## Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers |

Optics Express, Vol. 19, Issue 24, pp. 23965-23980 (2011)

http://dx.doi.org/10.1364/OE.19.023965

Acrobat PDF (2044 KB)

### Abstract

We investigate the effect of temperature gradients in high-power Yb-doped fiber amplifiers by a numerical beam propagation model, which takes thermal effects into account in a self-consistent way. The thermally induced change in the refractive index of the fiber leads to a thermal lensing effect, which decreases the effective mode area. Furthermore, it is demonstrated that the thermal lensing effect may lead to effective multi-mode behavior, even in single-mode designs, which could possibly lead to degradation of the output beam quality.

© 2011 OSA

## 1. Introduction

1. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express **12**, 6088–6092 (2004). [CrossRef] [PubMed]

3. J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb^{3+}-doped double-clad fiber lasers,” J. Mod. Opt. **55**, 459–471 (2008). [CrossRef]

4. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express **11**, 2982–2990 (2003). [CrossRef] [PubMed]

5. S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express **14**, 6091–6097 (2006). [CrossRef] [PubMed]

6. J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. **15**, 159–169 (2009). [CrossRef]

7. 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**, 94–96 (2010). [CrossRef] [PubMed]

9. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express **19**, 13218–13224 (2011). [CrossRef] [PubMed]

10. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express **19**, 3258–3271 (2011). [CrossRef] [PubMed]

11. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**, 10180–10192 (2011). [CrossRef] [PubMed]

12. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. **29**, 1739–1741 (2004). [CrossRef] [PubMed]

## 2. Numerical method

**u**

*is the polarization unit vector,*

_{s}*E*is the slowly varying envelope of the signal field,

_{s}*β*is an estimate of the propagation constant of the fundamental mode and

*ω*is the carrier angular frequency of the signal. In the scalar approximation, the field envelope

_{s}*E*obeys the paraxial wave equation The first terms on the right hand side of Eq. (2) describe the propagation of the signal field, with vacuum wavenumber

_{s}*k*

_{0}, in a fiber with a relative permittivity distribution given by

*ɛ*. The perturbation of this permittivity distribution due to heating of the core is given by Δ

*ɛ*. The last term describes the effect of the Ytterbium doping, which gives rise to an induced polarization

**P**

*given by a slowly varying envelope*

_{Yb}*p*as In this paper we assume that the fiber is cladding pumped at a wavelength of 975 nm and that the signal wavelength is close to 1030 nm. In this case the Yb

^{3+}ions can be modeled as the quasi-three-level system shown in Fig. 1.

*σ*are [13]: where

_{μν}*μ*are the matrix elements of the dipole moment operator,

_{μν}*E*is the pump field envelope,

_{p}*γ*are spontaneous emission rates from state

_{μν}*μ*into state

*ν*and the non-radiative transition rates are denoted

*γ*̄

*. The complex dephasing rates Γ̃*

_{μν}*of the coherences are given by where*

_{μν}*γ*̃

*are the real dephasing rates of the coherences*

_{μν}*σ*, Δ

_{μν}*=*

_{p}*ω*

_{p}*–*

*ω*

_{21}is the pump detuning, Δ

*=*

_{s}*ω*–

_{s}*ω*

_{23}is the signal detuning and Δ

_{2}= Δ

*– Δ*

_{p}*is the two-photon detuning of the pump and signal. Since the dephasing rates*

_{s}*γ*̃

*in Eq. (4) are large, we can adiabatically eliminate the coherences [14], which leads to a rate equation for the excited state population where Analogous to the McCumber theory [15*

_{μν}15. D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. **136**, A954–A957 (1964). [CrossRef]

*ρ*

_{1}=

*σ*

_{11}+

*σ*

_{33}and the excited state population as

*ρ*

_{2}=

*σ*

_{22}, we arrive at a rate equation for the excited state population. In the steady-state, this equation becomes where Φ

*and Φ*

_{p}*denote the pump and signal photon flux density, respectively, and*

_{s}*γ*=

*γ*

_{21}+

*γ*

_{23}is the total spontaneous emission rate of the excited state. In our simplified 3-level model, the absorption and emission cross sections at the signal wavelength,

*σ*and

_{as}*σ*, are given by while at the pump wavelength, the absorption and emission cross sections are Here Δ

_{es}*E*is the energy difference between states 1 and 3.

