## Thermally induced mode distortion and its limit to power scaling of fiber lasers |

Optics Express, Vol. 21, Issue 12, pp. 14272-14281 (2013)

http://dx.doi.org/10.1364/OE.21.014272

Acrobat PDF (983 KB)

### Abstract

A general model is proposed to describe thermal-induced mode distortion in the step-index fiber (SIF) high power lasers. Two normalized parameters in the model are able to determine the mode characteristic in the heated SIFs completely. Shrinking of the mode fields and excitation of the high-order modes by the thermal-optic effect are investigated. A simplified power amplification model is used to describe the output power redistribution under various guiding modes. The results suggest that fiber with large mode area is more sensitive on the thermally induced mode distortion and hence is disadvantaged in keeping the beam quality in high power operation. The model is further applied to improve the power scaling analysis of Yb-doped fiber lasers. Here the thermal effect is considered to couple with the optical damage and the stimulated Raman scattering dynamically, whereas direct constraint from the thermal lens is relaxed. The resulting maximal output power is from 67kW to 97kW, depending on power fraction of the fundamental mode.

© 2013 OSA

## 1. Introduction

1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B **27**(11), B63–B92 (2010). [CrossRef]

2. Y. Y. Fan, B. He, J. Zhou, J. T. Zheng, H. K. Liu, Y. R. Wei, J. X. Dong, and Q. H. Lou, “Thermal effects in kilowatt all-fiber MOPA,” Opt. Express **19**(16), 15162–15172 (2011). [CrossRef] [PubMed]

4. 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**(13), 6091–6097 (2006). [CrossRef] [PubMed]

4. 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**(13), 6091–6097 (2006). [CrossRef] [PubMed]

5. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express **19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

4. 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**(13), 6091–6097 (2006). [CrossRef] [PubMed]

7. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express **11**(22), 2982–2990 (2003). [CrossRef] [PubMed]

5. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express **19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

10. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express **16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

10. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express **16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

11. J. J. Zhu, P. Zhou, Y. X. Ma, X. J. Xu, and Z. J. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express **19**(19), 18645–18654 (2011). [CrossRef] [PubMed]

10. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express **16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

11. J. J. Zhu, P. Zhou, Y. X. Ma, X. J. Xu, and Z. J. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express **19**(19), 18645–18654 (2011). [CrossRef] [PubMed]

## 2. Theoretical model

### 2.1 Normalized parameters governing mode characteristics

*ψ*(

*r*) can be described with the scalar wave equation:where

*k = 2π/λ*is the wave vector with

*λ*as the wavelength,

*ν*is the azimuthal order number,

*n*(

*r*) is the radial distribution of refractive index,

*n*is the effective refractive index of mode. Within high power fiber amplifiers,

_{eff}*n*(

*r*) deviates from the initial step function due to the thermal load induced by quantum defect in doped zone. By assuming that there is a uniform heat power density

*q*in the core of fiber,

*n*(

*r*) induced by radial temperature gradient is as following [12

12. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. **37**(2), 207–217 (2001). [CrossRef]

*n*and

_{co}*n*are the original core and cladding refractive index respectively,

_{cl}*r*and

_{co}*r*are the core and the inner cladding radius respectively,

_{cl}*dn/dT*= 11.8 × 10

^{−6}/K is the thermal-optic coefficient of silica,

*T*-

_{co}*T*

_{0}counts the temperature rising at the boundary of the core, ∆

*T*=

_{co}*qr*

_{co}^{2}/4

*κ*is the temperature difference of the fiber core with

*κ*as the thermal conductivity.

*T*/

_{co}dn*dT*always keeps small value in the fiber lasers, we may ignore its second order terms. Then by introducing variable

*x*=

*r*/

*r*, Eq. (1) can be rewritten aswherewith

_{co}*T*

_{co}-T_{0})

*dn/dT*, has been ignored. This is because its major effect is shifting the effective index of the mode and it affects the mode profile lightly.

*V*and

*Z*.

