Open Access
Add to BibSonomy
Abstract
A phase shift between the modal interference pattern and the thermally-induced refractive index grating is most likely the ultimate trigger for the damaging effect of transverse mode instabilities (TMI) in high-power fiber laser systems. By using comprehensive simulations, the creation and evolution of a thermally-induced phase shift is explained and illustrated in detail. It is shown that such a phase shift can be induced by a variation of the pump power. The gained knowledge about the generation and evolution of the phase shift will allow for the development of new mitigation strategies for TMI.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-power fiber laser systems are known to deliver high average output powers with nearly diffraction-limited beam quality [1]. However, the good reputation that these systems have gained over many years is currently at stake due to the detrimental effect of transverse mode instabilities (TMI) [2,3]. The phenomenon of TMI is observed once that a certain average power threshold is exceeded and manifests itself in a strong and dynamic energy transfer between different transverse modes. Thus, TMI are currently preventing the use of fiber lasers in many applications and they represent the main limitation for the further power scaling of fiber lasers and amplifiers with nearly diffraction-limited beam quality.
For this reason, intense research has been conducted in the past years to understand the physics behind TMI [4–13]. In this context it was found that the origin of TMI is a thermally-induced refractive index grating (RIG), which is written by the interference of two or more transverse fiber modes [4]. Additionally, it was stated that a phase shift between the modal interference pattern (MIP) and the RIG must exist to enable energy coupling between the modes [5,14]. Recently, it was shown that, by modulating the pump power of a fiber amplifier with an appropriate frequency and amplitude, a washing out of the RIG along the fiber can be achieved [15]. This resulted in a decrease of the energy coupling and, thus, it led to an increase of the TMI threshold. Systematic investigations with a high-speed camera have revealed that a change in the pump power of the system can trigger energy transfer between different transverse modes [16]. Thereby a decrease in pump power resulted in energy transfer from the fundamental mode into higher-order modes, whereas a pump power increase led to an energy flow from the higher-order modes into the fundamental mode. Interestingly, according to theory, the energy transfer induced with the pump modulation must be accompanied by a phase shift between the MIP and the RIG. Furthermore, such an energy transfer can already take place below the TMI threshold, demonstrating that the RIG is strong enough to allow for transverse mode coupling. At higher average powers the RIG is much stronger and, as a consequence, it becomes more and more sensitive to even small phase shifts, which can eventually suffice to trigger TMI.
As mentioned above, the modal coupling induced with the pump modulation implies that there is a phase shift between the MIP and the RIG. That such a phase shift can occur due to a pump power change was briefly mentioned in [6] and [12], but a detailed explanation and discussion about this phenomenon has not been given yet. The crux of the problem is that, because both the change of the MIP and the one of the RIG are thermally driven processes and, thus, their response times are similar, it is not trivial to understand how a phase shift between the two patterns can arise due to a pump power variation. Crucially, since this phase shift is most likely the ultimate trigger for TMI, the understanding of its evolution is essential to develop effective mitigation strategies. Thus, the aim of this work is to explain and illustrate the creation and evolution of this phase shift in detail by using comprehensive simulations. Furthermore, with this knowledge new ways to improve existing mitigation strategies are discussed and new approaches to suppress TMI are proposed.
