## A theoretical treatment of two approaches to SBS mitigation with two-tone amplification

Optics Express, Vol. 16, Issue 18, pp. 14233-14247 (2008)

http://dx.doi.org/10.1364/OE.16.014233

Acrobat PDF (242 KB)

### Abstract

A technique that employs two seed signals for the purpose of mitigating stimulated Brillouin scattering (SBS) effects in narrow-linewidth Yb-doped fiber amplifiers is investigated theoretically by constructing a self-consistent model that incorporates the laser gain, SBS, and four-wave mixing (FWM). The model reduces to solving a two-point boundary problem consisting of an 8x8 system of coupled nonlinear differential equations. Optimal operating conditions are determined by examining the interplay between the wavelength separation and power ratio of the two seeds. Two variants of this ‘two-tone’ amplification are considered. In one case the wavelength separation is precisely twice the Brillouin shift, while the other case considers a greater wavelength separation. For the former case, a two-fold increase in total output power over a broad range of seed power ratios centered about a ratio of approximately 2 is obtained, but with fairly large FWM. For the latter case, this model predicts an approximately 100% increase in output power (at SBS threshold with no signs of FWM) for a ‘two-tone’ amplifier with seed signals at 1064nm and 1068nm, compared to a conventional fiber amplifier with a single 1068nm seed. More significantly for this case, it is found that at a wavelength separation greater than 10nm, it is possible to appreciably enhance the power output of one of the laser frequencies.

© 2008 Optical Society of America

## 1. Introduction

1. D. P. Machewirth, Q. Wang, B. Samson, K. Tankala, M. O’Connor, and M. Alam, “Current developments in high-power, monolithic, polarization maintaining fiber amplifiers for coherent beam combining applications,” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE **6453**, 64531F (2007).

2. J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B. Roh, “Comparison of Stimulated Brillouin Scattering thresholds and spectra in non-polarization maintaining and polarization-maintaining passive fibers,” Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE **5709**, 147–156 (2005).

4. M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. 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**, 8290–8299 (2007). [CrossRef] [PubMed]

5. S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express **15**, 17044–17050 (2007). [CrossRef] [PubMed]

*λ*

_{1}and

*λ*

_{2}.

*λ*is large enough to seriously mitigate four-wave mixing (FWM) which is the next lowest-threshold nonlinear effect. It has been thought previously that while an approximately 100% increase in total power can be obtained, enhancement of either of the two laser signals is not possible [7

7. P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express **12**, 4443–4448 (2004). [CrossRef] [PubMed]

*λ*>10 nm, appreciable power enhancement of one of the tones is possible through power transfer between the two tones and an overall reduction of the integrated SBS gain. Another two-tone technique which we investigate was previously demonstrated experimentally by Wessels et al. [7

7. P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express **12**, 4443–4448 (2004). [CrossRef] [PubMed]

*λ*twice the Brillouin shift. This enabled roughly a one-fold improvement in the amplified power of one of the seed signals through additional interactions among the input and Stokes signals, but also transferred large amount of power into numerous FWM-generated sidebands. This broadening of the optical power spectrum precludes the application of this method to fiber laser applications that require well-defined spectra such as electronically phased coherent arrays [8

8. J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE **6102**, 61020U (2006). [CrossRef]

9. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express **24**, 12188–12195 (2006). [CrossRef]

10. J. E. Kansky, C. X. Yu, D. V. Murphy, S. R. Shaw, R. C. Lawrence, and C. Higgs, “Beam control for a 2D polarization maintaining fiber optic phased array with a high-fiber count,” Proc. SPIE. **6306**, 63060G (2006). [CrossRef]

11. V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, “Spectral beam combining of a broad-stripe diode laser array in an external cavity,” Opt. Lett. **25**, 405–407 (2000). [CrossRef]

12. T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “ Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. **13**, 487–497 (2007). [CrossRef]

9. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express **24**, 12188–12195 (2006). [CrossRef]

*λ*case is discussed in Section 2 along with the simulations and results. In Section 3, we investigate the technique proposed by Wessels et al.