*V*is nothing but the conventional

*V*-parameter of SIF. The definition of

*Z*-parameter is analogous to

*V*whereas the

*n*part of

_{co}-n_{cl}*V*is replaced by the thermally induced refractive index difference in the core, i.e., ∆

*T*. The physical origination of

_{co}dn/dT*Z*-parameter is that the mode fields are deformed by local heat load. In other words, the thermal-optic effect in fibers is not cumulated with the propagation of laser beam. This is essentially different from the bulk SSLs.

*q*=

*ηαP*/

_{p}*πr*

_{co}^{2}, where

*η*denotes the quantum defect,

*α*is the absorption coefficient of pumping light and

*P*is the pumping power. Consequently

_{p}*Z*is proportional to

*r*(

_{co}*ηαP*)

_{p}^{1/2}. In order to maintain the same mode deformation and mode number,

*Z*must be fixed. It implies that for the same pump configuration the absorbed pump power

*αP*should be decreased as the core diameter of fiber increases. Consequently decreasing pump power or extension of fiber length is necessary in LMA fibers in order to avoid serious thermal induced mode distortion. Therefore a tradeoff design between thermal and nonlinear effects should be carefully considered in the high power LMA fiber lasers. The same conclusion can be obtained by the ABCD transformation of the Gaussian beam given in the appendix of [10

_{p}**16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

*V*-parameter.

### 2.2 Simplified power amplification model

5. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express **19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

13. A. Liu, X. Chen, M. Li, J. Wang, D. T. Walton, and L. A. Zenteno, “Comprehensive modeling of single frequency fiber amplifiers for mitigating stimulated Brillouin scattering,” J. Lightwave Tech. **27**(13), 2189–2198 (2009). [CrossRef]

**16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

*P*is launched into the active fiber, the input mode field can be expanded aswhere

_{in}*ψ*denotes all guiding modes of the heated fiber. Because slight heat can result in a sufficient refractive index difference over a large inner cladding, a large effective

_{k}*V*value shows up from the viewpoint of equivalent step-index fiber to support a lot of guiding modes [14

14. A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. **69**(12), 1663–1671 (1979). [CrossRef]

*NA*= 0.035,

*V*= 2.0) for an instance, nine guiding modes are allowed as only temperature difference ∆

*T*= 1K (i.e.

_{co}*Z*= 0.35) is applied on the core, though they spread their intensity in cladding mainly, so that ∑

*|*

_{k}*c*|

_{k}^{2}≈1 is true for most of the cases.

## 3. Mode field properties in high power SIF lasers

### 3.1 Thermal effect on guiding modes

*V*and

*Z*. In order to understand

*Z*-parameter more intuitively, we consider a

*λ*= 1064nm laser in the silica SIF as an example. Taking

*n*= 1.45,

_{co}*dn/dT*= 11.8 × 10

^{−6}/K,

*κ*= 1.38W/m/K, we have the linear relationship

*Z*≈0.0083 ×

*r*(

_{co}*ηαP*)

_{p}^{1/2}, where

*r*is in µm and the heat power per unit fiber length

_{co}*ηαP*is in W/m. Then

_{p}*Z*= 1 corresponds to a massive heat load of 145.3W/m in a fiber with the core diameter

*d*of 20µm but an acceptable heat load of 36.3W/m in a

_{co}*d*= 40µm fiber. Three guiding modes, namely LP

_{co}_{01}(fundamental mode), LP

_{11}(the first HOM) and LP

_{02}(the second HOM with

*ν*= 0), are considered here for different values of

*V*-parameter, which governs the unheated SIF to be single mode or multi-mode.

*Z*-parameter for

*V*= 2.4 (single mode) and

*V*= 3.8 (double modes). It can be seen that the mode fields shrink gradually with the increase of

*Z*. Moreover,

*Z*affects weakly on those modes supported by the original cooling fiber, but strongly inñuences the modes excited by the temperature gradient. The fields of thermally excited modes are mainly located in cladding for

*Z*below 0.4 and would rapidly shrink into core as Z increases to 1~2. A critical value of 0.33 for the confinement factor may be used to judge the excited HOM as the core modes or not [4

**14**(13), 6091–6097 (2006). [CrossRef] [PubMed]

*Z*-range when the

*V*value is large (e.g.

