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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20785–20798
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Four-wave mixing analysis of Brillouin dynamic grating in a polarization-maintaining fiber: theory and experiment

Da-Peng Zhou, Yongkang Dong, Liang Chen, and Xiaoyi Bao  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20785-20798 (2011)
http://dx.doi.org/10.1364/OE.19.020785


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Abstract

We investigate the Brillouin dynamic grating generation and detection process in polarization-maintaining fibers for the case of continuous wave operation both theoretically and experimentally. The four interacting light waves couple together through the material density variation due to stimulated Brillouin scattering. The four coupled equations describing this process are derived and solved analytically for two cases: moving fiber Bragg grating approximation and undepleted pump and probe waves approximation. We show that the conventional grating model oversimplifies the Brillouin dynamic grating generation and detection process, since it neglects the coupling between all the four waves, while the four-wave mixing model clearly demonstrates this coupling process and it is verified experimentally by measuring the reflection of the Brillouin dynamic grating. The trends of the theoretical calculation and experimental results agree well with each other confirming that the Brillouin dynamic grating generation and detection process is indeed a four-wave mixing process.

© 2011 OSA

1. Introduction

Since stimulated Brillouin scattering (SBS) was first observed in 1964 [1

1. R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964). [CrossRef]

], it has remained an intensive research topic. Although SBS limits the channel power in optical fiber transmission systems since it could occur at a low input power in optical fibers, many applications could benefit from this nonlinear phenomenon, such as light storage [2

2. Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007). [CrossRef] [PubMed]

], slow and fast light [3

3. L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

], Brillouin lasers and amplifiers [4

4. G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.

], as well as distributed fiber sensors [5

5. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011). [CrossRef]

]. More recently, dynamic gratings generated in polarization-maintaining fibers (PMFs) via SBS, called Brillouin dynamic gratings (BDGs) [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

], have attracted much attention. For example, it could be used to realize high-spatial resolution distributed sensing [7

7. Y. Dong, X. Bao, and L. Chen, “Distributed temperature sensing based on birefringence effect on transient Brillouin grating in a polarization-maintaining photonic crystal fiber,” Opt. Lett. 34, 2590–2592 (2009). [CrossRef] [PubMed]

10

10. S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011). [CrossRef]

], to completely discriminate temperature and strain for Brillouin based sensors [11

11. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009). [CrossRef] [PubMed]

14

14. W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19, 2363–2370 (2011). [CrossRef] [PubMed]

], to achieve tunable optical delays [15

15. K. Y. Song, K. Lee, and S. B. Lee, “Tunable optical delays based on Brillouin dynamic grating in optical fibers,” Opt. Express 17, 10344–10349 (2009). [CrossRef] [PubMed]

], and to measure birefringence of PMFs in a distributed manner [16

16. Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010). [CrossRef] [PubMed]

].

SBS occurs when two pump waves, whose frequency difference equals to Brillouin frequency, counter-propagate in an optical fiber. The two pump waves beat together giving rise to density variation associated with an acoustic wave through electrostriction effect [17

17. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

]. The acoustic wave introduces a moving periodically modulated refractive index acting like a moving fiber Bragg grating (FBG) which diffracts more Stokes wave reinforcing the SBS process. The property of the moving grating could be measured or detected by monitoring the diffracted wave from a third probe wave which is used to illuminate the grating. In order to discriminate the probe and the pump waves, a direct way is to decouple the polarization state of the pump and probe beams, so that most of the experiments are carried out in PMFs. This moving FBG was given the name of BDG in Ref. [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

] where separation of the grating generation and diffracting processes was achieved: two pump waves with frequency difference of a Brillouin frequency polarized along the same axis of the PMF generates the BDG, while a probe wave polarized along the orthogonal axis is diffracted strongly when phase matching condition is satisfied. The frequency difference between the diffracted and the probe waves also equals to the Brillouin frequency. So far, it is widely accepted that the fundamental nature of the BDG is similar to that of an FBG so that conventional FBG theory [18

18. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

] could be directly applied. Recent experiments [19

19. Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express 18, 18960–18967 (2010). [CrossRef] [PubMed]

, 20

20. K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010). [CrossRef] [PubMed]

] also tend to confirm this by rigorously measuring the intrinsic spectra of the BDG showing that the linewidth of the BDG appears inversely proportional to the BDG length. Note that BDG spectrum observed at the beginning with a broad bandwidth of ∼ 80 MHz [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

] or ∼ 320 MHz [11

11. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009). [CrossRef] [PubMed]

] were contributed mainly due to the birefringence non-uniformity of PMFs as well as the relative frequency fluctuations between the multiple sources [14

14. W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19, 2363–2370 (2011). [CrossRef] [PubMed]

].

However, as discussed above, the BDG generation and detection process actually involves four waves interacting with each other through the material density variations; moreover, the diffracted wave has a Brillouin frequency difference with respect to the probe wave. Obviously, this would introduce a second SBS process between diffracted and probe waves, since these two waves have the same polarization state. More specifically, in most of the experiments [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

,11

11. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009). [CrossRef] [PubMed]

,20

20. K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010). [CrossRef] [PubMed]

], the diffracted wave would experience an SBS amplification since its frequency is downshifted by a Brillouin frequency from that of the probe beam whose propagation direction is the same as that of the BDG. Since the power level of the probe wave is relatively high in practice [11

11. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009). [CrossRef] [PubMed]

,19

19. Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express 18, 18960–18967 (2010). [CrossRef] [PubMed]

,20

20. K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010). [CrossRef] [PubMed]

] in order to have strong diffracted wave for detection, this second SBS process is expected to play an important role. Physically speaking, this is a four-wave mixing (FWM) process that the longitudinal acoustic wave couples the four optical waves with different polarization states together. In this work, we will theoretically study this FWM process which is equivalent to the BDG generation and detection process in the case of continuous wave (CW) operation by deriving the four coupled light wave equations. These equations could be solved analytically for two cases: moving FBG model approximation and undepleted pump 1 (see Fig. 1) and probe waves approximation; showing that the FBG model clearly deviates from some of the reported experimental results [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

]. In Section 3, experimental results will be provided to compare with our theoretical calculation confirming that the BDG generation and detection process is indeed an FWM process, and the moving FBG model oversimplifies this process. It is worth mentioning that even though this work is intent on studying the CW case, we believe that in the transit case (probe pulse follows one of the pump pulses) [7

7. Y. Dong, X. Bao, and L. Chen, “Distributed temperature sensing based on birefringence effect on transient Brillouin grating in a polarization-maintaining photonic crystal fiber,” Opt. Lett. 34, 2590–2592 (2009). [CrossRef] [PubMed]

, 16

16. Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010). [CrossRef] [PubMed]

, 21

21. K. Y. Song, W. Zou, Z. He, and K. Hotate, “Optical time-domain measurement of Brillouin dynamic grating spectrum in a polarization-maintaining fiber,” Opt. Lett. 34, 1381–1383 (2009). [CrossRef] [PubMed]

], when the power of the probe is high, the second SBS process is also expected to have influences on the reflectivity of the BDG characteristics.

Fig. 1 Diagram of the theoretical model. BDG generation and detection scheme: an FWM process.

2. Theoretical model and discussion

The diagram of the BDG generation and detection process is shown in Fig. 1. Two pump waves 1 and 2, whose frequencies are locked near the Brillouin frequency, are propagating oppositely inside a PMF with the same polarization state along slow () axis. The frequency of pump 1 wave is assumed to be higher than that of the pump 2 wave, so that the generated BDG through SBS is propagating along +z direction. A probe wave 3 polarized along fast (ŷ) axis has the same propagation direction as pump 1 wave; when phase-matching condition is satisfied, a diffracted wave 4 would be generated whose frequency is downshifted from the frequency of 3 by an amount of about a Brillouin frequency which is determined by the frequency difference between the two pump waves. This diffracted wave would experience a second SBS amplification by 3 along fast axis, since it has the same polarization state with the probe wave. Therefore, inside the PMF, the material density variation (acoustic wave) is contributed by two SBS processes which in turn couple the four optical waves together which could be considered as an FWM process.