2. Theoretical model and experimental observations
As already mentioned, experiments have shown that a modulation of the pump power of a fiber amplifier can induce an energy transfer between different transverse modes [16]. This, according to theory, must be accompanied by a phase shift between the MIP and the RIG [5]. In order to study the evolution of this phase shift, we have used the theoretical model presented in [15], in which we solve the full 3D-resolved rate-equations. In this model, the temporal response of the 3D inversion pattern and the 3D temperature profile to changes in the system, e.g. a pump power variation, is calculated using semi-analytic formulas, which provide the temporal evolution of the resulting MIP and RIG along the fiber. To validate our theoretical model, we have compared the energy transfer measured in the experiments to the one calculated by our model. In both cases the ~1 m long Yb-doped fiber amplifier with a mode field diameter of ~65 µm was operated at an average output power of 150 W. The pump was modulated with a frequency of 50 Hz and a modulation depth of ± 50 W. In this situation the output power stays below the TMI threshold at all times. The TMI threshold was measured experimentally, according to [10], to be 233 W. Both systems were seeded with 5 W at 1030 nm and counter-pumped at 976 nm. In the simulations we took photodarkening into account and assumed a higher-order mode content of 0.3% at the fiber input. We have simulated a temporal window of 40 ms (which corresponds to two modulation periods) with a temporal resolution of ~63 µs. This value is close to the temporal resolution of the high-speed camera (100 µs), which recorded the output beam in the experiments. The recorded images were analyzed with a mode reconstruction algorithm that finally gives the time-resolved relative modal content. Details about the mode reconstruction algorithm can be found in [16].
Figure 1 illustrates the results of the experiments [1(a) and 1(b)] and the ones of the simulations [1(c) and 1(d)]. The upper plots represent the temporal evolution of the signal output power normalized to the average signal power and the bottom plots represent the temporal evolution of the relative modal content of the fundamental mode (FM - blue) and the higher-order modes (HOMs - red). As can be seen by comparing Fig. 1(b) and 1(d), the simulations based on our theoretical model are able to reproduce the modal energy transfer observed in the experiments. Even the onset time of the energy transfer and its specific features are qualitatively well replicated. A hundred percent reproduction was not possible because of the different fiber types used in the simulations (step index fiber - SIF) and in the experiments (large pitch fiber - LPF). However, this experimental validation of the model shows that it is able to describe the dynamic processes of TMI in detail and, hence, it can be used to study the evolution of the phase shift between the MIP and the RIG, which is presented in the next section.
Fig. 1 Energy transfer induced by a pump power modulation - comparison between experiment and simulations (average signal power 150 W, modulation frequency 50 Hz, modulation depth ± 50 W). a) Experiment: temporal evolution of the normalized signal output power. b) Experiment: temporal evolution of the relative modal content (FM - blue, HOMs - red). c) Simulations: temporal evolution of the normalized signal output power. d) Simulations: temporal evolution of the relative modal content (FM - blue, HOMs - red).
3. Simulations
To simplify the analysis and to clearly illustrate the creation and evolution of the phase shift between the MIP and the RIG, the modal energy coupling is switched off in the simulations. On the one hand this ensures that the MIP has distinct maxima and minima at all times, which enables the continuous tracking of the MIP maxima and, thus, the calculation of the phase shift. On the other hand this also reduces the computation time. However, we have confirmed that the simulations with energy coupling deliver qualitatively similar results to the ones obtained without energy coupling. The evolution of the phase shift is simulated in an 80 µm core-diameter fiber (MFD ~65 µm) with a length of 1 m, 228 µm pump-cladding diameter, 1.2 mm outer fiber diameter, a V-parameter of 7 and doped with 3.25∙1025 Yb-ions/m3. The fiber was pumped at 976 nm in the counter-propagating direction and seeded at 1030 nm with 5.5 W of average power (10% higher-order mode content: LP11 mode). The spatial resolution of the simulations is 1.25 µm transversally and 300 µm in the longitudinal direction.
