## Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles

Optics Express, Vol. 17, Issue 18, pp. 15685-15699 (2009)

http://dx.doi.org/10.1364/OE.17.015685

Acrobat PDF (460 KB)

### Abstract

A numerical investigation is presented of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles. The equation determining the acoustic displacement in response to the electrostriction caused by the pump and Stokes waves reduces to the non-homogeneous Helmholtz equation for fibers with a uniform acoustic velocity profile. In this special case the acoustic displacement and subsequently the Brillouin gain are calculated using a Green's function. These results are then used to validate a finite-element solution of the same equation. This finite element method is then used to analyze a standard large mode area fiber as well as fibers incorporating four different acoustic velocity profiles with 5% variation in the acoustic velocity across the core. The profiles which suppress the peak Brillouin gain most effectively exhibit a maximum acoustic gradient near the midpoint between the center and boundary of the fiber core. These profiles produce 11 dB of suppression relative to standard large mode area fibers.

© 2009 OSA

## 1. Introduction

## 2. Brillouin gain

*u*is the longitudinal acoustic displacement and

_{z}*n,*the permittivity of free space

*λ*, the vacuum speed of light

*c*, and the effective index of the fundamental optical mode

*u*must satisfy the non-homogeneous acoustic wave Eq. (22).where

*η*is the viscosity,

*ρ*is the mass density and are the transverse and longitudinal acoustic velocities given in terms of the Young’s modulus

*E =*73 GPa, the Poisson ratio ν = 0.17, and ρ = 2200 kg/m

^{3}where we have given the values for bulk fused silica. Thus the bulk acoustic velocities are

*V*5972 m/s and

_{l}=*V*= 3766 m/s. We have retained the usual symbol for the Young’s modulus; however, it is not to be confused with the symbol for the electric field.

_{t}*V*. This wave equation is non-homogeneous in the sense that the term on the right hand side is a source term [24,25]. Substituting Eq. (3) and (4) into Eq. (6) yieldswhere again

_{t}*G*with the source function Φ [25]where the fiber cross-section is the domain of integration. Substituting Eq. (14) in Eq. (5) and expressing the Stokes and pump powers in terms of their amplitudesyields the equation for the evolution of the Stokes power in the absence of pump depletionwhere andis the non-linear effective area of the fundamental optical mode. In general the Brillouin gain is a function of the Stokes frequency which enters through the dependence of

*κ*on

*ω*expressed through

_{s}*Ω*. In the case that the optical mode takes the form of a uniform plane wave, then the classic result [3,26] for the peak Brillouin gain coefficient is recovered.

## 3. Brillouin gain in a fiber with uniform acoustic profile

*K*is the modified Bessel function. We consider a step-index large mode area fiber whose fundamental mode field intensity profile is well approximated by a Gaussianwhere

_{0}*d*is the 1/

*e*mode field diameter. The integral required to evaluate the acoustic displacement field is thenand the Brillouin gain is given byEven for this simple case, carrying out the integral over

^{2}*θ*yields a 2-dimensional integral that must be evaluated numerically. This motivates the use of other methods to treat cases where the fiber has a non-uniform acoustic velocity profile.

## 4. Finite element analysis of Brillouin gain

*κ*and

*Φ*may be treated as constants within each element as shown in Fig. 1. The discrete field vector

*φ*corresponds to the values of

*x*and

_{k}*y*are the global coordinates of the nodes and

_{k}*L*and

_{1}*L*range from 0 to 1. The spatial derivatives of

_{2}*φ*at the integration points are then obtained by taking the derivatives of the shape functions. The Jacobian matrix relating the derivatives of the local and global coordinates is used to evaluate the derivatives of the shape functions in terms of the local coordinates of the integration points.withEquation (27) and (28) are then used to construct matrices

**N**,

**N**

_{x}and

**N**

_{y}such that where

*φ*within element

*i*at local coordinates

*L*and

_{1}*L*, likewise for its derivatives, and

_{2}*φ*at node

*j*when the nodes are numbered globally throughout the entire mesh. Thus the matrices

**N**,

**N**

_{x}and

**N**

_{y}have a number of rows equal to the number of elements in the mesh and a number of columns equal to the number of nodes in the mesh. These matrices are constructed by inserting the values given by Eq. (27) and (28) in the rows corresponding to each element in the six columns corresponding to the global indices of the six nodes on the boundary of that element. These matrices may then be used to write Eq. (25) in terms of the global vector

**φ**whereandThe sums are taken over the integration points within each element,

*T*denotes the transpose of a matrix or vector,

*W*is the weighting factor of integration point

_{k}*k*,

*J*is a diagonal matrix containing the determinant of the Jacobian, Eq. (30), at the local coordinates of integration point

