## Stimulated thermal Rayleigh scattering in optical fibers |

Optics Express, Vol. 21, Issue 3, pp. 2642-2656 (2013)

http://dx.doi.org/10.1364/OE.21.002642

Acrobat PDF (1499 KB)

### Abstract

Recently, mode instability was observed in optical fiber lasers at high powers, severely limiting power scaling for single-mode outputs. Some progress has been made towards understanding the underlying physics. A thorough understanding of the effect is critical for continued progress of this very important technology area. Mode instability in optical fibers is, in fact, a manifestation of stimulated thermal Rayleigh scattering. In this work, a quasi-closed-form solution for the nonlinear coupling coefficient is found for stimulated thermal Rayleigh scattering in optical fibers. The results help to significantly improve understanding of mode instability.

© 2013 OSA

## 1. Introduction

1. 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]

3. 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 laser amplifiers,” Opt. Express **19**(14), 13218–13224 (2011). [CrossRef] [PubMed]

3. 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 laser amplifiers,” Opt. Express **19**(14), 13218–13224 (2011). [CrossRef] [PubMed]

4. F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. **36**(23), 4572–4574 (2011). [CrossRef] [PubMed]

5. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express **20**(1), 440–451 (2012). [CrossRef] [PubMed]

10. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

5. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express **20**(1), 440–451 (2012). [CrossRef] [PubMed]

7. 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]

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

9. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

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

9. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

10. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

## 2. Some historic background

11. C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. **18**(4), 107–109 (1967). [CrossRef]

21. H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B **3**(2), 253–273 (1986). [CrossRef]

11. C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. **18**(4), 107–109 (1967). [CrossRef]

12. R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. **19**(15), 824–828 (1967). [CrossRef]

13. D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. **19**(15), 828–830 (1967). [CrossRef]

14. I. L. Fabelinskii and V. S. Starunov, “Some studies of the spectra of thermal and stimulated molecular scattering of light,” Appl. Opt. **6**(11), 1793–1804 (1967). [CrossRef] [PubMed]

18. L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am. **63**(1), 13–16 (1973). [CrossRef]

19. R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A **27**(4), 1968–1976 (1983). [CrossRef]

21. H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B **3**(2), 253–273 (1986). [CrossRef]

_{11}modes for simplicity. The basic model can be modified for other fibers. It can also handle other higher order modes. Threshold condition is developed for amplifiers with constant gain along its length in this work. This is again for simplicity only. It can be developed for other amplifier designs.

## 3. Fields in optical fibers

_{mn}, where m and n is azimuthal and radial mode numbers respectively. The electric field of LP

_{mn}mode can be written as:

_{m}represents Bessel functions of the first kind and K

_{m}represents modified Bessel function of the second kind. U and W are defined as in [23] and determined by the eigenvalue equation. N

_{mn}is normalization factor.

_{mn}is optical power in LP

_{mn}mode. It is now clear that the normalization used previously enables the possibility of writing field with only modal power in the amplitude. Using Eq. (6), intensity in the optical fiber can be found. Ignoring interfering terms which do not include fundamental modes,

_{mn}-β

_{01}) and angular frequency ± (ω

_{mn}-ω

_{01}).

## 4. Steady-state solution for traveling temperature waves

^{i(qz-Ωt)}, it becomes

_{s}and λ

_{p}are signal and pump wavelength respectively. Note that q and Ω are dependent on mode numbers m and n (not expressed explicitly for clarity). Note also the ¼ reduction comparing to Eq. (7) for converting the two cosines to exponential. Also, ρ is density; C is specific heat; and κ is thermal conductivity. UsingEquation (10) can be transformed toFirstly, solutions to the non-driven equation will be sought,where q

*is the eigenvalue and*

_{ml}*l*is the spatial temperature mode number of the heat transportation equation. The solution is Bessel function of the first kind.

## 5. Coupled nonlinear equations

_{T}= dn/dT is thermal optics coefficient, i.e. index change per K. Change in permittivity can be written as,

_{mn}is introduced to account for the often higher loss of the higher order mode. The gains of the two modes g

_{01}and g

_{mn}(m>0) are given as, The nonlinear coupling coefficient can then be obtained.where k is vacuum wave vector and d is the radius of the active region. The damping factor is given by,

*Γ*.