## 2. Large wavelength separation

### 2.1 Derivation of the two-tone model

*ω*

_{1}and

*ω*

_{2}the interaction of the waves is mediated through the third-order susceptibility of the medium

*χ*

^{(3)}. Two sidebands oscillating at

*ω*

_{3}=

*ω*

_{1}-Δ

*ω*and

*ω*

_{4}=

*ω*

_{1}+2Δ

*ω*, where Δ

*ω*=

*ω*

_{2}-

*ω*

_{1}, are generated.

*i*represents the frequency component of the electric field and

*P*

_{i}^{(nl)}is the nonlinear polarization. The electric field is expressed as

*j*represents the mode,

*A*,

_{i}*(*

_{j}*z*) is the amplitude,

*β*

_{i},

*is the propagation constant, and*

_{j}*ϕ*,

_{i}*(*

_{j}*x*,

*y*) is the transverse profile.

*j*and it is understood that our equations describe the lowest-order mode for each frequency component. Furthermore, since the frequency separation among the waves is much smaller than the optical wavelength, the modal profiles of all waves are set to be equal. In the limit of the slowly-varying envelope approximation,

*d*

^{2}

*A*/

_{i}*dz*

^{2}<< 2

*iβ*(

_{i}*dA*/

_{i}*dz*) <<

*β*

^{2}

*, and using*

_{i}A_{i}*cβ*

_{1}≈

*n*

_{1}

*ω*

_{1}, where

*n*

_{1}is the linear index of refraction at

*ω*

_{1}, one can reduce through coupled mode theory the wave equation for the wave oscillating at

*ω*

_{1}to:

*β*

^{(2)}the group-velocity dispersion (GVD) parameter. The overlap integral for the SPM and XPM,

*κ*, and the overlap integral for the acoustic and optical wave interaction,

_{pm}*κ*, are given by:

_{ao}*κ*we assumed the transverse acoustic profile is described by |

_{ao}*ϕ*|

^{2}as can be inferred from the form of the electrostrictive force. If one were to subscribe to the notion of guided acoustic modes [4

4. M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. 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**, 8290–8299 (2007). [CrossRef] [PubMed]

*will be of the form:*

_{ao}κ*ψ*represents the acoustic mode.

*g*

_{1}, is given by

*N*

_{2}and

*N*

_{1}are the population densities of the upper and lower energy levels, respectively, and where

*σ*

^{(e)}

_{1}and

*σ*

^{(a)}

_{1}a represent the emission and absorption cross sections for the seed frequency

*ω*

_{1}, respectively. The integration in the numerator of Eq. (6) is carried out within the core.

*A*

_{1}

*, is initiated from noise at the opposite end of the fiber and travels in the backward direction. The evolution of its amplitude along its direction of propagation is given by*

_{S}*g*

_{1S}is the laser gain coefficient at the Stokes wavelength and has a similar form to that in Eq. (6) except that the emission and absorption cross sections correspond to the Stokes wavelength. Note that the noise contribution is incorporated into Eq. (7) as we employ a localized source model as proposed by Smith [15

15. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. **11**, 2489–2494 (1972). [CrossRef] [PubMed]

*ω*

_{2}, and its associated Stokes waves. These equations are given by:

*g*

_{2}and

*g*

_{2}

*have similar expressions to Eq. (6). We neglect the SBS interaction for the FWM sidebands*

_{S}*ω*

_{3}and

*ω*

_{4}. This is justified as long as their amplitudes are much smaller than the laser signals. The evolution of their amplitudes along

*z*is given by:

*N*

_{ο}represents the density of Yb ions in the fiber core,

*τ*is the lifetime of the upper laser level, and the subscripted

*I*’s represent the intensities of the various waves. The intensity of the pump which in our simulation is taken to propagate in the same direction evolves according to:

*d*and

_{core}*d*are the diameters of the core and the cladding, respectively.

