## SBS threshold for single mode and multimode GRIN fibers in an all fiber configuration

Optics Express, Vol. 13, Issue 6, pp. 2019-2024 (2005)

http://dx.doi.org/10.1364/OPEX.13.002019

Acrobat PDF (111 KB)

### Abstract

We investigate stimulated Brillouin scattering (SBS) threshold in single mode and multimode fibers in an all fiber network. The pump is a single mode fiber pigtail attached to a diode. We find the theory and experiment agree for both single mode and multimode GRIN fibers. We modify the bulk SBS threshold equation for use with fibers by properly accounting for mode sizes and modal dispersion.

© 2005 Optical Society of America

## 1. Introduction

1. V. I. Kovalev, R. G. Harrison, and A. M. Scott, “The build-up of stimulated Brillouin scattering excited by pulsed pump radiation in a long optical fibre,” Opt. Commun. **185**, 185–189 (2000). [CrossRef]

2. K. Tei, Y. Tsuruoka, T. Uchiyama, and T. Fujioka, “Critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. **40**, 3191–3194 (2001). [CrossRef]

3. Y. Tsuruoka, K. Tei, and T. Fujioka, “Influence of mode dispersion on critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. **41**, 5166–5169 (2002). [CrossRef]

4. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. **27**, 2022–2025 (2002). [CrossRef]

5. R. G. Harrison, V. I. Kovalev, W. Lu, and Dejin Yu, “SBS self-phase conjugation of CW Nd”YAG laser radiation in an optical fiber,” Opt. Commun. **163**, 208–211 (1999). [CrossRef]

6. V. I. Kovalev and R. G. Harrison, “Diffraction limited output form a CW Nd”YAG master oscillator/power amplifier with fibre phase conjugate SBS mirror,” Opt. Commun. **166**, 89–99 (1999). [CrossRef]

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

4. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. **27**, 2022–2025 (2002). [CrossRef]

*et al*. [4

4. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. **27**, 2022–2025 (2002). [CrossRef]

2. K. Tei, Y. Tsuruoka, T. Uchiyama, and T. Fujioka, “Critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. **40**, 3191–3194 (2001). [CrossRef]

5. R. G. Harrison, V. I. Kovalev, W. Lu, and Dejin Yu, “SBS self-phase conjugation of CW Nd”YAG laser radiation in an optical fiber,” Opt. Commun. **163**, 208–211 (1999). [CrossRef]

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

1. V. I. Kovalev, R. G. Harrison, and A. M. Scott, “The build-up of stimulated Brillouin scattering excited by pulsed pump radiation in a long optical fibre,” Opt. Commun. **185**, 185–189 (2000). [CrossRef]

**27**, 2022–2025 (2002). [CrossRef]

**27**, 2022–2025 (2002). [CrossRef]

*et al*. [2

2. K. Tei, Y. Tsuruoka, T. Uchiyama, and T. Fujioka, “Critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. **40**, 3191–3194 (2001). [CrossRef]

*et al*. [3

3. Y. Tsuruoka, K. Tei, and T. Fujioka, “Influence of mode dispersion on critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. **41**, 5166–5169 (2002). [CrossRef]

## 2. Experiment

*e*

^{2}radius of 7.8µm and an

*M*of 1.15. Thus, only the lowest-order mode in the multimode (MM) fiber is excited under butt coupling. Using the same technique the mode radius of the single-mode (SM) fiber was found to be 4.9µm. These results agree with the theoretical predictions, see Table 1 and the following section.

^{2}*R*(0) as a function of the input pump power

_{SM}=P_{c}/P_{p}*P*(0), see Fig. 2, where

_{p}*P*is the Stokes power exiting circulator. For the multimode fiber we calculate the reflectivity

_{c}*R*=

_{MM}*P*(0)/

_{s}*P*(0)=(

_{p}*P*(0)-

_{p}*P*(

_{p}*L*))/

*P*(0) from

_{p}*P*(

_{p}*z*) which is the pump power measured at both ends of the fiber at

*z*=0, and

*z=L*with

*P*(

_{s}*L*)=0. Note, that

*R*depends only on the measured pump inside the fiber at z=0, L, and hence does not represent a conjugation property. Further, we determine the multimode fidelity from

_{MM}*F*=

_{MM}*P*(0)=

_{c}/P_{s}*P*(0)/

_{c}/P_{p}*R*where

_{MM}*P*is the Stokes power exiting the circulator.