_{k}*k*,

**N**

_{k},

**N**

_{xk}and

**N**

_{yk}are the matrices defined in Eq. (31) evaluated at the local coordinates of integration point

*k*, and the diagonal matrix

**M**contains the values of κ

^{2}on each element in the mesh. Minimizing Eq. (32) with respect to

**φ**yields the set of linear equationswhich has the solutionThe Brillouin gain is then given bywhere ×denotes elementwise multiplication,

*E*is the scalar electric field defined on the nodes of the mesh and the matrixfacilitates the integration of the dot product of any two vectors defined on the nodes of the mesh over the fiber cross-section

## 5. Validation of the finite element method

## 6. Fibers with an acoustic guiding layer and a linearly-ramped acoustic profile

## 7. Improved fiber designs

^{−12}GHz-m/W. This suggests that further suppression may be achieved by designing a fiber with a flat-top Brillouin gain spectrum. Combining a more rapid acoustic velocity variation mid-radius relative to the core with a less rapid variation at the center and boundary serves to flatten out the Brillouin gain spectrum thus achieving a smaller maximum Brillouin gain.

## 8. Discussion and conclusion

## Acknowledgments

## References and links

1. | Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. |

2. | Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. |

3. | G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (Academic, New York, 2001). |

4. | C. G. Carlson, P. D. Dragic, B. W. Graf, R. K. Price, J. J. Coleman, and G. R. Swenson, “High power Ybdoped Fiber Laser-Based LIDAR for Space Weather,” Proc. SPIE |

5. | K. Takeno, T. Ozeki, S. Moriwaki, and N. Mio, “100 W, single-frequency operation of an injection-locked Nd:YAG laser,” Opt. Lett. |

6. | E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. |

7. | J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. |

8. | M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express |

9. | M. D. Mermelstein, A. D. Yablon, and C. Headley, “Suppression of Stimulated Brillouin Scattering in an Er-Yb Fiber Amplifier Utilizing Temperature Segmentation,” in |

10. | V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. |

11. | J. E. Rothenberg, P. A. Thielen, M. Wickham, and C. P. Asman, “Suppression of Stimulated Brillouin Scattering in Single-Frequency Multi-Kilowatt Fiber Amplifiers,” Proc. SPIE |

12. | S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express |

13. | M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express |

14. | M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Modea Area Yb-doped Optical Fiber,” Proc. SPIE |

15. | P. D. Dragic, C. Liu, G. C. Papen, and A. Galvanauskas, “Optical Fiber with an Acoustic Guiding Layer for Stimulated Brillouin Scattering Suppression,” in |

16. | A. H. McCurdy, “Modeling of Stimulated Brillouin Scattering in Optical Fibers with Arbitrary Radial Index Profile,” J. Lightwave Technol. |

17. | V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express |

18. | A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express |

19. | P. D. Dragic, “Acoustical-optical fibers for control of stimulated Brillouin scattering,” in 2006 Digest of the LEOS Summer Topical Meetings, pp. 3–4. (2006). |

20. | M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm |

21. | J. Nilsson, “SBS Supression at the kilowatt level,” presented at SPIE Photonics West, San Jose, California, USA, 24–29 Jan. (2009). |

22. | E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. |

23. | R. D. Cook, |

24. | S. J. Farlow, |

25. | M. D. Greenberg, |

26. | C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. |

27. | P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. |

28. | B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. |

29. | B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express |

30. | W. Torruellas, M. Alam, J. Edgecumbe, K. Tankala, J. Rothenberg, and M. Wickham, “Spectral SBS model for Yb:DCF with Discrete acoustic core designs,” presented at the SPIE Defense and Security Symposium, Orlando, Florida, USA, 13–17 Apr. 2009. |

31. | W. Zou, Z. He, and K. Hotate, “Acoustic modal analysis and control in w-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express |

32. | W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. |

33. | Ward and Spring, “Brillouin gain in optical fibers with inhomogeneous acoustic velocity”, Proc. SPIE, |

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

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2310) Fiber optics and optical communications : Fiber optics

(140.3510) Lasers and laser optics : Lasers, fiber

(190.2640) Nonlinear optics : Stimulated scattering, modulation, etc.