_{clad}### 2.2 ‘Two-tone’ simulations and results

4. M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. 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**, 8290–8299 (2007). [CrossRef] [PubMed]

5. S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express **15**, 17044–17050 (2007). [CrossRef] [PubMed]

**15**, 8290–8299 (2007). [CrossRef] [PubMed]

5. S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express **15**, 17044–17050 (2007). [CrossRef] [PubMed]

*λ*=4.0 nm. For Δ

*λ*<3nm the FWM increased considerably and became comparable to the Stokes light. For Δ

*λ*>10nm, the FWM was extremely small. Most remarkably for this wavelength separation, considerable enhancement in the power output of the lower laser frequency was obtained. For example, at Δ

*λ*=14nm and input seeds with wavelengths 1064 nm and 1050 nm, 46 W of output power was obtained for the 1064 nm light. The power ratio needed to obtain this output was approximately 9:1 with the 1050 nm having the higher input power. The 46 W output power represented a 64% enhancement over a 1064 nm single tone amplifier as shown in Fig. (6). Higher power output in one of the tones is possible to the point where almost all the output power would be in a single frequency. This can be achieved through an optimal ratio of seed and wavelength separation, or by selecting a more suitable fiber configuration; the details will be discussed in a future publication.

*N*

_{2}, decreases. Immediately past the point where the maximum power for the 1050 nm light is obtained, the population inversion is such that the 1050 nm light will experience negative gain. The 1064 nm light which has an appreciably lower absorption cross section will, however, continue to experience positive laser gain. As a consequence, power transfer occurs from the 1050 nm light and into the 1064 nm light. The SBS threshold is raised because the spatially integrated Stokes light gain for the two-tone 1064 nm light will be close to the 1064 nm single tone case even though more 1064 nm output power is obtained in the former. This is made possible because, for a significant portion of the fiber, the power in the 1064 nm light for the two tone case is less than that for the single tone case as can be seen from Fig. (6). We examined the total gain for the electric field amplitude of the Stokes light as a function of position. Referring to Eq. (7) in Section 2, this total gain is due to the total of the laser and Brillouin gain. It is worthwhile to point out here that the amplitude gain is half that of the intensity or power gain. Figure (7) represents a comparison of the amplitude gain for the two tone case pumped such that the power output at 1064 nm is equal to the power output in a single tone amplifier at threshold. Note that the spatially integrated Stokes gain for two-tone is reduced significantly, thus allowing for higher pumping power and consequently higher output at 1064 nm. As mentioned in the Section 1, this power enhancement was not thought to be possible. Thus, this is a novel way to increase power in CW narrow linewidth Yb-doped amplifiers.

## 3. Wavelength separation of twice the Brillouin shift

7. P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express **12**, 4443–4448 (2004). [CrossRef] [PubMed]

*ν*, i.e. twice the frequency of the phonon field. In optical fibers, this value is approximately 34 GHz (Δ

_{B}*λ*≈0.1 nm). It is important to point out here if the frequency separation does not lie within the range defined by 2(

*ν*±Δ

_{B}*ν*, where Δν

_{B}_{B}is the SBS gain linewidth, then there will be a one fold maximum increase in total amplifier output power as we described in Section 2. For Δ

*ν*=2Δ

*ν*, the equations describing the spatial evolution of

_{B}*A*

_{1}and

*A*

_{2}

*are modified to become:*

_{S}*ω*

_{2}is SBS-scattered into the input frequency

*ω*

_{1}, thus effectively raising the SBS threshold for

*ω*

_{2}. In order to gain maximum benefit from such a system, an optimal power ratio between the two input beams should be selected. To theoretically determine this ratio, we define

*r*=

*P*

_{1}/

*P*

_{2}and neglect laser gain and FWM. For small SBS signal gain, we can work in the undepleted pump limit to obtain from Eq. (7) and Eq. (16):

*g*

_{B1}=

*g*

_{B2}=

*g*, and

_{B}*n*

_{1}=

*n*

_{2}=

*n*. In order to achieve the largest suppression of SBS, the SBS small signal gain for each of the Stokes lights has to be approximately equal. Therefore

*r*=1/2, i.e. the input power of

*ω*

_{2}is twice that of

*ω*

_{1}. This is the power ratio used by Wessels et al. and our analysis above is further borne out by the numerical simulations presented below.

**12**, 4443–4448 (2004). [CrossRef] [PubMed]

**12**, 4443–4448 (2004). [CrossRef] [PubMed]

*λ*=14 nm, we note that the total power for the latter is one third less, but that fairly comparable outputs are obtained at the wavelength possessing the higher power. Therefore, if certain applications require the use of single frequency, the case of large wavelength separation will have a higher efficiency.

