## Numerical approximation for Brillouin fiber ring resonator |

Optics Express, Vol. 20, Issue 5, pp. 5783-5788 (2012)

http://dx.doi.org/10.1364/OE.20.005783

Acrobat PDF (980 KB)

### Abstract

A new method for describing the Stimulated Brillouin Scattering (SBS) generated in a fiber ring resonator in dynamic regime is presented. Neglecting the time derivatives of the fields amplitudes, our modeling method describes the lasers steady-state operations as well as their transient characteristics or pulsed emission. The developed approach has shown a very good agreement between the theoretical predictions given by the SBS model and the experimental results.

© 2012 OSA

## 1. Introduction

1. A. A. Fotiadi and P. Mégret, “Self-Q-switched Er-Brillouin fiber source with extra-cavity generation of a raman supercontinuum in a dispersion shifted fiber,” Opt. Lett. **31**, 1621–1623 (2006). [CrossRef] [PubMed]

2. Z. Pan, L. Meng, Q. Ye, H. Cai, Z. Fang, and R. Qu, “Repetition rate stabilization of the SBS Q-switched fiber laser by external injection,” Opt. Express **17**, 3124–3129 (2009). [CrossRef] [PubMed]

3. L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. **152**, 65–70 (1998). [CrossRef]

4. Z. Ou, J. Li, L. Zhang, Z. Dai, and Y. Liu, “An approximate analytic solution of the steady state Brillouin scattering in single mode optical fiber without neglecting the attenuation coefficient,” Opt. Commun. **282**, 3812–3816 (2009). [CrossRef]

5. V. Babin, A. Mocofanescu, V. I. Vlad, and M. J. Damzen, “Analytical treatment of laser-pulse compression in stimulated Brillouin scattering,” J. Opt. Soc. Am. B **16**, 155–163 (1999). [CrossRef]

6. I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A **71**, 043810–043814 (2005). [CrossRef]

7. H. Li and K. Ogusu, “Instability of stimulated brillouin scattering in a fiber ring resonator,” Opt. Rev. **7**, 303–308 (2000). [CrossRef]

## 2. Theory and computational method

8. A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A **44**, 3205–3209 (1991). [CrossRef] [PubMed]

9. S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A **20**, 1132–1137 (2003). [CrossRef]

_{p}(z, t) represents the slowly-varying complex field amplitude of the pump wave with the frequency

*ω*

_{p}and the wave number

*κ*

_{p}. E

_{s}(z, t) represents the slowly-varying complex field amplitude of Stokes wave with the frequency

*ω*

_{s}and the wave number

*κ*

_{s}.

*ρ*(z, t) is the complex amplitude of the variation of the material density from the mean value

*ρ*

_{0}. We have introduced the acoustic frequency Ω =

*ω*

_{p}−

*ω*

_{s}and the acoustic wave number q =

*κ*

_{p}+

*κ*

_{s}. The various constants used are:

*α*the linear fiber loss coefficient, n the refraction index, c the speed of the light in a vacuum,

*γ*the electrostrictive constant and Γ = 1/

*τ*

_{p}the phonon decay rate with

*τ*

_{p}the photon life time. Here, f(z, t) is a Langevin noise source that describes the thermal fluctuations in the density of the fiber that leads to spontaneous Brillouin scattering. This noise source is modeled as a centered Gaussian noise having the auto-correlation function of type 〈f(z, t), f

^{*}(z′, t′)〉 = Y

*δ*(z − z′)

*δ*(t − t′). As in [8

8. A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A **44**, 3205–3209 (1991). [CrossRef] [PubMed]

*κ*

_{B}is Boltzmann’s constant, T is the temperature, S

_{eff}is the effective core area and v

_{A}is the acoustic velocity in this material.

10. N. Vermeulen, C. Debaes, A. A. Fotiadi, K. Panajotov, and H. Thienpont, “Stokes-anti-Stokes iterative resonator method for modeling Raman lasers,” IEEE J. Quantum Electron. **42**, 1144–1156 (2006). [CrossRef]

11. I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. **35**, 1812–1816 (1999). [CrossRef]

_{p}and E

_{s}with acoustic field

*ρ*, is reduced from three to two coupled equations for the complex field amplitudes of the pump and the Stokes wave: where the gain coefficient is given by g

_{E}= (nc/16

*π*)g

_{B}and the constant b by

_{B}the standard Brillouin gain coefficient of the medium.