3.1 Instantaneous pump power jump
To get a clear and simple illustration of the creation and evolution of the phase shift, an instantaneous pump power jump from 100 W to 200 W was applied to the system, keeping the pump power constant at 200 W afterwards. Thus, the simulated fiber amplifier was operated below the TMI threshold at all times. We have simulated a temporal window of 2 ms with a resolution of 5 µs to observe the temporal evolution of the thermal process as well as changes in the MIP and in the RIG along the fiber. This temporal resolution ensures that the change of the system between two temporal steps is negligible. Figure 2 shows the changes of the MIP and the RIG at the end of the fiber (MIP - top plots, RIG - bottom plots) from an instant in time directly before the pump power jump (t = 0 µs, Fig. 2(a) and 2(b)) compared to an instant 250 µs after it [Fig. 2(c) and 2(d)]. The RIG in these plots [Fig. 2(b) and Fig. 2(d)] is the radially anti-symmetric part of the refractive index profile. We used this part of the refractive index profile, since it is the one that leads to TMI involving the fundamental mode and the LP11 mode. In order to obtain the RIG, we have subtracted the radially symmetric part from the whole refractive index profile. Hence, a value of zero in the refractive index change plots corresponds to the value of the radially symmetric part of the refractive index profile at this position. The temporal evolution of the process is illustrated in Visualization 1. As can be seen from Fig. 2(a) and 2(b), directly before the pump power jump (t = 0 µs) the maxima and minima of the MIP and the RIG are aligned with each other along the fiber, which means that both patterns are in phase. Thus, according to theory, no energy transfer can take place in this situation [5]. We have marked one maximum of the MIP with a white line and the corresponding RIG maximum with a black line both in Fig. 2 and in Visualization 1 to illustrate the evolution of the phase shift, i.e. the difference in the position of both maxima.
Fig. 2 Phase-shift evolution caused by a pump power jump from 100 W to 200 W. The last 50 mm of the fiber are depicted. a) Modal interference pattern right before the pump power jump (t = 0 µs) with the white line indicating the position of the intensity maximum of the MIP that will be tracked. b) Refractive index grating right before the pump power jump (t = 0 µs) with the black line indicating the position of the maximum of the RIG that is aligned with the tracked MIP maximum. c) MIP after the pump power jump (t = 250 µs) with the white line indicating the new position of the tracked MIP maximum. d) RIG after the pump power jump (t = 250 µs) with the black line indicating the new position of the tracked RIG maximum.
After the pump power jump, Fig. 2(c) and Visualization 1 show that the MIP is strongly compressed, which is a result of the increased signal output power, the increased heat deposition and the modified transverse temperature profile at each position along the fiber. The latter modifies the transverse refractive index profile via the thermo-optic effect and, thus, it changes the local guiding properties of the fiber. These modifications of the refractive index profile lead to an increased difference between the effective refractive indices of the modes. In turn, this index difference determines the local beat length Lb of the MIP according to:
In Eq. (1) λ is the wavelength, neff,FM and neff,HOM represent the effective refractive indices of the fundamental mode (LP01) and one higher-order mode (usually the LP11), respectively and z is the position along the fiber. The temporal evolution of the local beat length at the position of the marked MIP maximum (i.e. towards the end of the fiber) is illustrated in Fig. 3. The graph shows that the local beat length undergoes a fast compression in the first few hundreds of microseconds after the pump power jump and, then, it slows down approaching its new final steady state value. On top of this local beat length change of the MIP, there is an accumulation of this effect over the whole fiber length, which enhances the shift of the last MIP maximum at the end of the fiber. This means that all beat-length changes happening in the fiber before a particular point (e.g. the tracked MIP maximum) add up and, thus, the MIP maximum at the considered position is shifted significantly more than the change of its local beat length would imply. Consequently, the higher the number of MIP periods in a fiber, the stronger this effect will be. Hence, as can be seen by comparing Fig. 2(a) and Fig. 2(c) the marked MIP maximum has undergone a position shift of 8 mm after 250 µs, which is much higher than the local beat-length change of this MIP period of 0.8 mm (indicated by the dashed black line in Fig. 3).
Fig. 3 Temporal evolution of the local beat length of a modal-interference-pattern period at the end of the fiber (at the position of the marked MIP maximum) after a pump power jump from 100 W to 200 W.