_{ml}*Γ*. It is easy to see that the there is no coupling at Ω = 0 rad/s.

_{ml}/2## 6. Characteristics of the nonlinear gain coefficient

24. M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fibers,” J. Lightwave Technol. **16**(6), 1013–1023 (1998). [CrossRef]

**= 1.2 × 10**

_{T}^{−5}K

^{−1}was used in [5

5. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express **20**(1), 440–451 (2012). [CrossRef] [PubMed]

10. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

**37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

_{T}. The bench marking results are summarized in Table 2 . It can be seen that the error is within 3% for all the cases if K

**= 1.2 × 10**

_{T}^{−5}K

^{−1}was used.

_{01}. A fiber amplifier is usually designed for a target fundamental mode gain. Varying some fiber parameters such doping radius d can impact target gain for the fundamental mode. This definition of χ, unlike that in [10

**37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

_{11}is studied thereafter in this paper. Other higher order modes are expected to have lower nonlinear gain and therefore higher STRS thresholds. The solutions for the spatial temperature modes are found first. The nonlinear coupling coefficients amplitude χ

*defined in Eq. (28) and damping factor Γ*

_{mnl}_{m}

*at each*

_{l}*l*are plotted in Fig. 1(a) . The largest χ

*is achieved at the mode number*

_{mnl}*l*where the deposited heat overlaps best with the spatial temperature mode. The damping factor Γ

_{m}

*increases with mode number*

_{l}*l*. This is due to a reduced physical dimension of the spatial temperature mode features at large

*l*. The simulated χ is shown in Fig. 1(b). The real part of χ is responsible for the nonlinear coupling. It becomes 0 at Ω/2π = 0 kHz and reaches a maximum at Ω/2π = ~3.4kHz. It decays slowly towards higher frequency as a Lorentzian function. The positive sign of real part of χ at Ω/2π<0kHz indicates gain for LP

_{11}mode at the Stoke frequency. The negative sign of real part of χ at Ω/2π>0kHz indicates loss for LP

_{11}mode at anti-Stoke frequency. This is opposite to what observed in [22] for STRS in liquid. This is largely due to the fact that index change in liquid and gas is dominated by thermally-induced expansion with a negative n

_{2}, while, in glass, index change is dominated by thermal optic effect with a possible n

_{2}.

_{01}and LP

_{11}mode at the maximum nonlinear coupling is Ω/2π = ~3.4kHz, which is much smaller than spectral width of almost all seed lasers. This implied LP

_{11}mode can be seeded by a seed laser at the required Stoke frequency.

_{mn}

*are plotted in Fig. 2(a) , while real part of χ and its phase is plotted in Fig. 2(b). The*

_{l}*l*for peak χ

*moves to larger number for smaller cores to account for the smaller active area. The damping factors do not change at all with a change of core diameters. The data for various core diameters essentially overlap in Fig. 2(a). This is due to the fact that the period of traveling wave is significantly larger than fiber diameter 2b. The second term in the bracket in Eq. (29), therefore, dominates, which only depends on m,*

_{mnl}*l*and b. The peak of the real part of χ moves towards larger frequency separation for smaller core diameter, accompanied by a broadening of the spectrum. This move of peak towards higher frequency is a reflection of movement towards larger temperature mode number

*l*with higher damping factor shown in Fig. 2(b). It will be shown later on that the smaller nonlinear coupling coefficient at the peak for smaller core diameter is largely a reflection of smaller V value of these fibers.

_{max}. When V is reduced from 5.5, the peak nonlinear coupling coefficient decreases initially very slowly and this decrease then accelerates near LP

_{11}mode cut-off at around 2.405. The absolute value of the peak frequency f

_{max}decreases at smaller V, reflecting the increasing delocalization of LP

_{11}mode while moving towards its cut_off. A reduction of the doped area also reduces the peak nonlinear coupling coefficient (see Fig. 3(b)). This reduction is, however, small when d/a is near 1. Reducing the nonlinear coefficient by 50% requires d/a≈0.45, i.e. a doped area reduction by ~80%. This would require a significant increase of doping level to maintain the same level of gain/absorption per unit length. Considering doping levels are already near their upper limits in many fibers, this may not be possible. Absolute value of the peak frequency f

_{max}increases with a reduction of d/a, reflecting the smaller active area.

