γ A ( t t 0 T z V g ) ] ) u ( t t 0 T z V g ) } .
A p ( z , t ) Q * ( z , t ) = i g 1 ( z ) I p 0 A s 0 Γ A * { [ α 2 α 2 exp ( Γ A * t 1 ) ]     u ( t 1 )     + [ | β | 2 α 2 + ( α β | β | 2 ) exp ( Γ A * t 2 ) + ( α β + α 2 ) exp ( Γ A * t 1 ) ]     u ( t 2 )     + [ | γ | 2 | β | 2 + ( | γ | 2 + γ β * ) exp ( Γ A * t 3 ) + ( γ β * + | β | 2 + α γ α β ) exp ( Γ A * t 2 )     + ( γ α + β α ) exp ( Γ A * t 1 ) ]     u ( t 3 ) } ,
t 1 = t z / V g ,       t 2 = t t 0 z / V g ,     t 3 = t t 0 T z / V g .
a ˜ s ( z , s ) z s V g a ˜ s ( z , s ) = i 2 g 2 ( z ) LT [ A p ( z , t ) Q * ( z , t ) ] .
a ˜ s ( z , s ) z s V g a ˜ s ( z , s ) = g ( z )H ( s, Γ A ( z ) ) exp ( z V g s ) ,
C ˜ s ( z 1 , s ) = C ˜ s ( z 2 , s ) + z 2 z 1 g ( ζ )H ( s, Γ A ( ζ ) ) exp ( 2 s ζ V g ) d ζ .
a s ( z 1 , t ) = a s ( z 2 , t z 2 z 1 V g ) u ( t z 2 z 1 V g ) + ILT { z 2 z 1 g ( ζ )   H ( s, Γ A ( ζ ) ) exp ( ( 2 ζ z 1 ) s V g ) d ζ } .
a s ( [ z s t a r t z e n d ] , t ) = i = 1 N ILT ( z i + 1 z i g ( ζ )   H ( s, Γ A ( ζ ) ) exp [ ( 2 ζ z s t a r t ) s V g ] d ζ ) .
C ˜ s ( z z 0 , s ) = g ( z 0 )   H ( s, Γ A ( z 0 ) ) V g 2 s exp ( 2 z 0 s V g ) [ 1 exp ( 2 Δ z s V g ) ] .
a s ( z z 0 , Δ z , t ) = 1 2 g ( z 0 ) I p 0 A s 0 V g R 2 Γ A * u ( z 0 z ) ,
R = [ R 1 ( D 1 ) + R 2 ( D 1 ) + R 3 ( D 1 ) ] [ R 1 ( D 2 ) + R 2 ( D 2 ) + R 3 ( D 2 ) ] ,
R 1 ( D ) = α 2 [ ( t D )     1/ Γ A * ( 1 exp [ Γ A * ( t D ) ] ) ] u ( t D ), R 2 ( D ) = [ ( | β | 2 α 2 ) ( t t 0 D +   ( α β | β | 2 )/ Γ A * ( 1 exp [ Γ A * ( t t 0 D ) ] )                        + ( α 2 α β ) / Γ A * exp [ Γ A * t 0 ] ( 1 exp [ Γ A * ( t t 0 D ) ] ) ] u ( t t 0 D ), R 3 ( D ) = [ ( | γ | 2 | β | 2 ) ( t t 0 ( D + T ) )   + ( γ β * | γ | 2 ) / Γ A * ( 1 exp [ Γ A * ( t t 0 ( D + T )) ] )                    + ( α β * ) ( γ β )   exp [ Γ A * T ] / Γ A * ( 1 exp [ Γ A * ( t t 0 ( D + T )) ] )                    + α ( β γ )   exp [ Γ A * ( t 0 + T ) ] / Γ A * ( 1 exp [ Γ A * ( t t 0 ( D + T )) ] ) ] u ( t t 0 ( D + T )),
D 1 = ( 2 z 0 z ) / V g ,               D 2 = [ 2 ( z 0 + Δ z ) z ] / V g .
C ˜ s ( z z 0 , s ) = Δ z g ( z 0 )   H ( s, Γ A ( z 0 ) ) exp ( 2 z 0 s V g ) .
a ˜ s ( z z 0 , s ) = Δ z g ( z 0 )   H ( s, Γ A ( z 0 ) ) exp [ ( 2 z 0 z ) s V g ] .
a s s h o r t ( z z 0 , Δ z , t ) g ( z 0 ) F s h o r t [ ( t z / V g ) z 0 / ( V g / 2 ) , Δ z ] = g ( z 0 ) I p 0 Α s 0 2 Γ A * Δ z { [ ( α 2 α 2 exp [ Γ A * ( t 2 z 0 z V g ) ] )     u ( t 2 z 0 z V g ) α 2 u ( t t 0 2 z 0 z V g ) ]       + β [ β * ( β * α ) exp [ Γ A * ( t t 0 2 z 0 z V g ) ] ]     [ u ( t t 0 2 z 0 z V g ) u ( t t 0 T 2 z 0 z V g ) ]       + α ( α β ) exp [ Γ A * ( t 2 z 0 z V g ) ] u ( t t 0 2 z 0 z V g )       + γ [ γ * [ β * ( 1 exp [ Γ A * T ] ) ( α γ * exp [ Γ A * T ] ) ] exp [ Γ A * ( t t 0 2 z 0 z V g ) ] u ( t t 0 T 2 z 0 z V g )         + α * ( β γ ) exp [ Γ A * ( t 2 z 0 z V g ) ] u ( t t 0 T 2 z 0 z V g ) } .
a s S H O R T ( z z 0 , t | t 0 > > τ A , t t 0 ) =               g ( z 0 ) I P 0 A S 0 2 Γ A * Δ z { α 2 [ u ( t 2 z 0 z V g ) u ( t t 0 2 z 0 z V g ) ] ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​   ​ ​   + β [ β * ( β * α ) exp [ Γ A * ( t t 0 2 z 0 z V g ) ] ]     [ u ( t t 0 2 z 0 z V g ) u ( t t 0 T 2 z 0 z V g ) ]       + γ [ γ * [ β * ( 1 exp [ Γ A * T ] ) ( α γ * exp [ Γ A * T ] ) ] exp [ Γ A * ( t t 0 2 z 0 z V g ) ] ] u ( t t 0 T 2 z 0 z V g ) } .
a s ( z , t ) = i = 1 N g ( z 1 + ( i 1 ) Δ z ) F s h o r t [ t z / V g ( z 1 + ( i 1 ) Δ z ) / ( V g / 2 ) , Δ z ] = i = 1 N g ( z 1 + V g t i / 2 ) F s h o r t [ ( t z / V g z 1 / ( V g / 2 ) ) t i , Δ z ] ,

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