_{r}) in the ring cavity and superior to the phonon life time in this environment. Such a model, usually used to obtain the solutions in the steady-state regime, can be also used in the transient or pulsed regime when the time scales are longer than the round-trip time.

*κ*. In our model, we consider that the time evolution of the incident signal is periodic and has a secant-hyperbolic shape with a FWHM ∼ 500 t

_{r}. With these considerations, the role of the pulse shape is taken into account by using for each new m round-trip a different value E

_{in}(mt

_{r}) of the incident field (Fig. 1(a)). After several recirculation pass of the pump wave in the cavity, we obtain a concentration of energy inside the ring due to the incident power accumulated inside. Thus, using a weak incident power, one can obtain in this way high values of the power generating non-linear effects in the ring as the SBS.

^{th}round-trip of pump and Stokes waves in the resonator, the rate equations system (3) describing the spatial distribution of the pump wave amplitude

_{p}, ℑE

_{p}, ℜE

_{s}and ℑE

_{s}the real and imaginary part of the pump and Stokes fields. Then, the pump propagation is described by the following two equations, coupled through the values of Stokes field

12. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. **7**, 288–290 (1982). [CrossRef] [PubMed]

13. F. E. Seraji, “Steady-state performance analysis of fiber-optic ring resonator,” Prog. Quantum. Electron. **33**, 1–16 (2009). [CrossRef]

_{r}and the circulating field of the precedent round-trip which are transmitted through the coupler (Fig. 1(a)): where Δ

*ϕ*

_{p}is detuning from resonance which replaces the linear phase shift for the pump wave

*ϕ*

_{p}and is defined by: Δ

*ϕ*

_{p}= 2

*π*nL/

*λ*

_{p}− 2p

*π*, p being an integer. We may observe that, for the first pass, the boundary conditions depend only on the value of the incident field of the pump.

*ϕ*

_{s}is the linear phase shift per round-trip of the Stokes wave. The linear phase shift of the Stokes wave is different of the pump linear phase shift Δ

*ϕ*

_{p}and it is defined according to the following expression [7

7. H. Li and K. Ogusu, “Instability of stimulated brillouin scattering in a fiber ring resonator,” Opt. Rev. **7**, 303–308 (2000). [CrossRef]

*ϕ*

_{s}= Δ

*ϕ*

_{p}− 2

*πν*

_{B}nL/c, where

*ν*

_{B}is the acoustic frequency.

_{r}duration, we obtain the spatial distribution of fields and the new initial conditions computed using Eqs. (5) and (7). Simultaneously we obtain the expressions for the output fields after m

*round-trip: By Eq. (8) one computes the time evolution (at each*

^{th}*mt*) of the resonator transmission as :

_{r}*Q*inside the resonator :

## 3. Experimental setup and results

*ν*

_{B}(≈ 14.5 MHz at 1550 nm) of the Brillouin gain curve. This condition restricts the Stokes oscillation to a single longitudinal mode [14

14. A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B **18**, 556–567 (2001). [CrossRef]

*ν*

_{B}= 11 GHz, the effective mode area S

_{eff}= 85

*μ*m

^{2}, the linear refractive index n = 1.454, the density

*ρ*

_{0}= 2210 kg/m

^{3}, the acoustic velocity v

_{A}= 5960 m/s, the photon live time

*τ*

_{p}= 22 ns, the linear fiber loss coefficient

*α*= 2.3 × 10

^{−5}m

^{−1}and the pump wavelength

*λ*= 1550 nm. For both situations, experiment and simulation, we observe a phenomena of depletion in the transmitted pump that corresponds to the SBS signal emission. The shape of the pulses is qualitatively the same with a pulse duration of about 350 ns (FWHM) and its repetition period is of about 140

*μ*s which was imposed by the repetition period of the pumping signal (notice that it can be modified).