_{max}is also studied for various core diameters while V is kept constant for step index fibers with 2b = 400μm and 2a = 2d. NA is varied to keep V constant. The results are shown in Fig. 4 . Absolute value of the peak frequency f

_{max}increases with a reduction in core diameter as expected. The nonlinear coupling coefficient remains almost constant at various core diameters. The overlap integrals in Eq. (28) are essentially dependent only on V. As core diameter gets smaller, peak mode number

*l*moves to higher mode number with a larger Γ

*(see Fig. 2(a)), accompanied by a broadening of the distribution (see Fig. 2(a)). These effects cancel each other out overall to keep χ at the peak frequency f*

_{ml}_{max}largely constant.

## 7. STRS power threshold

_{11}

^{N}(z)<<P

_{01}

^{N}(z) and P

_{01}

^{N}(z)≈P

_{01}(0), assuming uniform gain along the fiber and threshold is reached at P

_{mn}(L) =

*x*P

_{01}(L) where

*x*<<1.The threshold power can then be obtained,For typical LMA fibers, the second term in the bracket is very small, Eq. (33) is reduced to,

_{mn}and weakly on input condition P

_{01}(0)/P

_{mn}(0) once

*x*is known in this high gain regime and independent of any other amplifier parameters. In the low gain regime, the assumption P

_{01}

^{N}(z)≈P

_{01}(0) is no longer true. The lower P

_{01}

^{N}(z) leads to a higher threshold power. It can be seen in Eq. (33) that smaller higher-order-mode gain and larger higher-order-mode loss can also increase the threshold power. This is, however, very limited, due to the fact that the first term in the bracket in Eq. (33) dominates in most cases.

_{11}= 0. Both g

_{01}and g

_{11}are considered with α

_{11}= 0.The results versus total gain factor g

_{01}L (plotted in dBs) are summarized in Fig. 5 . The predicated threshold powers from Eq. (34) are plotted as solid red lines in Fig. 5(a). The threshold power are independent of gain when g

_{01}L>4, i.e. ~17dB. Below this, threshold power increases with a reduction in gain. For g

_{01}L>4, Eq. (34) fits the numerical data very well. Equation (34) can be modified slightly to account for the threshold increases at lower gains.

_{11}(0)/P

_{01}(0) as expected from Eq. (35). The only two amplifier parameters which can be used to increase the threshold powers are lower gain and lower P

_{11}(0)/P

_{01}(0).

^{−28}W. In practice, the seed to the higher order mode are likely from the input signal when the input signal spectrum is broader than just few kHz for core diameter over ~30μm, and, can be, therefore, much higher than this quantum limit.

## 8. Mode coupling dynamics

_{11}= 0. It can be clearly seen in Fig. 6(a) that the LP

_{01}mode is amplified normally at the first part of the fiber, while the LP

_{11}mode experiences significant nonlinear gain of >250dB in this case. The fraction of LP

_{11}mode power over the total power is also plotted in Fig. 6(a), showing the rapid switch over at threshold. Once over the STRS threshold, the power continues to couple from LP

_{01}mode to LP

_{11}mode, because the sign of nonlinear coupling coefficient remains the same in Eq. (30). This coupling coefficient diminishes as power in LP

_{01}mode decreases. The LP

_{11}mode undergoes continued linear amplification in the second part of the fiber amplifier. In practice, it is expected that LP

_{01}mode can be re-seeded when its power is below the noise level in the fiber. This can be simulated by switching the sign of the nonlinear coupling coefficient in Eq. (30). This is done in Fig. 6(b), showing the reversing of coupling from LP

_{11}to LP

_{01}after the first coupling cycle. This behavior is repeated with an increasing spatial frequency, driven by the increased total power. This is confirmed to some extent by the observed chaotic behavior when operating well above the threshold power [9

9. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

## 9. Spectral considerations

_{01}and ω

_{mn}. In practice, seed lasers can have much broader spectrum than that of the STRS gain. In case where the input power spectrum is broader than that of the STRS gain spectrum, if the power in the higher order mode at a given frequency is seeded by the input signal at frequency very close by (less than few kHz frequency separation for core diameter over 30μm), it reasonable to assume that phases of the fields in LP

_{01}and LP

_{11}modes with a small frequency separation of Ω are identical at the amplifier input. This may in fact be true across the entire input signal spectrum even for the case where the seed is an ASE source as in [9