In order to obtain a clear view of four waves interacting with each other in a PMF, we need to solve the Maxwell’s equations with the following material density equation [17

17. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

]:
2ρ˜t2ΓA2ρ˜tvA22ρ˜=12ɛ0γe2E˜tot2,
(1)
where ρ̃ = ρρ0 is a change in the local density from its average value ρ0; ΓA is the damping coefficient; γe = ρ0(/)ρ=ρ0 is the electrostrictive constant; vA is the sound velocity inside the fiber; ɛ0 is the vacuum permittivity. The total electric field, tot, inside the PMF could be expressed as tot = x + y = (1 + 2) + ŷ(3 + 4), where
E˜x=12x^[F(x,y)A˜1(z,t)ei(k1zω1t)+F(x,y)A˜2(z,t)ei(k2zω2t)]+c.c.,
(2a)
E˜y=12y^[F(x,y)A˜3(z,t)ei(k3zω3t)+F(x,y)A˜4(z,t)ei(k4zω4t)]+c.c.,
(2b)
where Ãj(z,t) (j = 1 – 4) and F(x,y) are the slowly varying fields and dimensionless fundamental mode profile for the four interacting waves, respectively. Note that the mode profiles are assumed to be the same for all the four interacting waves for simplicity. ωj and kj are, respectively, frequencies and propagation constants of optical waves; c.c. represents complex conjugate.

Generally speaking, Eq. (3) needs to be solved numerically. However, with the approximation that pump 1 and probe waves are undepleted, these equations could be solved analytically which would give a clear physical insight for the FWM process. The analysis will be given in Sec. 2.2. Before that, we would like to discuss briefly the moving FBG model which is broadly accepted in all the reports so far, and show that such model oversimplifies the BDG generation and detection process.

2.1. Moving FBG model

Fig. 2 Properties of the tanh2(x) and tanh2(x) functions.

2.2. FWM process with undepleted pump 1 and probe waves

In most of the BDG experiments, the pump 1 power is much higher than pump 2 power [6

6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

, 16

16. Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010). [CrossRef] [PubMed]

, 20

20. K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010). [CrossRef] [PubMed]

]. Moreover, in order to have strong diffracted signal from the BDG, probe power is also chosen to be on the same order of the pump 1 power. Therefore, a more reasonable approximation compared to moving FBG model is that pump 1 and probe power are undepleted, since the length of PMF used in the experiments is short, usually less than tens of meters. In addition, since the probe power is also high in most of experiments, the diffracted signal would also experience an SBS amplification. Under these assumptions, Eq. (3) could be reduced to
A2z=κ(|A1|2A2+A1A3*A4eiΔkz),
(9a)
A4z=κ(|A3|2A4+A1*A2A3eiΔkz),
(9b)
with A1 and A3 being constants. Equations (9a) and (9b) could be solved analytically along with the boundary conditions, A2|z=L = A2(L) which is a known quantity and A4|z=L = 0, as
A4(z)=2κA1*A2(L)A3sinh[Φ2+4Ψ(Lz)/2]Φ2+4ΨeΦ(Lz)/2+iΔkL,
(10)
where Φ = κ(|A1|2 +|A3|2) – iΔk and Ψ = |A3|2Δk. Under phase-matching condition, Δk = 0, the reflectivity of the BDG could be expressed as (a more detailed derivation could be found in Appendix B),
RFWM=|A4(0)A3|2=|A1|2|A2(L)|2[eκ(|A1|2+|A3|2)L1|A1|2+|A3|2]2.
(11)

Fig. 3 Theoretically calculated BDG reflectivity with respect to (a) pump 1 power |A1|2 for different pump 2 and probe powers (b) pump 2 power |A2|2 for different probe powers and fixed pump 1 power (c) probe power |A3|2 for different pump 1 powers and fixed pump 2 power.

