We demonstrate an approach to actively stabilize the beam profile of a fiber amplifier above the mode instability threshold. Both the beam quality and the pointing stability are significantly increased at power levels of up to three times the mode instabilities threshold. The physical working principle is discussed at the light of the recently published theoretical explanations of mode instabilities.
© 2013 OSA
Fiber laser systems have rapidly evolved into light sources able to deliver output powers beyond the kilowatt level  and mode-field diameters larger than 100 µm to sufficiently reduce the influence of unwanted nonlinear effects . However, the conjunction of high average powers and large mode areas (which, with the current technological limitations, implicitly leads to multimode operation) has led to the observation of a new effect called mode instabilities (MI) . Herein, a stable Gaussian-like beam profile starts to fluctuate rapidly after a certain average output power threshold is reached. Besid
s the pointing instability, the beam quality is also significantly degraded due to the strong energy transfer to higher-order modes (HOM). Consequently, the onset of MI currently limits the further power scaling of fiber laser systems with diffraction-limited beam quality. Due to the far-reaching scientific and economic impact of MI, the first mention of this effect triggered the publication of several further articles dealing with its experimental and theoretical study. The result of all this research effort has been a rapidly evolving understanding of MI over the last two years. The origin of MI has been identified to be a thermally-induced index grating ultimately created by the beating of two or more guided fiber modes. It has been shown that this grating is potentially able to transfer energy between these interfering modes . However, a phase-lag between the modal interference pattern and the grating is needed for this energy transfer to actually take place [5,6]. One explanation of the origin of this phase-lag is the presence of thermally-induced non-adiabatic waveguide changes [7–9] caused by the strong transverse thermal gradients found in an active fiber under high power operation . Another explanation relays on the different guided modes having slightly different frequencies. Thus their interference will give rise to a traveling interference pattern that automatically produces a phase-lag due to the finite thermal diffusion time [6,11]. Currently, at least for very large mode area short fibers (i.e. those typically used for pulse amplification), the first explanation seems to fit the experimental observations more accurately. Moreover, the simulations carried out by Ward et al.  also support this thesis. In fact we believe that the physical mechanism used by Smith et al.  to explain the origin of the phase-lag is physically viable, but our experimental evidence suggests that it might not be the dominant one in short very large mode area fibers.
Apart from the theoretical study of MI, the first proposals to increase the MI threshold by using optimized fiber designs [13, 14] have been published. These optimized fiber designs have increased the MI threshold at least by a factor of two. Another approach to mitigate MI employed the build-up time of the thermal grating created by a pulsed pump [15, 16]. However, in this case the average output power was decreased and only the intra-pulse power could be demonstrated to be instantaneously higher than the MI threshold.
In this article we present the first proof-of-principle experiment of an active control scheme that is able to stabilize the beam profile at output power levels higher than three times the MI threshold. It is shown that both the stability and beam quality of the output beam can be significantly increased. Moreover, the experimental observations are discussed in the light of the newest theoretical explanations of mode instabilities [7, 17].
The paper is organized as follows: After a description of the experimental setup in section 2 the operating principle of the stabilization scheme is explained in section 3. The characterization of the free-running and stabilized system is presented in section 4 and 5. Furthermore the obtained results are discussed in section 6 and section 7 summaries the article.
2. Experimental setup
A sketch of the main components of the setup is depicted in Fig. 1. For our experiments we used a fiber CPA system based on that described in . The seed source delivers stretched femtosecond pulses at a center wavelength of 1040 nm and with an average output power of 10 W. A 1.2 m long rod-type large-pitch fiber (LPF) is used as the main amplifier. This fiber possesses a core diameter of 90 µm. The center wavelength of the cw-pump source is approximately 976 nm. Finally, the amplified beam is characterized by a high speed camera, a low speed camera and a photodiode (globally labeled as beam diagnostics). A detailed description of the beam diagnostic setup can be found in .