_{c}*F*depends on the Stokes power

_{MM}*P*measured external to the fiber compared to

_{c}*P*(0) which is internal to the fiber. Thus,

_{p}*F*represents a measurement of phase conjugation. This definition of phase fidelity is consistent with the standard definition based on overlap integrals. These two quantities are shown in Fig. 2 as a function of input pump power. In both figures we also show, as the solid line, our theoretical calculations for the threshold and the reflectivity

_{MM}*R=P*(0)/

_{s}*P*(0) obtained from the well known plane wave model [11].We note that for the single mode fiber the reflectivities agree within 3% and for the multimode GRIN fiber the agreement in reflectivities is within 10%. We now discuss the theory which incorporates the effects of the mode area and inhomogeneous gain which we used for the curves in Fig. 2.

_{p}## 3. Theory

**27**, 2022–2025 (2002). [CrossRef]

*g*for inhomogeneous broadening due to waveguiding is derived as

_{i}*F*(

_{B}*ϕ*)=2

*n*sin(

_{co}v*ϕ*/2)/

*λ*,

*v*is the velocity of sound, λ is the pump wavelength in vacuum,

*n*is the core index-of-refraction, and ϕ is the backward Stokes angle with respect to the pump. The on-axis frequency is

_{co}*F*(

_{0}=F_{B}*π*), and the frequency at the critical angle is

*θ*=sin

_{c}^{-1}((

*NA)/n*). Kovalev and Harrison [4

_{co}**27**, 2022–2025 (2002). [CrossRef]

_{0}=36MHz for the Brillouin linewidth, λ=1.06

*µ*m,

*v*=5.96km/

*s, n*=1.46 and the bulk gain

_{co}*g*

_{0}=5×10

^{-11}m/W for their calculations. The above equations are applied to step index fibers where all ray directions are assumed to be equally probable up to the critical angle. This assumption is based on the uniformity of the index profile which is not valid for GRIN fibers. They find that the bandwidth in Eq. (2) compares favorably with their experiment.

*g*(

_{i}*f*) is the inhomogeneous gain at the SBS return frequency

_{S}*f*. However, Eq. (3) differs from previous work in that we replace the core area by the optimum Gaussian mode area of radius

_{S}*ω*. The effective length remains as

*l*=(1-exp(-

_{eff}*αl*))/

*α*, where α=0.046/km.

*v*=5960m/s,

*λ*=1.5

*µ*m,

*n*=1.4682,

_{co}*F*

_{0}=2

*n*

_{co}v/λ, g_{0}=2.5×10

^{-11}m/W this gain is half of the bulk gain value given in reference (10). Since our wavelength is λ=1.5

*µ*m the value of Γ

_{0}is determined from the scaling Γ

_{0}=38.4/λ

^{2}MHz [10

10. D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3*µ*m,” Electron. Lett. **18**, 495–496 (1982). [CrossRef]

*µ*m.

*ω*for both step index and GRIN fibers. He finds for a step index fiber of core radius

*a*, the optimum beam radius

*ω*is given by

*e*

^{-2}value of the measured intensity. In these equations the

*V*-number is

*V=ka*(

*NA*)

*, where (*

_{SI}, k=2π/λ*NA*)

*is the step index numerical aperture. For the GRIN fiber the numerical aperture is*

_{SI}*a*=4.0µm, (

*NA*)SI=0.14, thus

*V*=2.34 and Eq. (4) yields a mode radius of

*r*=4.5µm. The fiber length is

*l*=1km. For the GRIN fiber

*V*-number is

*V*≈20. Thus, Eq. (5) gives a mode radius

*ω*=8µm for a core radius

*α*=25µm. The length is

*l*=4.4km. Next we consider the radial dependence of the GRIN numerical aperture and we show how it affects Eq. (1). We first note that the distribution of Stokes frequencies, or all ray directions, are not equally distributed within

*θ*as was previously assumed for the step index fiber [4

_{c}**27**, 2022–2025 (2002). [CrossRef]

*NA*in Eq. (1) with the spatial average

*g*(

_{i}*F*

_{0})=5.64×10

^{-12}m/W for the single mode fiber, and a gain of

*g*(

_{i}*F*

_{0})=9.93×10

^{-12}m/W for the GRIN fiber; these are reduced from

*g*

_{0}=2.5×10

^{-11}m/W. Also, Eqs. (4),(5) for the optimum Gaussian radii leads to the same conclusion. For large

*V*, and we have

*V*=20, the step index fiber has a waist determined by

*ω/α*=.65 while for the GRIN fiber

*NA*)

_{SI}.

*ω*are obtained using Eqs. (4),(5). The corrected threshold uses Eq. (3). However, the uncorrected threshold retains the same mode size, but uses only the gain

*g*

_{0}to illustrate the effect of dispersion. If, on the other hand, the core area is used instead of the mode area, along with

*g*

_{0}, the threshold increases to 400mW.

