## Four-wave mixing analysis of Brillouin dynamic grating in a polarization-maintaining fiber: theory and experiment |

Optics Express, Vol. 19, Issue 21, pp. 20785-20798 (2011)

http://dx.doi.org/10.1364/OE.19.020785

Acrobat PDF (1364 KB)

### 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

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]

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]

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]

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]

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]

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]

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. 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. 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]

**33**, 926–928 (2008). [CrossRef] [PubMed]

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

**17**, 1248–1255 (2009). [CrossRef] [PubMed]

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]

**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]

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. 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]

## 2. Theoretical model and discussion

*Ẽ*

_{1}and

*Ẽ*

_{2}, whose frequencies are locked near the Brillouin frequency, are propagating oppositely inside a PMF with the same polarization state along slow (

*x̂*) 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

*Ẽ*

_{3}polarized along fast (

*ŷ*) axis has the same propagation direction as pump 1 wave; when phase-matching condition is satisfied, a diffracted wave

*Ẽ*

_{4}would be generated whose frequency is downshifted from the frequency of

*Ẽ*

_{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

*Ẽ*

_{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.

*ρ̃*=

*ρ*–

*ρ*

_{0}is a change in the local density from its average value

*ρ*

_{0}; Γ

*is the damping coefficient;*

_{A}*γ*=

_{e}*ρ*

_{0}(

*dɛ*/

*dρ*)

_{ρ=ρ0}is the electrostrictive constant;

*v*is the sound velocity inside the fiber;

_{A}*ɛ*

_{0}is the vacuum permittivity. The total electric field,

**Ẽ**

_{tot}, inside the PMF could be expressed as

**Ẽ**

_{tot}=

**Ẽ**+

_{x}**Ẽ**=

_{y}*x̂*(

*Ẽ*

_{1}+

*Ẽ*

_{2}) +

*ŷ*(

*Ẽ*

_{3}+

*Ẽ*

_{4}), where where

*Ã*(

_{j}*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.

*ω*and

_{j}*k*are, respectively, frequencies and propagation constants of optical waves; c.c. represents complex conjugate.

_{j}### 2.1. Moving FBG model

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]

**35**, 2958–2960 (2010). [CrossRef] [PubMed]

*A*

_{1}and

*A*

_{2}being constants. Equations (6a) and (6b) could then be solved analytically for the diffracted wave along with the boundary conditions,

*A*

_{3}|

_{z}_{=0}=

*A*

_{3}(0) which is a known quantity and

*A*

_{4}|

_{z=L}=

*A*

_{4}(

*L*) = 0, where

*L*is the length of the PMF [18

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

*k*= 0, the reflectivity of the BDG could be expressed as Note that |

*A*

_{1}| and |

*A*

_{2}| are the square roots of the pump 1 and pump 2 powers, respectively. Figure 2 shows the properties of the functions of tanh

^{2}(

*x*) and

^{2}(

*x*) and

*A*

_{1}|

^{2}and |

*A*

_{2}|

^{2}are fixed, the reflectivity of the BDG varies with the length of the PMF following the trend showing as the solid curve. This has been confirmed by a recent experiment observing the intrinsic BDG spectrum [20

**35**, 2958–2960 (2010). [CrossRef] [PubMed]

*L*and pump 2 power |

*A*

_{2}|

^{2}are kept at constant, the BDG reflectivity versus pump 1 power |

*A*

_{1}|

^{2}should follow the trend showing as the dashed curve. However, in the early experiment [6

**33**, 926–928 (2008). [CrossRef] [PubMed]

### 2.2. FWM process with undepleted pump 1 and probe waves

**33**, 926–928 (2008). [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]

**35**, 2958–2960 (2010). [CrossRef] [PubMed]

*A*

_{1}and

*A*

_{3}being constants. Equations (9a) and (9b) could be solved analytically along with the boundary conditions,

*A*

_{2}|

_{z=L}=

*A*

_{2}(

*L*) which is a known quantity and

*A*

_{4}|

_{z=L}= 0, as where Φ =

*κ*(|

*A*

_{1}|

^{2}+|

*A*

_{3}|

^{2}) –

*i*Δ

*k*and Ψ =

*iκ*|

*A*

_{3}|

^{2}Δ

*k*. Under phase-matching condition, Δ

*k*= 0, the reflectivity of the BDG could be expressed as (a more detailed derivation could be found in Appendix B),

*L*, by varying pump or probe powers. The results are shown in Fig. 3. The parameters used to plot these curves are as follows:

*γ*= 0.902,

_{e}*c*= 3 × 10

^{8}m/s;

