## Impact of modal interference on the beam quality of high-power fiber amplifiers |

Optics Express, Vol. 19, Issue 4, pp. 3258-3271 (2011)

http://dx.doi.org/10.1364/OE.19.003258

Acrobat PDF (1470 KB)

### Abstract

Recent work on high-power fiber amplifiers report on a degradation of the output beam quality or even on the appearance of mode instabilities. By combining the transversally resolved rate equations with a 3D Beam propagation method we have managed to create a model able to provide an explanation of what we believe is at the root of this effect. Even though this beam quality degradation is conventionally linked to transversal hole burning, our simulations show that this alone cannot explain the effect in very large mode area fibers. According to the model presented in this paper, the most likely cause for the beam quality degradation is an inversion-induced grating created by the interplay between modal interference along the fiber and transversal hole burning.

© 2011 OSA

## 1. Introduction

1. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. **49**(25), F71–F78 (2010). [CrossRef] [PubMed]

3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. **35**(2), 94–96 (2010). [CrossRef] [PubMed]

4. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. Barty, “Ultimate power limits of optical fibers, ” in *Optical Fiber Communication Conference*, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMO6, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMO6.

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express **15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

15. N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express **16**(24), 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038. [CrossRef] [PubMed]

## 2. Simulation model

### 2.1. Beam propagation method

*n(x,y,z)*is the three-dimensional refractive index of the material, and

*k*is the wavenumber. Considering that the complex index of refraction

*n(x,y,z)*typically varies slowly in the direction of propagation (z direction in this case), the vectorial Helmholtz equation can be rewritten as a function of the transverse electric field components

*n*is a reference refractive index close to the actual effective index of the beam in the fiber (i.e. it should be chosen so that the envelope varies slowly in the propagation direction). Introducing Eq. (3) into Eq. (2) it is possible to obtain the so-called one-way wave equation:where the operator

_{o}*P*is given by:

*Δz*is the longitudinal step size and

*α*is a weighting factor that controls the finite difference scheme. Thus,

*α*=0 corresponds to an explicit scheme, whereas

*α*=1 is implicit. Additionally,

*α*=0.5 represents the well-known Crank-Nicholson scheme. As can be appreciated, Eq. (7) relates the electric field at one longitudinal step with the electric field at the previous step, i.e. is a beam propagation equation.

*P*in Eq. (7), it is possible to obtain a system of linear equations which can be expressed in matrix form as:where

*l+1)Δz*and

*lΔz*, respectively. Additionally,

**and**

*A***are non-symmetric complex band matrixes. These matrixes can be efficiently inverted using the BiCG-STAB method [16**

*B*16. H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. **13**(2), 631–644 (1992). [CrossRef]

17. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. **16**(9), 624–626 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=ol-16-9-624. [CrossRef] [PubMed]

### 2.2. Transversally-resolved steady state rate equations

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express **15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

*(x,y,z)*. Additionally,

*P*and

_{p}(z)*P*are the pump and signal powers along the propagation direction

_{s}(z)*z*, respectively. The signs + and – on the powers represent the propagation direction (either +

*z*or –

*z*). On the other hand,

*σ*and

_{ap}*σ*are the absorption cross-sections at the pump and signal wavelengths, respectively. Similarly,

_{as}*σ*and

_{ep}*σ*are the emission cross-sections at the pump and signal wavelengths, respectively. Besides,

_{es}*h*is the Planck constant,

*τ*is the lifetime in the excited state,

*υ*and

_{p}*υ*are the pump and signal frequencies, respectively. In addition,

_{s}*α*and

_{p}*α*are the attenuation coefficients of the pump and signal due to their propagation through the fiber, respectively. It is also worth noting that the integration limits

_{s}*x*,

_{1}*x*, y

_{2}*, y*

_{1}*are chosen to sweep the complete core area. Finally,*

_{2}*Γ*and

_{p}(x,y)*Γ*are the power filling distributions of pump and signal, which can be expressed as follows:where

_{s}(x,y)*A*is the area of the pump core, and

_{clad}*ψ(x,y)*is the transversal intensity distribution of the signal beam.

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express **15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

*P*can only propagate in the forward direction and, therefore, it is only represented by a + sign in Eq. (9). Additionally, only one signal beam is considered (instead of one per fiber mode as in [14

_{s}**15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

**15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

*Δx*and

*Δy*in the x- and y-directions respectively. Thus, the discrete transversally resolved steady state rate equations are:

*N*represents the total ion concentration at the transversal point

_{(m,k)}*(mΔx, kΔy)*. We have programmed these equations in a computer and solved them using the Runge-Kutta methods.