3. Experiment validation

Figure 4 shows the experiment setup. Two narrow linewidth (∼ 3 kHz) fiber lasers operating at 1550 nm are used to provide pumps 1 and 2, and their frequency difference is locked by a phase-locking loop in a frequency counter via a photodetector (PD1). The frequency of Laser 1 is higher than Laser 2. A tunable laser (Agilent 81940A) whose linewidth is ∼ 100 kHz with a continuous sweep mode is used as the probe wave. Three polarization controllers (PCs) are used to adjust the polarization states of the three waves. A variable attenuator is used to adjust the power level of the pump 2 wave, while two erbium-doped fiber amplifiers (EDFAs) are employed to adjust power levels of pump 1 and probe waves. Pump 1 and pump 2 waves are launched into the PMF along slow axis in opposite direction to each other through polarization beam splitter/combiner (PBS/PBC), and the probe wave is injected along the orthogonal axis. Power meter 2 (PM2) is used to monitor pump 2 after passing through the PMF in order to determine whether a strong pump depletion happens. In the detection end, a 125 MHz photodetector (PD2) and a high performance oscilloscope (Agilent Infiniium DSO81204B) are used to detect the back diffracted wave via a narrow bandwidth (0.1 nm) tunable filter (TF), whose function is to eliminate the leakage of pump 1 and 2 waves along the slow axis. After the frequency of the two pump waves are locked at the Brillouin frequency of the PMF to generate the BDG, the probe wave is fast swept and the diffracted spectrum of the BDG is recorded by the oscilloscope (A trigger is given to the oscilloscope when the sweep starts for synchronization).

Fig. 4 Experiment Setup. PC: polarization controller; PD: photodetector; EDFA: erbium-doped fiber amplifier; ISO: isolator; PBS/PBC: polarization beam splitter/combiner; PM: power meter; TF: tunable filer; PMF: polarization-maintaining fiber.

Fig. 5 Typical BDG spectra for (a) 13-m Panda and (b) 10-m Bow-Tie fiber.
Fig. 6 BDG spectra for different pump 1 power, with the pump 2 power of 10 mW and probe power of 100 mW.

Fig. 7 Experimentally obtained BDG reflectivity with respect to (a) pump 1 power |A1|2 for different pump 2 and probe powers (b) pump 2 power |A2|2 for different probe powers and fixed pump 1 power (c) probe power |A3|2 for different pump 1 powers and fixed pump 2 power.

4. Conclusion

Appendix A: derivation of the four coupled wave equations

Appendix B: analytical solution of the four coupled wave equations under undepleted pump 1 and probe waves approximation

We calculate z derivative of Eq. (9b) with the help of (9a), considering that A1 and A3 are constants, and then the resultant equation for A4 could be expressed as follows:
2A4z2+ΦA4zΨA4=0,
(B.1)
where Φ = κ(|A1|2 + |A3|2) – iΔk and Ψ = |A3|2Δk. The general solution for it is
A4(z)=Cep+z+Depz,
(B.2)
with p±=12Φ±12Φ2+4Ψ and two constants C and D which could be determined by boundary conditions:
A4|z=L=0,
(B.3a)
A4z|z=L=κA1*A2(L)A3eiΔkL.
(B.3b)
Equation (B.3b) is obtained from Eq. (9b) after insertion of Eq. (B.3a). Substituting Eq. (B.2) into Eq. (B.3), the two constants could be determined as
C=κA1*A2(L)A3eiΔkLep+Lpp+,
(B.4a)
D=κA1*A2(L)A3eiΔkLepLpp+.
(B.4b)
Therefore, substituting Eq. (B.4) into Eq. (B.2), the diffracted wave A4(z) could be obtained as expressed in Eq. (10). Under phase-matching condition, Δk = 0, so that Φ = κ(|A1|2 + |A3|2) and Ψ = 0; the reflectivity could be expressed in Eq. (11). Note that the expression for pump 2 wave A2 could also be obtained in the similar way.