*σ*

_{23}and is given in terms of the signal field and population inversion as where

*n*is the core refractive index and

_{c}*ρ*is the density of Ytterbium ions. As discussed in [10

_{Yb}10. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express **19**, 3258–3271 (2011). [CrossRef] [PubMed]

*≠ 0), and an estimate of the magnitude of this effect when Δ*

_{s}*=*

_{s}*γ*̃

_{23}/2 and

*ρ*

_{2}= 0.5 yields Δ

*ɛ*≈ 10

^{−6}, which is quite small compared to the thermally induced changes in the refractive index. We therefore ignore this effect in our calculations.

*| ≪*

_{s}*γ*̃

_{23}yields a beam propagation equation which includes the signal gain due to the Ytterbium doping The radial coordinate is discretized using a second order finite-difference scheme, and a transparent boundary condition [16

16. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. **16**, 624–626 (1991). [CrossRef] [PubMed]

*z*from an initial beam profile

*E*(0,

_{s}*r*) using a split-operator approach where the operators

*R̂*and

*N̂*are given by and

*w*is the characteristic radius of the input beam, which is chosen to be equal to the core radius, and

*P*(0) is the input signal power.

_{s}*ɛ*due to heating of the fiber under high-power operation, we solve the steady-state heat equation under the assumption that the longitudinal temperature gradient is much smaller than the radial temperature gradient Here Δ

*T*is the temperature increase relative to the temperature of the coolant,

*κ*is the thermal conductivity of the fiber and the thermal load

*Q*is given by where

*ω*is the pump angular frequency. The first term in this expression represents heat generated by non-radiative relaxation following stimulated emission of signal photons and is the dominant heat source, whereas the last, much smaller, term is due to non-radiative relaxation following spontaneous emission at the signal wavelength. We assume uniform efficient water cooling of the fiber and thus apply a simple Dirichlet boundary condition Δ

_{p}*T*= 0 at the outer boundary when solving the heat equation. The temperature induced change of the refractive index of the fiber is given by the simple linear relationship Δ

*ɛ*=

*η*Δ

*T*, where the thermal sensitivity

*η*= 3.5 × 10

^{−5}K

^{−1}.

*P*is the pump power,

_{p}*A*is the area of the inner cladding and

_{cl}*R*is the radius of the doped core. The positive sign applies to forward pumping whereas the negative sign applies to backward pumping. The pump power can thus be propagated in the

_{c}*z*direction from its initial value

*P*(0) by the equation The beam propagation algorithm can be summarized by the following steps: These steps are carried out until the signal has been propagated the desired distance. In the case of backward pumping an initial guess for the initial condition of the pump power

_{p}*P*(0) must be provided, and the entire beam propagation algorithm is run iteratively until the desired input pump power

_{p}*P*(

_{p}*L*) is obtained. After each run the pump initial condition

*P*(0) is adjusted according to a simple secant method [17]. With a tolerance of 1% of the desired input pump power and a reasonable estimate of the pump initial condition, only a few iterations of the beam propagation algorithm are required.

_{p}## 3. Numerical results

### 3.1. Thermal lensing

6. J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. **15**, 159–169 (2009). [CrossRef]

*n*

_{2}is the non-linear refractive index of Silica,

*P*is the average signal power and

_{s}*A*is the effective area of the signal beam

_{eff}*μ*m, an inner cladding diameter of 200

*μ*m and an outer diameter of 400

*μ*m. The core-cladding index step is 10

^{−4}, which gives

*V*≈ 2.08 at the signal wavelength of 1030 nm, and the Yb doping in the core is 10

^{8}

*μ*m

^{−3}. The fiber is cladding pumped at 975 nm, and we have varied the input pump power between approximately 50 W −5 kW. The signal input power was kept fixed at 1 W, and both forward and backward pumping was simulated. Figure 2 shows the pump and signal power as a function of distance along the fiber for forward and backward pumping with 1 kW input pump power and 1 W input signal power. The pump is efficiently depleted due to the high Yb doping which results in a high efficiency close to the quantum limit.

*ρ*

_{2}of Yb as a function of longitudinal and radial distance for Fiber A at 1 kW pump power. The effect of transverse hole burning is clearly seen, especially in the backward pumped case.

*z*and

*r*for Fiber A at 1 kW pump power. The heating of the core creates a large radial thermal gradient in the fiber, which leads to a thermal lensing effect that causes the beam area to decrease, as shown in Fig. 5. It is also clear from Fig. 4 that the temperature increase is much larger in the backward pumped case due to the large gain near the output end of the fiber.