**20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

_{mn}mode at an adjacent frequency Ω away to produce an intensity traveling wave described by exp(i(qz-Ωt)). In another word, power at any frequency within the input power spectrum can interfere with its corresponding power in the LP

_{mn}mode to add to the intensity of the traveling temperature wave described by exp(i(qz-Ωt)). This is only true when the phase difference of the two fields in the interfering modes is constant across the power spectrum. This is not the case, for example, for SBS, where the counter-propagating wave is seeded from quantum noise without any fixed phase relationship to the input signal. This collaborative effect can lead to the possibility that total power of the input signal contributes towards nonlinear coupling at any local frequency within the input signal spectrum in STRS, despite the fact that the power spectrum of the signal is much larger than the STRS gain spectrum. This effect can lead to threshold being independent of input signal bandwidth but more dependent on the total power of the input signal as experimentally observed in [9

**20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

_{mn}mode is seeded by the signal power in LP01 mode), the integral in Eq. (36) is, in fact, square root of product of total powers in the two modes. It is easy to see in Eq. (36) that the total powers in the two modes are contributing towards nonlinear coupling at any frequency within the power density spectrum. This can also lead to uniform nonlinear gain across the power spectrum for the LP

_{mn}mode.

## 10. Discussions and conclusions

## Acknowledgments

## References and links

1. | 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. |

2. | 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. |

3. | 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 laser amplifiers,” Opt. Express |

4. | F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. |

5. | C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express |

6. | 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 |

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

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

9. | B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express |

10. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. |

11. | C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. |

12. | R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. |

13. | D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. |

14. | I. L. Fabelinskii and V. S. Starunov, “Some studies of the spectra of thermal and stimulated molecular scattering of light,” Appl. Opt. |

15. | C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. |

16. | W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. |

17. | N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A |

18. | L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am. |

19. | R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A |

20. | H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron. |

21. | H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B |

22. | R. W. Boyd, “Nonlinear Optics,” third edition, Elsevier, 2008. |

23. | A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” Chapman and Hall, 1983. |

24. | M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fibers,” J. Lightwave Technol. |

25. | C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express |

26. | A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” 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

(060.3510) Fiber optics and optical communications : Lasers, fiber

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 29, 2012

Revised Manuscript: December 27, 2012

Manuscript Accepted: January 17, 2013

Published: January 28, 2013

**Citation**

Liang Dong, "Stimulated thermal Rayleigh scattering in optical fibers," Opt. Express **21**, 2642-2656 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2642

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### References

- 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]
- 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(5), 689–691 (2011). [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 laser amplifiers,” Opt. Express19(14), 13218–13224 (2011). [CrossRef] [PubMed]
- F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett.36(23), 4572–4574 (2011). [CrossRef] [PubMed]
- C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20(1), 440–451 (2012). [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(4), 3258–3271 (2011). [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]
- A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011). [CrossRef] [PubMed]
- B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012). [CrossRef] [PubMed]
- K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett.37(12), 2382–2384 (2012). [CrossRef] [PubMed]
- C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett.18(4), 107–109 (1967). [CrossRef]
- R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett.19(15), 824–828 (1967). [CrossRef]
- D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett.19(15), 828–830 (1967). [CrossRef]
- I. L. Fabelinskii and V. S. Starunov, “Some studies of the spectra of thermal and stimulated molecular scattering of light,” Appl. Opt.6(11), 1793–1804 (1967). [CrossRef] [PubMed]
- C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev.175(1), 271–274 (1968). [CrossRef]
- W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett.22(18), 915–918 (1969). [CrossRef]
- N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A3(1), 404–412 (1971). [CrossRef]
- L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am.63(1), 13–16 (1973). [CrossRef]
- R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A27(4), 1968–1976 (1983). [CrossRef]
- H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron.22(4), 552–562 (1986). [CrossRef]
- H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B3(2), 253–273 (1986). [CrossRef]
- R. W. Boyd, “Nonlinear Optics,” third edition, Elsevier, 2008.
- A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” Chapman and Hall, 1983.
- M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fibers,” J. Lightwave Technol.16(6), 1013–1023 (1998). [CrossRef]
- C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express20(12), 12912–12925 (2012). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express20(22), 24545–24558 (2012). [CrossRef] [PubMed]

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