*V*= 6). It means that the fibers with large

*V*value are more insensitive to the thermal effect.

### 3.2 Thermally induced mode competition

_{0}

*modes are excited due to axial symmetry. The number of LP*

_{k}_{0}

*modes used in the calculation is large enough to assure that ∑*

_{k}*|*

_{k}*c*|

_{k}^{2}>0.99. The gain

*G*of the amplifier is set as 10.

_{01}and LP

_{02}at the output port of the amplifier as functions of

*Z*for several

*V*-values. It can be seen that the fibers with larger

*V*-value is more robust in keeping power in LP

_{01}content. This is because a stronger mode field confinement leads to insensitivity to the thermal effect. Therefore, in certain cases the fibers with high

*V*-value may provide better beam quality due to their weaker LP

_{01}mode deformation and fewer thermally induced HOMs. Being accompanied by the decrease of LP

_{01}content, the power of LP

_{02}mode increases gradually. It will reach a maximum of about 30% and the power transfers to LP

_{03}mode further and so on.

**19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

_{01}and LP

_{02}was found to be around 75% and 25% at

*z*= 0.5m, respectively. The

*V*-value of the fiber was 2.28, and Z≈6 is obtained according to Fig. 17 in [5

**19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

*T*≈20K). Meanwhile, our model show that

_{co}*V*= 2.3 and

*Z*= 6 leads to the LP

_{01}fraction of 75% and the LP

_{02}fraction of 21%. The similar results indicate that the overlap between the input mode and the thermally induced HOMs in the heated fibers may be an important reason of which to cause the power transferring. It may further enhance the mode instability caused by the longitudinal refractive index grating which is induced by thermal load and mode beating [9].

*Z*= 2 can be regarded as a critical point where power fraction of the LP

_{01}mode starts to decrease rapidly from almost 100%. For the fiber with a smaller core size,

*Z*= 2 corresponds to a high thermal or pumping load. For example, a 3.71kW/m of pump absorption is needed to produce

*Z*= 2 in a fiber with

*d*= 25µm,

_{co}*η*= 0.1,

*α*= 1.26dB/m. Therefore for an ideal cylindrically symmetric fiber, the input of pure fundamental mode can guarantee approximate single mode output at a high power situation because the non-symmetric mode cannot be excited. Unfortunately, in real fiber lasers, the fibers are always bent and there is always a slight mismatch at the juncture of fibers. All ingredients cause the LP

_{11}mode excitation from LP

_{01}. It can be seen from Fig. 2 in section 3.1 that the confinement factor of LP

_{11}is around 0.7 for

*Z*= 2. Therefore the LP

_{11}mode may be amplified effectively due to its complementary pattern compared to LP

_{01}[1

1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B **27**(11), B63–B92 (2010). [CrossRef]

**19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

*Z*>2 is easy to reach. For example, 1.45kW/m and 3.27kW/m of pump absorption corresponds to

*Z*= 4 and

*Z*= 6 respectively for a LMA fiber with

*d*= 80µm,

_{co}*α*= 2.2dB/m (

*η*= 0.1 again). It indicates that the LMA fibers face a serious multi-mode problem though they may take advantage of avoiding strong nonlinear effect.