Furthermore, it can be inferred from Fig. 2(c) and 2(d) that the marked maxima of the MIP and the RIG have significantly separated after 250 µs. Thus, a phase shift between the MIP and the RIG has occurred. This phase shift is strongest at the end of the fiber due to the counter-pumping configuration of our simulated amplifier. Figure 4 illustrates the temporal evolution of this phase shift. In particular, Fig. 4(a) depicts the shift in position of the marked maxima of the MIP (blue) and of the RIG (red) and Fig. 4(b) shows the phase shift between both, e.g. the difference of the two positions normalized to the local beat length and expressed in rad.
Fig. 4 Phase-shift evolution between the modal interference pattern and the refractive index grating at the end of the fiber after a pump power jump from 100 W to 200 W. a) Temporal evolution of the shifts in position of the marked MIP maximum (blue) and of the corresponding RIG maximum (red). b) Temporal evolution of the phase shift, e.g. the relative difference in position normalized to the local beat length, between these two maxima.
In Fig. 4(a) the marked MIP maximum has strongly shifted towards the seed side of the fiber after only a few hundreds of microseconds, whereas the marked RIG maximum follows with a certain delay. This leads to a difference in position of the two maxima and, thus, to a phase shift. As can be inferred from Fig. 4(b), this phase shift reaches its maximum at 250 µs. At this point of time the MIP maximum and the RIG maximum have separated by 4.43 mm, which corresponds to a phase shift of 1.17 rad at a local MIP beat length of 23.8 mm. This phase shift is rather high, taking into account that the strongest energy coupling between different transverse modes in transmission gratings occurs for a phase shift of π/2 = 1.57 [17]. Afterwards, the RIG catches up with the MIP and the phase shift decreases continuously approaching zero. Thus, in the equilibrium state the MIP and the RIG will be in phase again.
As already mentioned above, one reason for the occurrence of this phase shift is the strong and fast shift of the MIP maxima at the end of the fiber, which is mainly caused by the accumulation of the beat-length changes of all MIP periods before. Therefore, the further downstream the fiber a MIP maximum is, the stronger and faster its position shift will be. This process is illustrated by the blue dots in Fig. 5, where the shift in position of each MIP maximum along the fiber after 250 µs is displayed. It can be seen that the MIP maxima at the seed side of the fiber (low numbers) undergo almost no shift in position, whereas the shift increases nonlinearly towards the pump side of the fiber (MIP maxima with high numbers). The red dots in Fig. 5 illustrate the phase shift that has developed for each MIP maximum after 250 µs compared to the corresponding RIG maxima. It can be seen that this phase shift also increases nonlinearly towards the pump side of the fiber. This demonstrates that the accumulation of the beat-length changes of all MIP periods along the fiber has a significant influence on the phase shift at the end of the fiber, which finally determines a potential energy transfer.
Fig. 5 Accumulation of the shift in position of the modal-interference-pattern maxima along the fiber (blue dots), 250 µs after the pump power jump from 100 W to 200 W. Low numbers correspond to MIP maxima at the seed side of the fiber and high numbers to the ones at the pump side of the fiber. The corresponding phase shift (normalized to the local beat length) is illustrated by the red dots.
The second key point to explain the occurrence of a phase shift between the MIP and the RIG arises from the thermal diffusivity of the fiber material (silica in this case). A shift in position of the MIP leads, in turn, to a shift of the heat source that ultimately gives rise to the RIG. Consequently, the temperature at the former position of the MIP starts to diffuse and, thus, the RIG at this position starts to decay (we will refer to this as old RIG in the following). At the same time a new RIG starts to arise in phase with (i.e. at the new position of) the MIP. However, our simulations demonstrate that the old RIG is still strong even when the MIP has been compressed significantly.
Figure 6 displays the temporal change of the radially anti-symmetric part of the refractive index profile over time at the initial position of the marked RIG maximum (see black dot in Fig. 2(b)). Thus, Fig. 6 can be used to track the decay of the old RIG. As already described above, a value of zero corresponds to the value of the radially symmetric part of the refractive index profile at this position.