Acknowledgments

The authors would like to thank Natural Science and Engineering Research Council of Canada (NSERC) Discovery Grants and Canada Research Chair Program for the financial support. D. P. Zhou would like to acknowledge the Province of Ontario Ministry of Research and Innovation and the University of Ottawa for the financial support of the Vision 2020 Postdoctoral Fellowship. The authors would like to thank Dr. Wenhai Li for helpful discussions.

References and links

1.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964). [CrossRef]

2.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007). [CrossRef] [PubMed]

3.

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

4.

G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.

5.

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011). [CrossRef]

6.

K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008). [CrossRef] [PubMed]

7.

Y. Dong, X. Bao, and L. Chen, “Distributed temperature sensing based on birefringence effect on transient Brillouin grating in a polarization-maintaining photonic crystal fiber,” Opt. Lett. 34, 2590–2592 (2009). [CrossRef] [PubMed]

8.

K. Y. Song and H. J. Yoon, “High-resolution Brillouin optical time domain analysis based on Brillouin dynamic grating,” Opt. Lett. 35, 52–54 (2010). [CrossRef] [PubMed]

9.

K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010). [CrossRef]

10.

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011). [CrossRef]

11.

W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009). [CrossRef] [PubMed]

12.

W. Zou, Z. He, and K. Hotate, “Demonstration of Brillouin distributed discrimination of strain and temperature using a polarization-maintaining optical fiber,” IEEE Photon. Technol. Lett. 22, 526–528 (2010). [CrossRef]

13.

Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22, 1364–1366 (2010). [CrossRef]

14.

W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19, 2363–2370 (2011). [CrossRef] [PubMed]

15.

K. Y. Song, K. Lee, and S. B. Lee, “Tunable optical delays based on Brillouin dynamic grating in optical fibers,” Opt. Express 17, 10344–10349 (2009). [CrossRef] [PubMed]

16.

Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010). [CrossRef] [PubMed]

17.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

18.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

19.

Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express 18, 18960–18967 (2010). [CrossRef] [PubMed]

20.

K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010). [CrossRef] [PubMed]

21.

K. Y. Song, W. Zou, Z. He, and K. Hotate, “Optical time-domain measurement of Brillouin dynamic grating spectrum in a polarization-maintaining fiber,” Opt. Lett. 34, 1381–1383 (2009). [CrossRef] [PubMed]

22.

A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bichham, and R. Mishra, “Design concept for optical fibers with enhaced SBS threshold,” Opt. Express 13, 5338–5346 (2005). [CrossRef] [PubMed]

23.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2, 1–59 (2010). [CrossRef]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(290.5900) Scattering : Scattering, stimulated Brillouin
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 11, 2011
Revised Manuscript: September 4, 2011
Manuscript Accepted: September 25, 2011
Published: October 4, 2011

Citation
Da-Peng Zhou, Yongkang Dong, Liang Chen, and Xiaoyi Bao, "Four-wave mixing analysis of Brillouin dynamic grating in a polarization-maintaining fiber: theory and experiment," Opt. Express 19, 20785-20798 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20785


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References

  1. R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett.12, 592–595 (1964). [CrossRef]
  2. Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science318, 1748–1750 (2007). [CrossRef] [PubMed]
  3. L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics2, 474–481 (2008). [CrossRef]
  4. G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.
  5. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors11, 4152–4187 (2011). [CrossRef]
  6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett.33, 926–928 (2008). [CrossRef] [PubMed]
  7. Y. Dong, X. Bao, and L. Chen, “Distributed temperature sensing based on birefringence effect on transient Brillouin grating in a polarization-maintaining photonic crystal fiber,” Opt. Lett.34, 2590–2592 (2009). [CrossRef] [PubMed]
  8. K. Y. Song and H. J. Yoon, “High-resolution Brillouin optical time domain analysis based on Brillouin dynamic grating,” Opt. Lett.35, 52–54 (2010). [CrossRef] [PubMed]
  9. K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol.28, 2062–2067 (2010). [CrossRef]
  10. S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011). [CrossRef]
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