## 4. Application on power scaling

*A*was always set as Γ

_{eff}^{2}

*πr*

_{co}^{2}in the original calculation. However, it is clearly that

*A*is affected by thermal load and the change is notable especially in high power amplifier. Thus these two power limits should be coupled with thermal effect. According to the assumption of the scaling model, the thermal load

_{eff}*q*can be estimated by the laser output power

*P*as

*q*=

*η*/

_{heat}*η*∙

_{laser}*P*/

*πr*

_{co}^{2}

*L*with

*L*as the fiber amplifier length. Consequently

*A*can be calculated with

_{eff}*A*=

_{eff}*f*(

*P*), where

*f*can be proved to be a monotonic decreasing function. If the power limit of a constraint condition (say optical damage and SRS here) is given by

*g*the monotonic increasing function, the inequation

*A*to be larger than half of the core area in [10

_{eff}**16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

11. J. J. Zhu, P. Zhou, Y. X. Ma, X. J. Xu, and Z. J. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express **19**(19), 18645–18654 (2011). [CrossRef] [PubMed]

*Z*is the critical value of

_{c}*Z*-parameter for a given

*V*-value and a required power content of the fundamental mode (e.g. 90%). It should be noted that the original thermal lens criterion (the last equation of Eq. (13) in Appendix A of this paper) can be obtained by setting

*Z*= 2/Γ

_{c}^{2}≈4 in Eq. (12).

*V*= 2.4, 3.8 and 6.0 together with LP

_{01}power fraction of 90%, Figs. 4(d), 4(e) and 4(f) correspond to LP

_{01}power fraction of 80%. Here we ignore the fiber bending and alignment error at the fiber junctures; hence LP

_{11}mode cannot be excited. It can be seen that the maximal power are always limited by optical damage (yellow region), SRS (orange region) and thermal effect (brown region). The intersection of three regions gives the maximal output power achieved by the minimal core diameter and the shortest fiber length. For

*V*= 2.4 and LP

_{01}power fraction of 90%, the maximal output power allowed is about 67.6kW. When the constraint of LP

_{01}content is relaxed, the thermal effect area shifts to right-hand side and the maximal power increases as shown in Fig. 4(d). For the cases of the multi-mode fibers shown in Figs. 4(b), 4(c), 4(e) and 4(f), the maximal powers are much higher compared to single mode case with the same LP

_{01}content constraint though the required core diameter is larger as well. For example, the maximal power of Fig. 4(f) is around 97kW while the minimal core diameter required exceeds 200μm (which may only be implemented in photonic crystal fibers (

*V*= 6.0)). That is because multi-mode fibers can carry higher thermal load as shown in section 3.2. However, for the lower output power requirement, single mode fibers are able to achieve it with a smaller core. For example,

*d*= 50µm is sufficient to achieve 20kW output for a fiber with

_{co}*V*= 2.4 while for the fiber with

*V*= 6,

*d*= 70µm is required. It can be understood that the weaker field confinement of single mode fibers leads to the larger effective mode area and consequently allows higher output power.

_{co}## 5. Conclusion

*V*,

*Z*, are found to be able to determine the mode properties in high power SIF lasers. We show that the

*Z*-parameter governs the waveguide capability of the radial temperature gradient, similar to the

*V*-parameter for unheated SIFs. The thermal effect on fiber guiding modes is investigated with these parameters in general. A simplified power amplification model is then set up to describe the power distribution on different modes at the output port of fiber amplifier. The results are found to be consistent with those obtained by the BPM method. It indicates that the overlap between the input mode and the HOMs of heated fiber may be a considerable factor to cause the output power transferring. The model also implies that the LMA fibers face a serious multi-mode problem though they may take advantage in avoiding strong nonlinear effect. The power scaling of YDFL is investigated with an improved model, where the optical damage and SRS limitations are coupled with thermal effect dynamically and the original thermal lens criterion is replaced with a more practical one based on the power content of fundamental mode. It predicts that the maximal output power is from 67kW to 97kW, depending on power fraction of the fundamental mode and

*V*-parameter of SIFs.

## Appendix A. Formulae and parameters of the power scaling model

**16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

**19**(19), 18645–18654 (2011). [CrossRef] [PubMed]

*L*is the fiber length,

*n*= 1.45,

_{co}*κ*= 1.38W/m/K,

*dn/dT*= 11.8 × 10

^{−6}/K,

*λ*= 1064nm. Definition of other symbols used in Eq. (13) and their values are further listed in Table 1. Here the tandem-pumping scheme is assumed. The maximal allowed power is the minimum of the above six power limits.