Fig. 6 Temporal evolution of the refractive index grating at the initial position of the marked RIG maximum (see black dot in Fig. 2(b)).
It can be seen that in the first 80 µs after the pump power jump, the refractive index at the initial position of the marked RIG maximum increases due to the increased heat and temperature deposition in the fiber and, in turn, the old RIG first gets stronger. However, after this, the compression of the MIP and, thus, the shift of the heat source plays the major role for the evolution of the old RIG. Consequently, the temperature at the initial position of the marked RIG maximum diffuses and the refractive index decreases. However, the old RIG is still strong when the MIP maximum has shifted by 8 mm after 250 µs (i.e. when the maximum phase shift is reached in Fig. 4(b)). It can be inferred from Fig. 6 that, at this point in time, the old RIG has still about 60% of its initial strength (see dashed line). According to the superposition theory of gratings, to define an effective RIG, this old RIG can be superposed with the new RIGs arising along the fiber as the MIP shifts. This effective RIG is shifted with respect to the new MIP, since the old RIG adds a component at the initial position of the MIP. Consequently, a phase shift between the MIP and the effective RIG occurs (plotted in Fig. 4), which can lead to energy transfer between different transverse modes, provided that the RIG is strong enough [16].
Following the temporal evolution of the graph in Fig. 6 beyond t = 250 µs, it can be seen that the strength of the RIG reaches a minimum before it starts to increase again. This behavior is a result of the next MIP maximum arriving at the initial position of the marked RIG maximum. This, in turn, leads to a new heat deposition and, thus, to an increase in temperature and refractive index. Consequently, a new RIG is inscribed at this position.
3.2 Slow pump power increase
To investigate the dependence of the phase shift on the temporal rate of change of the pump power, we have simulated a slow pump power increase from 100 W to 200 W within 10 s and with a temporal resolution of 20 ms. Once again, this temporal resolution ensures that the change of the system between two temporal steps is negligible. The results of the simulations are shown in Fig. 7. Similar to Fig. 4, Fig. 7(a) illustrates the shift in position of the marked maxima of the MIP (blue) and of the RIG (dashed red) and Fig. 7(b) depicts the relative phase shift arising between both over time. The scales of the y-axes are identical to the ones of Fig. 4 to ensure a better comparability of the results.
Fig. 7 Phase-shift evolution between the modal interference pattern and the refractive index grating at the end of the fiber after a slow pump power increase from 100 W to 200 W (within 10 s). a) Temporal evolution of the shifts in position of the marked MIP maximum (blue) and of the corresponding RIG maximum (dashed red). b) Temporal evolution of the phase shift (relative difference in position) between these two maxima.
As can be seen, in the case of a slow pump power increase, the RIG can follow the MIP changes. Thus, both profiles do not really separate and almost no phase shift develops, which intrinsically excludes any potential modal energy transfer. The remaining phase shift is related to the spatial resolution of our simulations. Consequently, our simulations have shown that the evolution of a phase shift between the MIP and the RIG depends on the temporal rate of change of the pump power. This is in good agreement with the experimental investigations presented in [16], where the dependence of the induced energy transfer on the modulation frequency of a pump modulation has been studied. It was found that, at an average output power of 150 W (below the TMI threshold) and with a modulation frequency of 30 Hz, a strong energy transfer occurs, whereas at lower frequencies the strength of the energy transfer decreases, since the phase shift between the MIP and the RIG is lower. This correlates very well with the simulations presented in this work.
However, these findings are not in contradiction with the occurrence of TMI in free-running systems (with no pump modulation). As shown in [16], at the TMI threshold the RIG is much stronger than actually necessary to start an energy coupling between different transverse modes. Hence, the RIG is highly sensitive to even slight phase shifts, which suffice to trigger modal energy transfer. In this situation the noise of the pump or the seed source would be enough to induce such a small phase shift and, therefore, to trigger TMI.