10. D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3*µ*m,” Electron. Lett. **18**, 495–496 (1982). [CrossRef]

*µ*m. Our simulation gives within 15% agreement with a threshold of 6.5mW. The experiments of Harrison

*et al*. [5

5. R. G. Harrison, V. I. Kovalev, W. Lu, and Dejin Yu, “SBS self-phase conjugation of CW Nd”YAG laser radiation in an optical fiber,” Opt. Commun. **163**, 208–211 (1999). [CrossRef]

*et al*. [7

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

*µ*m. All of these experiments use a focused pump.

## Acknowledgments

## References and links

1. | V. I. Kovalev, R. G. Harrison, and A. M. Scott, “The build-up of stimulated Brillouin scattering excited by pulsed pump radiation in a long optical fibre,” Opt. Commun. |

2. | K. Tei, Y. Tsuruoka, T. Uchiyama, and T. Fujioka, “Critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. |

3. | Y. Tsuruoka, K. Tei, and T. Fujioka, “Influence of mode dispersion on critical power of stimulated Brillouin scattering in multimode optical fibers,” Jpn. J. Phys. |

4. | V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. |

5. | R. G. Harrison, V. I. Kovalev, W. Lu, and Dejin Yu, “SBS self-phase conjugation of CW Nd”YAG laser radiation in an optical fiber,” Opt. Commun. |

6. | V. I. Kovalev and R. G. Harrison, “Diffraction limited output form a CW Nd”YAG master oscillator/power amplifier with fibre phase conjugate SBS mirror,” Opt. Commun. |

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

8. | Robert W. Boyd, |

9. | D. Marcuse “Loss analysis of single-mode fiber splices,” The Bell Sys. Tech. J. |

10. | D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 |

11. | W. Kaiser and M. Maier, |

**OCIS Codes**

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(290.5830) Scattering : Scattering, Brillouin

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 7, 2005

Revised Manuscript: February 28, 2005

Published: March 21, 2005

**Citation**

A. Mocofanescu, L. Wang, R. Jain, K. Shaw, A. Gavrielides, P. Peterson, and M. Sharma, "SBS threshold for single mode and multimode GRIN fibers in an all fiber configuration," Opt. Express **13**, 2019-2024 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-2019

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### References

- V. I. Kovalev and R. G. Harrison, A. M. Scott, "The build-up of stimulated Brillouin scattering excited by pulsed pump radiation in a long optical fibre,�?? Opt. Commun. 185, 185-189 (2000). [CrossRef]
- K. Tei, Y. Tsuruoka, T. Uchiyama, T. Fujioka, "Critical power of stimulated Brillouin scattering in multimode optical fibers,�?? Jpn. J. Phys. 40, 3191-3194 (2001). [CrossRef]
- Y. Tsuruoka, K. Tei, and T. Fujioka, "Influence of mode dispersion on critical power of stimulated Brillouin scattering in multimode optical fibers,�?? Jpn. J. Phys. 41, 5166-5169 (2002). [CrossRef]
- V. I. Kovalev and R. G. Harrison, "Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,�?? Opt. Lett. 27, 2022-2025 (2002). [CrossRef]
- R. G. Harrison, V. I. Kovalev, W. Lu, Dejin Yu, "SBS self-phase conjugation of CW Nd�??YAG laser radiation in an optical fiber,�?? Opt. Commun. 163, 208-211 (1999). [CrossRef]
- V. I. Kovalev, R. G. Harrison, "Diffraction limited output form a CW Nd�??YAG master oscillator/power amplifier with fibre phase conjugate SBS mirror,�?? Opt. Commun. 166, 89-99 (1999). [CrossRef]
- L. Chen, X. Bao, "Analytical and numerical solutions for steady state stimlulated Brillouin scattering in a single-mode fiber,�?? Opt. Commun. 152, 65-70 (1998). [CrossRef]
- Robert W. Boyd, Nonlinear Optics (Academic Press, Inc, 1992), pp. 339.
- D. Marcuse, "Loss analysis of single-mode fiber splices,�?? The Bell Sys. Tech. J. 56, No. 6, 703-718 (1977).
- D. Cotter, "Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 µm,�?? Electron. Lett. 18, 495-496 (1982). [CrossRef]
- W. Kaiser, M. Maier, Laser Handbook Vol. 2, ed. by F. T. Arecchi, E. O. Schulz-Dubois (Northland-Holland Publishing Company, 1972), pp. 1118.

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