*ρ*

_{0}= 2210 kg/m

^{3},

*/2*

_{B}*π*= 10875 MHz, Γ

*/2*

_{B}*π*= 30 MHz,

*λ*= 1550 nm,

*L*= 13 m. From Fig. 3(a), we could see that when the probe and pump 2 powers are fixed, when the pump 1 power is increased, the reflectivity increases in a way that is very different from the moving FBG model. The trend could be approximately considered as a cubic curve by expanding the exponential term in Eq. (11) and keeping to the second order within the power range used for calculation, since the gain is not too high. When the pump 1 power is further increased, from Eq. (11), one could expect that the exponential growth would dominate until strong pump depletion occurs where numerical solution is needed for completely solving Eq. (3). However, this similar trend has already been observed in Ref. [6

**33**, 926–928 (2008). [CrossRef] [PubMed]

## 3. Experiment validation

*ν*, is near 50 GHz for Panda fiber and 43 GHz for Bow-Tie fiber, strong diffracted signal could be observed. Typical spectra of the corresponding BDGs generated in these two kinds of fibers are shown in Figs. 5(a) and 5(b). We intently set the position of highest peak as 0 MHz for comparison. The sweep speed of the tunable laser is 40 nm/s corresponding to ∼ 5 MHz/

*μ*s. The noise is reduced in the way that high resolution sampling mode of the oscilloscope is chosen; therefore, each BDG spectrum could be obtained within 200

*μ*s for a 1 GHz sweep range. The frequency resolution of the measured BDG spectrum is ∼ 0.1 MHz (1 GHz/8003 points). Both spectra show multi-peaks due to the non-uniformity of the birefringence inside the PMF. This similar spectrum has been reported recently in a 5-m long PMF (Ref. [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]

**35**, 2958–2960 (2010). [CrossRef] [PubMed]

**33**, 926–928 (2008). [CrossRef] [PubMed]

**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]

**35**, 193–195 (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]

## 4. Conclusion

## Appendix A: derivation of the four coupled wave equations

**Ẽ**

_{tot}, on the right-hand side of Eq. (1) is expressed as Eqs. (2) and (3). Since acoustic wave is predominantly longitudinal, we neglect the transversal components of the acoustic wave for SBS process. Then, the right-hand of Eq. (1) takes the form, where Ω =

*ω*

_{1}−

*ω*

_{2}=

*ω*

_{3}−

*ω*

_{4},

*q*

_{1}=

*k*

_{1}+

*k*

_{2}and

*q*

_{2}=

*k*

_{3}+

*k*

_{4}. Then, we are looking for the solution of the Eq. (1) in the form, where

*Q*

_{1}(

*z*,

*t*) and

*Q*

_{2}(

*z,t*) are the slowly varying amplitudes of the acoustic waves corresponding to the two SBS processes.

*F*(

_{A}*x*,

*y*) is the dimensionless acoustic wave mode profile which is the solution of the unperturbed Eq. (1) and thus satisfies the eigenvalue equation [23

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

*is the eigenfrequency of the solution of the modal acoustic equation that for a given wave vector*

_{Bm}*q*which satisfies Eq. (A.3). Assuming steady-state condition with slowly varying envelope approximation, substituting Eqs. (A.1) and (A.2) into Eq. (1) and grouping the results by the exponential factors

_{m}*e*

^{i(q1z−Ωt)}and

*e*

^{i(q2z−Ωt)}, one could obtain two equations for

*Q*

_{1}and

*Q*

_{2}, respectively, where

*∂Q*/

_{m}*∂z*has been dropped since the hypersonic phonons are strongly damped and thus propagate only over very shot distances before being absorbed [17]; since the ratio

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

*F*(

_{A}*x*,

*y*) and integrating over the transverse plane, one could obtain that, where angular brackets denote averaging over transverse cross section of the fiber. Note that the eigenfrequency of the acoustic mode Ω

*corresponds to the full resonance occurring both for frequencies and for wave vectors. Strictly speaking, it is the solution of the set of equations and where*