4. Discussion and Conclusions
By using comprehensive simulations, we have been able to verify that a phase shift between the MIP and the RIG in high-power fiber laser systems can be induced by a variation of the pump power. This finding is in good agreement with the experimental investigations in [16], in which energy transfer between different transverse modes below the TMI threshold was induced by modulating the pump power of a fiber amplifier. Furthermore, it supports the conclusion of [16] that small fluctuations of the pump power can suffice to trigger TMI when the RIG is strong enough.
Additionally, it was found that the accumulation of the local beat-length changes of the MIP along the whole fiber plays a major role in the evolution of this phase shift. More specifically, the higher the number of MIP periods in a fiber, the stronger the phase shift will be. Thus, it might be beneficial to develop a fiber design with a low difference between the effective refractive indices of the transverse modes. This would increase the beat length of the MIP and, thus, reduce the number of MIP periods in the fiber. Accordingly, the shift of the last MIP maximum relative to the beat length will also be significantly lower. Consequently, the fiber will become less sensitive to pump power fluctuations, which could increase the TMI threshold. This would be particularly beneficial for rod-type fibers, in which it is inherently more complicated to suppress TMI. On the contrary, it can be concluded that the TMI mitigation strategy of modulating the pump power [15] would potentially require a much lower modulation depth to wash out the RIG in long fibers compared to short ones, since there are more periods of the MIP that contribute to a shift of the MIP maxima at the end of the fiber. Furthermore, another approach to mitigate TMI based on an induced phase shift is the operation of the fiber laser system in burst mode [18]. By carefully adjusting the modulation period and the duty cycle of the seed burst it can be achieved that only a positive phase shift is induced between the MIP and the RIG. According to [5] and [16] this will result in an energy flow from the higher-order modes into the fundamental mode. In turn, this permanent beam cleaning should allow for an intra-burst average power significantly above the TMI threshold. However, a detailed investigation of the proposed mitigation strategies is beyond the scope of this article.
In summary, this work illustrates the evolution of a phase shift between the MIP and the RIG in high-power fiber laser systems and, thus, it helps in understanding the ultimate trigger of TMI and in developing effective mitigation strategies.
Funding
German Research Foundation (DFG) (IRTG 2101); European Research Council (ERC) (ACOPS, 617173), Fraunhofer and Max Planck cooperation program (PowerQuant), German Federal Ministry of Education and Research (BMBF) (PT-VDI, TEHFA II).
References and links
1. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013). [CrossRef]
2. 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 amplifiers,” Opt. Express 19(14), 13218–13224 (2011). [CrossRef] [PubMed]
4. 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(4), 3258–3271 (2011). [CrossRef] [PubMed]
5. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011). [CrossRef] [PubMed]
6. 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]
7. A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013). [CrossRef] [PubMed]
8. K. R. Hansen and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22(9), 11267–11278 (2014). [CrossRef] [PubMed]
9. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012). [CrossRef] [PubMed]
10. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012). [CrossRef] [PubMed]
11. B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21(10), 12053–12067 (2013). [CrossRef] [PubMed]
12. S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013). [CrossRef] [PubMed]
13. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013). [CrossRef] [PubMed]
14. 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). [CrossRef] [PubMed]
15. C. Jauregui, C. Stihler, A. Tünnermann, and J. Limpert, “Pump-modulation-induced beam stabilization in high-power fiber laser systems above the mode instability threshold,” Opt. Express 26(8), 10691–10704 (2018). [CrossRef] [PubMed]
16. C. Stihler, C. Jauregui, A. Tünnermann, and J. Limpert, “Modal energy transfer by thermally-induced refractive index gratings in Yb-doped fibers,” Light Sci. Appl.submitted.
17. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]
18. C. Jauregui, C. Stihler, J. Limpert, and A. Tünnermann, “Transverse mode instabilities in burst operation of high-power fiber laser systems,” Proc. SPIE 10512, 1051207 (2018).