*η*is set to zero. For comparison, we have also simulated a fiber (Fiber B), identical to Fiber A except that the core, inner cladding and outer diameters are 80

*μ*m, 400

*μ*m and 800

*μ*m, respectively. It can be seen that as operating power on the order of 1 kW is reached, the thermal lensing effect starts to have a significant impact on the beam area and hence the severity of undesirable non-linear optical effects such as SPM, in particular for the backward pumped case, which is preferred over the forward pumped case due to the overall lower value of the B-integral and hence smaller impact of the aforementioned effects. Furthermore, the thermal lensing effect changes the pump power dependence of the B integral from linear to super-linear. Comparing Fiber A to Fiber B, we see that the B integral for Fiber B, when the thermal lensing effect is taken into account, can exceed the expected B integral of Fiber A in the absence of thermal lensing, despite the fact that Fiber B has a core area 4 times larger than Fiber A. This occurs at a pump power of approximately 4.5 kW for forward pumping and 1 kW for backward pumping. However, it is also clear that even in the presence of a strong thermal lensing effect, increasing the core diameter still results in a significant reduction of the B integral and hence the severity of SPM.

### 3.2. Thermally induced multi-mode behavior

18. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express **15**, 15402–15409 (2007). [CrossRef] [PubMed]

10. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express **19**, 3258–3271 (2011). [CrossRef] [PubMed]

*μ*m, 400

*μ*m and 800

*μ*m, respectively. The index step between core and cladding is Δ

*n*= 3×10

^{−5}. While such a small index step is difficult to achieve in actual SIFs, it is obtainable using PCFs. The Yb concentration is 5 × 10

^{7}

*μ*m

^{−3}and the pump and signal wavelength are 975 nm and 1030 nm. With these parameters, the V-parameter for the SIF is 2.28 and the fiber is thus single-mode by design. The pump power is 5 kW and the input signal power is 50 W. Figure 7 shows the signal and pump power as a function of

*z*for both forward and backward pumping. The lower Yb concentration compared to the simulations of Fiber A and B in section 3.1 reduces the efficiency of the amplifier. We also note the presence of small oscillations of the signal power near the output, which are likely due to excitation of radiation modes.

*z*shown in Fig. 8, we observe strong oscillations of the effective area, which would not occur if the signal was guided by a single mode and which therefore are a direct consequence of thermally induced multi-mode behavior. These oscillations are also seen in the temperature, shown in Fig. 9, and the excited state population, shown in Fig. 10.

*z*position by solving the eigenvalue problem and computing the overlap between the normalized modes Ψ and the signal field. This overlap is given by where the inner product is defined as The radial profiles of the fundamental and higher-order local modes at a given distance along Fiber C are calculated numerically from Eq. (23) using the thermally perturbed refractive index profiles, examples of which are shown in Fig. 11. The initial (

*z*= 0 m) mode profiles, plotted in Fig. 12, clearly show that several guided modes are present at the signal input. The launched signal can thus excite these higher-order local modes and the fiber is effectively multi-mode. The number of guided local modes may of course vary along the

*z*axis as the temperature profile, and thus the refractive index perturbation, changes. Since our model is limited to cylindrically symmetric fields, we do not consider non-symmetric local modes, although such modes would certainly be expected to become guided under the present conditions. For comparison, we also plot the local mode profiles at the output end (

*z*= 1 m) of the fiber in Fig. 13. In the backward pumped case, the modes are more strongly confined due to the strong thermal lensing effect compared to the forward pumped case.

*A*(

_{i}*z*) of the signal field is then defined to be and is plotted in Fig. 14. It is clear that in both the forward and backward pumped configurations, the input signal excites a small amount of the first higher-order local mode. The beating of these two modes creates the oscillations seen in the beam effective area shown in Fig. 8, and the associated intensity variation leads to the periodic variation of the population inversion seen in Fig. 10. Since the thermal load, given by Eq. (18), also depends on the signal intensity and population inversion, the temperature also exhibits a periodic variation as seen in Fig. 9. This temperature variation results in an index grating with a local period that matches the local beat length of the two local modes, and therefore provides coupling between the local modes. As is evident from Fig. 14, this leads to transfer of power from the fundamental to the higher-order local mode. We believe that this effective multi-mode behavior of the fiber amplifier under high-power operation can cause significant degradation of the output beam quality, since uncontrollable external perturbations such as temperature fluctuations and mechanical vibrations could conceivably change the relative phase of the local modes leading to significant changes in the output beam, as discussed in [18

18. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express **15**, 15402–15409 (2007). [CrossRef] [PubMed]

11. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**, 10180–10192 (2011). [CrossRef] [PubMed]

11. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**, 10180–10192 (2011). [CrossRef] [PubMed]

*z*, and therefore breaks the left/right mirror symmetry which is the core of the argument in [11

**19**, 10180–10192 (2011). [CrossRef] [PubMed]

^{8}

*μ*m

^{−3}and the length of the fiber is increased to 1.5 m. The input pump and signal power is 2000 W and 50 W, respectively. As can be seen from the plots of the signal and pump power, shown in Fig. 15, the pump is completely depleted in both the forward and backward pumped cases. An interesting observation is that the signal power decreases significantly beyond 1 m in the forward pumped case. To understand the mechanism responsible for this decrease, we again consider the decomposition of the signal field into local modes, shown in Fig. 16, and the temperature distribution of the fiber, shown in Fig. 17. It is clear that the first higher-order local mode is strongly excited in the forward pumped case, but also that the signal gradually returns to the fundamental local mode beyond 1 m, which coincides with the decrease in signal power and temperature. Recalling that we have employed a transparent boundary condition in our algorithm and considering the definition of the local mode content given in Eq. (26), it is clear that the reason for the loss of signal power is due to the fact that the higher-order local mode evolves from a guided mode into a radiation mode as the radial temperature gradient decreases. The higher-order local mode content of the signal is thus lost beyond 1 m. Another source of the loss of signal power beyond 1 m is of course spontaneous emission from the excited state of the Ytterbium ions. However, this loss is quite small due to the relatively long lifetime of the excited state. We have verified that the signal power loss is dominated by the evolution of the higher-order local modes from guided to non-guided by simulating Fiber D without the thermal effect, which shows negligible signal power loss.

7. 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**, 94–96 (2010). [CrossRef] [PubMed]

9. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express **19**, 13218–13224 (2011). [CrossRef] [PubMed]

**19**, 3258–3271 (2011). [CrossRef] [PubMed]

**19**, 10180–10192 (2011). [CrossRef] [PubMed]

## 4. Conclusion

18. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express **15**, 15402–15409 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express |

2. | D. Gapontsev, “6 kW CW single mode Ytterbium fiber laser in all-fiber format,” in Proc. Solid State and Diode Laser Technology Review (2008). |

3. | J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb |

4. | J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express |

5. | S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express |

6. | J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. |

7. | 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. |

8. | F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. |

9. | T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express |

10. | C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express |

11. | A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express |

12. | M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. |

13. | R. W. Boyd, |

14. | P. W. Milonni and J. H. Eberly, |

15. | D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. |

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

17. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

18. | S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express |

**OCIS Codes**

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

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(350.6830) Other areas of optics : Thermal lensing

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 5, 2011

Revised Manuscript: September 19, 2011

Manuscript Accepted: October 13, 2011

Published: November 10, 2011

**Citation**

Kristian Rymann Hansen, Thomas Tanggaard Alkeskjold, Jes Broeng, and Jesper Lægsgaard, "Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers," Opt. Express **19**, 23965-23980 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-23965

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### References

- Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express12, 6088–6092 (2004). [CrossRef] [PubMed]
- D. Gapontsev, “6 kW CW single mode Ytterbium fiber laser in all-fiber format,” in Proc. Solid State and Diode Laser Technology Review (2008).
- J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt.55, 459–471 (2008). [CrossRef]
- J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express11, 2982–2990 (2003). [CrossRef] [PubMed]
- S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express14, 6091–6097 (2006). [CrossRef] [PubMed]
- J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron.15, 159–169 (2009). [CrossRef]
- 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, 94–96 (2010). [CrossRef] [PubMed]
- F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett.36, 689–691 (2011). [CrossRef] [PubMed]
- T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19, 13218–13224 (2011). [CrossRef] [PubMed]
- C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express19, 3258–3271 (2011). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19, 10180–10192 (2011). [CrossRef] [PubMed]
- M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett.29, 1739–1741 (2004). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics, 3rd. ed. (Elsevier2008).
- P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons1988).
- D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev.136, A954–A957 (1964). [CrossRef]
- G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett.16, 624–626 (1991). [CrossRef] [PubMed]
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).
- S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express15, 15402–15409 (2007). [CrossRef] [PubMed]

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