## Acknowledgments

## References and links

1. | D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B |

2. | Y. Y. Fan, B. He, J. Zhou, J. T. Zheng, H. K. Liu, Y. R. Wei, J. X. Dong, and Q. H. Lou, “Thermal effects in kilowatt all-fiber MOPA,” Opt. Express |

3. | M. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 71951U (2009). |

4. | 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 |

5. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express |

6. | C. Jauregui, T. Eidam, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “On the thermal origin of mode instabilities in high power fiber lasers,” Proc. SPIE |

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

8. | T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. 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 |

9. | A. V. Smith and J. J. Smith, “Thermally induced mode instability in high power fiber amplifiers,” Proc. SPIE |

10. | J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express |

11. | J. J. Zhu, P. Zhou, Y. X. Ma, X. J. Xu, and Z. J. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express |

12. | D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. |

13. | A. Liu, X. Chen, M. Li, J. Wang, D. T. Walton, and L. A. Zenteno, “Comprehensive modeling of single frequency fiber amplifiers for mitigating stimulated Brillouin scattering,” J. Lightwave Tech. |

14. | A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. |

15. | B. Morasse, S. Chatigny, C. Desrosiers, É. Gagnon, and M. Lapointe, “Simple Design for Single mode High Power CW Fiber Laser using Multimode High NA Fiber,” Proc. SPIE |

**OCIS Codes**

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

(140.6810) Lasers and laser optics : Thermal effects

(140.3615) Lasers and laser optics : Lasers, ytterbium

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 25, 2013

Revised Manuscript: May 24, 2013

Manuscript Accepted: May 29, 2013

Published: June 7, 2013

**Citation**

Wei-Wei Ke, Xiao-Jun Wang, Xian-Feng Bao, and Xiao-Jian Shu, "Thermally induced mode distortion and its limit to power scaling of fiber lasers," Opt. Express **21**, 14272-14281 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14272

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### References

- D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B27(11), B63–B92 (2010). [CrossRef]
- Y. Y. Fan, B. He, J. Zhou, J. T. Zheng, H. K. Liu, Y. R. Wei, J. X. Dong, and Q. H. Lou, “Thermal effects in kilowatt all-fiber MOPA,” Opt. Express19(16), 15162–15172 (2011). [CrossRef] [PubMed]
- M. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 71951U (2009).
- 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(13), 6091–6097 (2006). [CrossRef] [PubMed]
- K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express19(24), 23965–23980 (2011). [CrossRef] [PubMed]
- C. Jauregui, T. Eidam, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “On the thermal origin of mode instabilities in high power fiber lasers,” Proc. SPIE 8237, 82370D (2012).
- J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express11(22), 2982–2990 (2003). [CrossRef] [PubMed]
- T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. 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(14), 13218–13224 (2011). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Thermally induced mode instability in high power fiber amplifiers,” Proc. SPIE 8237, 82370B (2012).
- J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express16(17), 13240–13266 (2008). [CrossRef] [PubMed]
- J. J. Zhu, P. Zhou, Y. X. Ma, X. J. Xu, and Z. J. Liu, “Power scaling analysis of tandem-pumped Yb-doped fiber lasers and amplifiers,” Opt. Express19(19), 18645–18654 (2011). [CrossRef] [PubMed]
- D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron.37(2), 207–217 (2001). [CrossRef]
- A. Liu, X. Chen, M. Li, J. Wang, D. T. Walton, and L. A. Zenteno, “Comprehensive modeling of single frequency fiber amplifiers for mitigating stimulated Brillouin scattering,” J. Lightwave Tech.27(13), 2189–2198 (2009). [CrossRef]
- A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am.69(12), 1663–1671 (1979). [CrossRef]
- B. Morasse, S. Chatigny, C. Desrosiers, É. Gagnon, and M. Lapointe, “Simple Design for Single mode High Power CW Fiber Laser using Multimode High NA Fiber,” Proc. SPIE 7195, 719505 (2009).

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