## 4. Conclusion

## References and links

1. | D. P. Machewirth, Q. Wang, B. Samson, K. Tankala, M. O’Connor, and M. Alam, “Current developments in high-power, monolithic, polarization maintaining fiber amplifiers for coherent beam combining applications,” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE |

2. | J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B. Roh, “Comparison of Stimulated Brillouin Scattering thresholds and spectra in non-polarization maintaining and polarization-maintaining passive fibers,” Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE |

3. | B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers.” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE |

4. | M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. 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 |

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

6. | M. D. Mermelstein, M. J. Andrejco, J. Fini, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Mode Area Yb-Doped Optical Fiber,” Fiber Lasers V: Technology, Systems, and Applications, Proc. SPIE |

7. | P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express |

8. | J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE |

9. | T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express |

10. | J. E. Kansky, C. X. Yu, D. V. Murphy, S. R. Shaw, R. C. Lawrence, and C. Higgs, “Beam control for a 2D polarization maintaining fiber optic phased array with a high-fiber count,” Proc. SPIE. |

11. | V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, “Spectral beam combining of a broad-stripe diode laser array in an external cavity,” Opt. Lett. |

12. | T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “ Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. |

13. | F. Patel, |

14. | L. G. Cohen, “Comparison of single-mode fiber dispersion measurement techniques,” J. Lightwave Technol. |

15. | R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. |

16. | B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(290.5900) Scattering : Scattering, stimulated Brillouin

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 9, 2008

Revised Manuscript: August 20, 2008

Manuscript Accepted: August 20, 2008

Published: August 27, 2008

**Citation**

Iyad Dajani, Clint Zeringue, T. J. Bronder, Thomas Shay, Athanasios Gavrielides, and Craig Robin, "A theoretical treatment of two approaches to SBS mitigation with two-tone amplification," Opt. Express **16**, 14233-14247 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14233

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### References

- D. P. Machewirth, Q. Wang, B. Samson, K. Tankala, M. O�??Connor, and M. Alam, "Current developments in high-power, monolithic, polarization maintaining fiber amplifiers for coherent beam combining applications," Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 64531F (2007).
- J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B. Roh, "Comparison of Stimulated Brillouin Scattering thresholds and spectra in non-polarization maintaining and polarization-maintaining passive fibers," Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE 5709, 147-156 (2005).
- B. G. Ward, C. Robin, and M. Culpepper, "Photonic crystal fiber designs for power scaling of single-polarization amplifiers," Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 645307 (2007).
- M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. 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, 8290-8299 (2007). [CrossRef] [PubMed]
- S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, "502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier," Opt. Express 15, 17044-17050 (2007). [CrossRef] [PubMed]
- M. D. Mermelstein, M. J. Andrejco, J. Fini, C. Headley, and D. J. DiGiovanni, "11.2 dB SBS Gain Suppression in a Large Mode Area Yb-Doped Optical Fiber," Fiber Lasers V: Technology, Systems, and Applications, Proc. SPIE 6873,68730N (2008).
- P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, "Novel suppression scheme for Brillouin scattering," Opt. Express 12, 4443-4448 (2004). [CrossRef] [PubMed]
- J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, "Coherently coupled high power fiber arrays," Proc. SPIE 6102, 61020U (2006). [CrossRef]
- T. M. Shay, "Theory of electronically phased coherent beam combination without a reference beam," Opt. Express 24, 12188-12195 (2006). [CrossRef]
- J. E. Kansky, C. X. Yu, D. V. Murphy, S. R. Shaw, R. C. Lawrence, and C. Higgs, "Beam control for a 2D polarization maintaining fiber optic phased array with a high-fiber count," Proc. SPIE. 6306, 63060G (2006). [CrossRef]
- V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, "Spectral beam combining of a broad-stripe diode laser array in an external cavity," Opt. Lett. 25, 405- 407 (2000). [CrossRef]
- T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, "Spectrally beam-combined fiber lasers for high-average-power applications," IEEE J. Sel. Top. Quantum Electron. 13, 487-497 (2007). [CrossRef]
- F. Patel, Ph.D. Dissertation, University of California, Davis (2000).
- L. G. Cohen, "Comparison of single-mode fiber dispersion measurement techniques," J. Lightwave Technol. LT-3, 958 (1985). [CrossRef]
- R. G. Smith, "Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering," Appl. Opt. 11, 2489-2494 (1972). [CrossRef] [PubMed]
- B. Ya. Zel�??dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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