### 2.2. Active BPM model

*ψ(x,y)*obtained for each iteration after the propagation process is used to calculate the new power filling factors

- 1. Define an input beam (i.e. transversal electric field distribution)
- 2. Propagate the beam using the BPM algorithm until the
*z*-point corresponding to the next iteration of the RK method is reached. - 3. Use these power filling factors to solve the system of Eqs. (11).
- 4. Determine the new
*z*-point for the next iteration of the RK method and repeat from 2.

## 3. Simulation results: Impact of modal interference on the beam quality of high-power fiber lasers

3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. **35**(2), 94–96 (2010). [CrossRef] [PubMed]

^{25}ion/m

^{3}. The fiber cross-sections employed are those measured in [18

18. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. **33**(7), 1049–1056 (1997). [CrossRef]

6. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express **13**(4), 1055–1058 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-4-1055. [CrossRef] [PubMed]

_{01}since the doped region is smaller than the whole core area. This tends to favor the amplification of the fundamental mode against the amplification of the higher order modes [14

**15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

19. T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B **19**(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539. [CrossRef]

### 3.1. Physical origin of the beam quality degradation

*x-z*plane. At the input of the fiber 95% of the energy was coupled in the fundamental LP

_{01}mode and 5% in the LP

_{11}mode (with the right orientation to shift the center of gravity of the beam in the x-z plane). As seen in Fig. 2(a), the evolution of the beam intensity along a fiber, when considering mode interference, creates periodic changes of the beam (in this example seen as a periodic shift of the center of gravity of the beam) (see e.g. Fig. 1). This gives rise to periodic core areas where the inversion (here defined as

*N*) has not been efficiently depleted (see Fig. 2(b)) which, in turn, via effects such as the resonantly induced index change of doped fibers discussed in [9

_{2}/N9. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. **3**(1), 44–64 (1997). [CrossRef]

12. L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. **11**(9), 1435–1446 (1993). [CrossRef]

_{01}and LP

_{11}), so that at the end of the fiber the HOM content can grow substantially. This alone can reduce the beam quality of the laser output and, additionally, it is our belief that it may trigger the mode instabilities reported elsewhere [3

3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. **35**(2), 94–96 (2010). [CrossRef] [PubMed]

6. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express **13**(4), 1055–1058 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-4-1055. [CrossRef] [PubMed]

### 3.2. Beam quality degradation: simulation results

_{01}and 5% in the LP

_{11}.

20. F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, “Modal fields calculation using the finite difference beam propagation method,” J. Lightwave Technol. **12**(12), 2066–2072 (1994). [CrossRef]

*c*of the i

_{i}^{th}fiber mode (with electric field

*z*was calculated using the following complex overlap integral:

_{01}content thanks to the effect of the preferential gain. Please note that even though this simulation does not consider the effect of the inversion grating it still includes THB. Thus, these results show that THB alone cannot explain the beam quality degradation observed in some experiments (at least when some amount of preferential gain is included in the fiber design).

_{11}mode in the beam as it propagates through the fiber. This increase of the LP

_{11}modal content is a direct effect of the energy transfer propitiated by the inversion-grating. The growth of the LP

_{11}mode is particularly relevant in this example, because, as discussed before, the fiber used in here has a preferential gain for the LP

_{01}. Thus, all the models not considering the effect of mode interference will predict the faster growth of the fundamental mode and, therefore, the progressive reduction of the relative modal content of the LP

_{11}as the beam propagates through the fiber. In this case, on the contrary, even though the inversion grating is in direct competition with the preferential gain, it is clear that it is able to overpower it.

_{02}mode, which even though not strictly guided in this fiber, it is close to its cut-off), which would be translated in extra losses in the practical case. This happens because there is not a perfect mode matching at the fiber input and some of the energy (~0.5%) is coupled to the radiation modes which, even though lossy, are able to also propagate and interfere with the beam along this short fiber length. Thus, an additional (weak) grating is also created, which favors the energy transfer to the radiation modes. The reason why there is an initial amount of energy coupled to the radiation modes has to do with the fact that the modes of the passive structure have been the ones used for the excitation. However, the active structure has a slightly different transversal index profile (due to the refractive index change generated by the inversion). This means that the modes of the passive and active structure are not identical and, therefore, there is some energy lost to the radiation modes due to this mode mismatch (this is also the reason why there are ripples in the traces of the relative modal content plot).