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. A. Fotiadi and P. Mégret, “Self-Q-switched Er-Brillouin fiber source with extra-cavity generation of a raman supercontinuum in a dispersion shifted fiber,” Opt. Lett. |

2. | Z. Pan, L. Meng, Q. Ye, H. Cai, Z. Fang, and R. Qu, “Repetition rate stabilization of the SBS Q-switched fiber laser by external injection,” Opt. Express |

3. | L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. |

4. | Z. Ou, J. Li, L. Zhang, Z. Dai, and Y. Liu, “An approximate analytic solution of the steady state Brillouin scattering in single mode optical fiber without neglecting the attenuation coefficient,” Opt. Commun. |

5. | V. Babin, A. Mocofanescu, V. I. Vlad, and M. J. Damzen, “Analytical treatment of laser-pulse compression in stimulated Brillouin scattering,” J. Opt. Soc. Am. B |

6. | I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A |

7. | H. Li and K. Ogusu, “Instability of stimulated brillouin scattering in a fiber ring resonator,” Opt. Rev. |

8. | A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A |

9. | S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A |

10. | N. Vermeulen, C. Debaes, A. A. Fotiadi, K. Panajotov, and H. Thienpont, “Stokes-anti-Stokes iterative resonator method for modeling Raman lasers,” IEEE J. Quantum Electron. |

11. | I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. |

12. | L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. |

13. | F. E. Seraji, “Steady-state performance analysis of fiber-optic ring resonator,” Prog. Quantum. Electron. |

14. | A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B |

15. | V. V. Spirin, C. A. López, P. Mégret, and A. A. Fotiadi, “Single-mode Brillouin fiber laser passively stabilized at resonance frequency with self-injection locked pump laser,“ Laser Phys. Lett. (to be published), 1–4 (2012). |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

(140.3560) Lasers and laser optics : Lasers, ring

(290.5900) Scattering : Scattering, stimulated Brillouin

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 13, 2012

Revised Manuscript: February 11, 2012

Manuscript Accepted: February 14, 2012

Published: February 24, 2012

**Citation**

Cristina Elena Preda, Andrei A. Fotiadi, and Patrice Mégret, "Numerical approximation for Brillouin fiber ring resonator," Opt. Express **20**, 5783-5788 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5783

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### References

- A. A. Fotiadi and P. Mégret, “Self-Q-switched Er-Brillouin fiber source with extra-cavity generation of a raman supercontinuum in a dispersion shifted fiber,” Opt. Lett.31, 1621–1623 (2006). [CrossRef] [PubMed]
- Z. Pan, L. Meng, Q. Ye, H. Cai, Z. Fang, and R. Qu, “Repetition rate stabilization of the SBS Q-switched fiber laser by external injection,” Opt. Express17, 3124–3129 (2009). [CrossRef] [PubMed]
- L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun.152, 65–70 (1998). [CrossRef]
- Z. Ou, J. Li, L. Zhang, Z. Dai, and Y. Liu, “An approximate analytic solution of the steady state Brillouin scattering in single mode optical fiber without neglecting the attenuation coefficient,” Opt. Commun.282, 3812–3816 (2009). [CrossRef]
- V. Babin, A. Mocofanescu, V. I. Vlad, and M. J. Damzen, “Analytical treatment of laser-pulse compression in stimulated Brillouin scattering,” J. Opt. Soc. Am. B16, 155–163 (1999). [CrossRef]
- I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A71, 043810–043814 (2005). [CrossRef]
- H. Li and K. Ogusu, “Instability of stimulated brillouin scattering in a fiber ring resonator,” Opt. Rev.7, 303–308 (2000). [CrossRef]
- A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A44, 3205–3209 (1991). [CrossRef] [PubMed]
- S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A20, 1132–1137 (2003). [CrossRef]
- N. Vermeulen, C. Debaes, A. A. Fotiadi, K. Panajotov, and H. Thienpont, “Stokes-anti-Stokes iterative resonator method for modeling Raman lasers,” IEEE J. Quantum Electron.42, 1144–1156 (2006). [CrossRef]
- I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron.35, 1812–1816 (1999). [CrossRef]
- L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett.7, 288–290 (1982). [CrossRef] [PubMed]
- F. E. Seraji, “Steady-state performance analysis of fiber-optic ring resonator,” Prog. Quantum. Electron.33, 1–16 (2009). [CrossRef]
- A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B18, 556–567 (2001). [CrossRef]
- V. V. Spirin, C. A. López, P. Mégret, and A. A. Fotiadi, “Single-mode Brillouin fiber laser passively stabilized at resonance frequency with self-injection locked pump laser,“ Laser Phys. Lett. (to be published), 1–4 (2012).

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