_{Bm}*n*

_{eff,x}and

*n*

_{eff,y}are the effective refractive indices of the optical mode along

*x̂*and

*ŷ*axis, respectively. Moreover, approximations have been used as

*n*

_{eff,x}(

*ω*

_{2}) ≈

*n*

_{eff,x}(

*ω*

_{1}) and

*n*

_{eff,y}(

*ω*

_{4}) ≈

*n*

_{eff,y}(

*ω*

_{3}), since Ω ≪

*ω*

_{1},

*ω*

_{3}. Thus, for a fixed frequency

*ω*

_{1}or

*ω*

_{3}, Ω

*could be calculated as the intersection of the two curves of*

_{Bm}*q*(Ω) obtained from Eqs. (A.6) and (A.7). However, since

_{m}*q*(Ω) is a very weak function because of Ω ≪

_{m}*ω*

_{1},

*ω*

_{3}, with

*n*and

_{x}*n*being the refractive index of the PMF along

_{y}*x̂*and

*ŷ*axis, respectively. The obtained

*q*value could be substituted into Eq. (A.7) to find Ω

_{m}*directly. Moreover, as discussed before, for BDG generation and detection process, phase matching is required as that Δ*

_{Bm}*k*= 0 which is equivalent to

*q*

_{1}=

*q*

_{2}=

*q*; therefore, we could take Ω

_{B1}= Ω

_{B2}= Ω

*, Γ*

_{B}_{B1}= Γ

_{B2}= Γ

*for further calculations. From Eqs. (A.2) and (A.5), the full material density variation could be expressed as,*

_{B}*Ẽ*(

_{j}*j*= 1 − 4) inside the PMF as follows: where

*n*=

*n*for

_{x}*j*= 1, 2 and

*n*=

*n*for

_{y}*j*= 3, 4. The total nonlinear polarization induced by the acousto-optic interaction which gives rise to the source term in this equation is given by where with By determining different parts of

**P̃**

_{tot}that can act as phase-matched source terms for the four waves, one could obtain where Δ

*k*=

*k*

_{3}+

*k*

_{4}−

*k*

_{1}−

*k*

_{2}is the phase mismatch. Neglecting the dependence of the optical modal profile on nonlinear effects, we can substitute the optical modal equation along with Eqs. (A.14) into Eq. (A.10), using slowly varying envelope approximation, and multiplying

*F*(

*x*,

*y*) on both sides of each resultant equation, integrating over the transverse cross section, to obtain four equations for four interacting optical waves under steady state, where with

*A*|

_{j}^{2}represents optical power by using |

*A*|

_{j}^{2}=

*ɛ*

_{0}

*cn*|

*Ã*|

_{j}^{2}〈

*F*

^{2}〉/2. In addition,

*ω*

_{1}≈

*ω*

_{2}and

*ω*

_{3}≈

*ω*

_{4}are used to obtain Eqs. (A.16) and (A.17), and

*n*≈

_{x}*n*is assumed for normalization. When the frequency difference of the two pump waves has been locked at Brillouin frequency, e.g., Ω = Ω

_{y}*, Eq. (A.16) reduces to Eq. (3) where a further approximation of*

_{B}*λ*

_{1}≈

*λ*

_{3}(wavelengths of pump 1 and probe waves) is considered.

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

*z*derivative of Eq. (9b) with the help of (9a), considering that

*A*

_{1}and

*A*

_{3}are constants, and then the resultant equation for

*A*

_{4}could be expressed as follows: where Φ =

*κ*(|

*A*

_{1}|

^{2}+ |

*A*

_{3}|

^{2}) –

*i*Δ

*k*and Ψ =

*iκ*|

*A*

_{3}|

^{2}Δ

*k*. The general solution for it is with

*C*and

*D*which could be determined by boundary conditions: 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 Therefore, substituting Eq. (B.4) into Eq. (B.2), the diffracted wave

*A*

_{4}(

*z*) could be obtained as expressed in Eq. (10). Under phase-matching condition, Δ

*k*= 0, so that Φ =

*κ*(|

*A*

_{1}|

^{2}+ |

*A*

_{3}|

^{2}) and Ψ = 0; the reflectivity could be expressed in Eq. (11). Note that the expression for pump 2 wave

*A*

_{2}could also be obtained in the similar way.

## Acknowledgments

## 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. |

2. | Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science |

3. | L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics |

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 |

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. |

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. |

8. | K. Y. Song and H. J. Yoon, “High-resolution Brillouin optical time domain analysis based on Brillouin dynamic grating,” Opt. Lett. |

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. |

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 |

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. |

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. |

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 |

15. | K. Y. Song, K. Lee, and S. B. Lee, “Tunable optical delays based on Brillouin dynamic grating in optical fibers,” Opt. Express |

16. | Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. |

17. | R. W. Boyd, |

18. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

19. | Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express |

20. | K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. |

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. |

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 |

23. | A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. |

**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

- 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]
- 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]
- L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics2, 474–481 (2008). [CrossRef]
- G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.
- X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors11, 4152–4187 (2011). [CrossRef]
- 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]
- 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]
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