_{01}and the LP

_{11}modes should increase. This is clearly confirmed by Fig. 7 , where it can be seen that the relative modal content of the LP

_{11}mode at the output of the fiber is ~50%. Additionally, it can also be seen that the coupling to radiation modes also becomes stronger (but being this also an effect of the inversion-grating it comes as no surprise).

_{11}relative modal content amounts to less than 10% at the output of the fiber. In fact, by closely looking at Fig. 9, it can also be seen how in the first half of the fiber the modal content of the LP

_{11}slightly decreases. This is the effect of the preferential gain. Thus, Fig. 9 shows the competition between preferential gain and inversion-induced grating at work.

## 4. Conclusions

_{11}grows as the beam propagates through the fiber, giving thus rise to a reduced beam quality at the output of the amplifier. The influence of this inversion grating can be so strong that it can even overpower the effect of a preferential gain design. This effect is, however, dependent on the signal wavelength, on the saturation characteristics of the amplifier and on the fiber length. Thus, our simulations indicate that a possible way to minimize the impact of this inversion grating is to use short very large mode area fibers and highly saturated amplifiers.

## Acknowledgments

## References and links

1. | A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. |

2. | D. Gapontsev and I. P. G. Photonics, “6kW CW single mode ytterbium fiber laser in all-fiber format,” in |

3. | T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. |

4. | J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. Barty, “Ultimate power limits of optical fibers, ” in |

5. | G. P. Agrawal, |

6. | J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express |

7. | C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express |

8. | N. Andermahr and C. Fallnich, “Optically induced long-period fiber gratings for guided mode conversion in few-mode fibers,” Opt. Express |

9. | M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. |

10. | J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. F. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in Ytterbium-doped fiber,” J. Lightwave Technol. |

11. | A. A. Fotiadi, O. L. Antipov and P. Megret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” Frontiers in Guided Wave Optics and Optoelectronics, 209–234 (2010). |

12. | L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. |

13. | C. Xu and W. Huang, “Finite-difference beam propagation method for guide-wave optics,” Prog. Electromagn. Res. |

14. | M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express |

15. | N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express |

16. | H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. |

17. | G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. |

18. | R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. |

19. | T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B |

20. | F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, “Modal fields calculation using the finite difference beam propagation method,” J. Lightwave Technol. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 14, 2010

Revised Manuscript: December 24, 2010

Manuscript Accepted: January 3, 2011

Published: February 3, 2011

**Virtual Issues**

February 18, 2011 *Spotlight on Optics*

**Citation**

Cesar Jauregui, Tino Eidam, Jens Limpert, and Andreas Tünnermann, "Impact of modal interference on the beam quality of high-power fiber amplifiers," Opt. Express **19**, 3258-3271 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3258

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### References

- A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010). [CrossRef] [PubMed]
- D. Gapontsev and I. P. G. Photonics, “6kW CW single mode ytterbium fiber laser in all-fiber format,” in Solid State and Diode Laser Technology Review (Albuquerque, 2008).
- T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010). [CrossRef] [PubMed]
- J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. Barty, “Ultimate power limits of optical fibers, ” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMO6, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMO6 .
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).
- J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-4-1055 . [CrossRef] [PubMed]
- C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-10-8476 . [CrossRef] [PubMed]
- N. Andermahr and C. Fallnich, “Optically induced long-period fiber gratings for guided mode conversion in few-mode fibers,” Opt. Express 18(5), 4411–4416 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-5-4411 . [CrossRef] [PubMed]
- M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3(1), 44–64 (1997). [CrossRef]
- J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. F. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in Ytterbium-doped fiber,” J. Lightwave Technol. 16(5), 798–806 (1998). [CrossRef]
- A. A. Fotiadi, O. L. Antipov and P. Megret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” Frontiers in Guided Wave Optics and Optoelectronics, 209–234 (2010).
- L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11(9), 1435–1446 (1993). [CrossRef]
- C. Xu and W. Huang, “Finite-difference beam propagation method for guide-wave optics,” Prog. Electromagn. Res. 11, 1–49 (1995) (PIER).
- M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]
- N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express 16(24), 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038 . [CrossRef] [PubMed]
- H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). [CrossRef]
- G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=ol-16-9-624 . [CrossRef] [PubMed]
- R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]
- T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539 . [CrossRef]
- F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, “Modal fields calculation using the finite difference beam propagation method,” J. Lightwave Technol. 12(12), 2066–2072 (1994). [CrossRef]

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