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 15, 2009

Revised Manuscript: July 29, 2009

Manuscript Accepted: August 14, 2009

Published: August 20, 2009

**Citation**

Benjamin Ward and Justin Spring, "Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles," Opt. Express **17**, 15685-15699 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15685

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### References

- Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-5-459 . [CrossRef] [PubMed]
- Y. Jeong, J. Nilsson, J. Sahu, D. Payne, R. Horley, L. Hickey, and P. Turner, “Power Scaling of Single-Frequency Ytterbium-Doped Fiber Master-Oscillator Power-Amplifier Sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (Academic, New York, 2001).
- C. G. Carlson, P. D. Dragic, B. W. Graf, R. K. Price, J. J. Coleman, and G. R. Swenson, “High power Ybdoped Fiber Laser-Based LIDAR for Space Weather,” Proc. SPIE 5891, 68,730K–1–68,730K–12 (2008).
- K. Takeno, T. Ozeki, S. Moriwaki, and N. Mio, “100 W, single-frequency operation of an injection-locked Nd:YAG laser,” Opt. Lett. 30(16), 2110–2112 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-16-2110 . [CrossRef] [PubMed]
- E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354 . [CrossRef] [PubMed]
- J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442 . [CrossRef]
- M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express 25, 11,071–11,076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071 .
- M. D. Mermelstein, A. D. Yablon, and C. Headley, “Suppression of Stimulated Brillouin Scattering in an Er-Yb Fiber Amplifier Utilizing Temperature Segmentation,” in Optical Amplifiers and Their Applications Topical Meeting Technical Digest (CD) (Optical Society of America, 2005), paper TuD3.
- V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. 31(2), 161–163 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-2-161 . [CrossRef] [PubMed]
- J. E. Rothenberg, P. A. Thielen, M. Wickham, and C. P. Asman, “Suppression of Stimulated Brillouin Scattering in Single-Frequency Multi-Kilowatt Fiber Amplifiers,” Proc. SPIE 6873, 68,730O–1–7 (2008).
- S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. R. J. A. De Meritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17,044–17,050 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-17044 . [CrossRef]
- M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8290 . [CrossRef] [PubMed]
- M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Modea Area Yb-doped Optical Fiber,” Proc. SPIE 6873, 68,730N–1–68,730N–7 (2008).
- P. D. Dragic, C. Liu, G. C. Papen, and A. Galvanauskas, “Optical Fiber with an Acoustic Guiding Layer for Stimulated Brillouin Scattering Suppression,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CThZ3. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2005-CThZ3
- A. H. McCurdy, “Modeling of Stimulated Brillouin Scattering in Optical Fibers with Arbitrary Radial Index Profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005). [CrossRef]
- V. I. Kovalev and R. G. Harrison, “Threshold for stimulated Brillouin scattering in optical fiber,” Opt. Express 15(26), 17,625–17,630 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17625 . [CrossRef]
- A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5338 . [CrossRef] [PubMed]
- P. D. Dragic, “Acoustical-optical fibers for control of stimulated Brillouin scattering,” in 2006 Digest of the LEOS Summer Topical Meetings, pp. 3–4. (2006).
- M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, “SBS gain efficiency measurements and modeling in a 1714 mm2 effective area LP08 higher-order mode optical fiber,” Opt. Express 15(24), 15,952–15,963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952 . [CrossRef]
- J. Nilsson, “SBS Supression at the kilowatt level,” presented at SPIE Photonics West, San Jose, California, USA, 24–29 Jan. (2009).
- E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999). [CrossRef]
- R. D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).
- S. J. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1993).
- M. D. Greenberg, Application of Green’s Functions in Science and Engineering (Prentice Hall, New Jersey,1971).
- C. Tang, “Saturation and Spectral Characteristics of the Stokes Emission in the Stimulated Brillouin Process,” J. Appl. Phys. 37(8), 2945–2956 (1966), http://link.aip.org/link/?JAPIAU/37/2945/1 . [CrossRef]
- P. C. Hammer, O. J. Marlowe, and A. H. Stroud, “Numerical Integration over Simplexes and Cones,” Math. Tables Other Aids Comput. 10(55), 130–137 (1956). [CrossRef]
- B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Inhomogeneous Anisotropic Refractive Index Tensor,” IEEE J. Quantum Electron. 44(2), 150–156 (2008). [CrossRef]
- B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-853 . [CrossRef] [PubMed]
- W. Torruellas, M. Alam, J. Edgecumbe, K. Tankala, J. Rothenberg, and M. Wickham, “Spectral SBS model for Yb:DCF with Discrete acoustic core designs,” presented at the SPIE Defense and Security Symposium, Orlando, Florida, USA, 13–17 Apr. 2009.
- W. Zou, Z. He, and K. Hotate, “Acoustic modal analysis and control in w-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10006 . [CrossRef] [PubMed]
- W. Zou, Z. He, and K. Hotate, “Two-dimensional finite element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006). [CrossRef]
- Ward and Spring, “Brillouin gain in optical fibers with inhomogeneous acoustic velocity”, Proc. SPIE, 7195, 71951J–1–11 (2009).
- V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(22), 2022–2024 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-22-2022 . [CrossRef]

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