For a dynamic excitation of the fiber modes in the main amplifier the seed signal is launched through an acousto-optic deflector (AOD). Based on an acoustically-induced grating, the seed signal can be deflected under different angles. Afterwards, the resulting angular spectrum is transferred into a spatial offset of the seed by passing the beam through a focusing lens. In this way the position of the seed signal on the core of the main-amplifier fiber can be controlled electronically. Additionally, the output of the photodiode is fed into a logic which, based on the information contained in this temporal trace, makes the appropriate decision on the deflection of the input beam and acts upon the acousto-optic deflector (AOD). A detailed description of both the logic and the optimization algorithm is given in sections 5 and 6.
3. Operating principle
In the following, the main idea behind this mitigation strategy is briefly presented. For illustration purposes and for the sake of simplicity a binary periodic function (i.e. a square wave) is considered to drive the AOD and, therefore, the dynamic mode excitation. Furthermore, for simplicity, the discussion will assume the interference of just two modes: the fundamental one and the lowest order radially anti-symmetric mode (i.e. LP11-like mode). Under these conditions only two deflection positions of the seed beam on the fiber core are possible. One of them corresponds to a position in which the center of the deflected incoming beam is higher than the fiber axis (illustrated in the upper part of Fig. 2), and the other one corresponds to a position at which the center of the deflected incoming beam is lower than the fiber axis (illustrated in the lower part of Fig. 2). For the sake of simplicity the LPF modes employed will be labeled after the similar LP modes of a step-index fiber. This way, for example, the fundamental mode (FM) of the LPF will be labeled LP01. Assuming a steady-state excitation (i.e. no switching between the low and high positions) MI will be observable for output powers higher than a certain threshold. If the deflected incoming beam positions are symmetric relative to the fiber center, the resulting MI threshold will be the same regardless of the excitation state. In other words, by exciting modes at symmetric positions with respect to the fiber center, the relative modal contents remain unchanged. However, when switching between the deflected beam positions there is an instantaneous π phase shift in the interference pattern (i.e. at the position of an intensity maximum there will be a minimum and vice versa). Due to this π phase shift, the thermally induced long period grating will, given enough time, also exhibit the π phase shift (see right-hand side in Fig. 2).
Now, if the switching between the deflected beam positions happens fast enough (this rate is determined by the thermal diffusion time), the thermally induced grating will not have enough time to completely mimic the alternating interference patterns. This way, if the deflection frequency is chosen appropriately, it can be achieved that the thermally-induced refractive index profile washes out in a kind of instable equilibrium. The resulting refractive index profile makes the waveguide changes more adiabatic and so weakens the thermally induced grating, which according to  should mitigate MI and stabilize the beam. Note that this explanation is based on the assumption of a strong quasi-static index grating below the threshold, as postulated by  and as seen in the simulations in [7,12].
In general this stabilization mechanism is not limited to the interference of the fundamental mode with radially anti-symmetric modes. Other low-order HOMs would require switching the seed beam between different spatial positions of the fiber core (e.g. for the LP0x-like modes the switching should happen between the center and an off-centered position) to achieve the desired π phase shift.
4. Characterization of the free-running mode instabilities
Using the setup presented in Fig. 1, and driving the AOD by a constant control signal with a disabled feedback loop, the beam was characterized regarding the onset and evolution of mode instabilities. In order to do this, photodiode traces were recorded at different power levels and their corresponding standard deviation were calculated. Based on this data, the mode instabilities threshold was determined with the method introduced in  as an average of multiple measurements. Moreover it is important to note that measuring the threshold several times also ensured that the used main amplifier fiber was totally degraded  and so the threshold of the laser system kept constant. The output power evolution of the standard deviation and the fit function used to calculate the threshold of the laser system are shown in Fig. 3. The threshold of the laser system was Pthr = (109 ± 5) W. Moreover, from Fig. 3 it can be seen that there is a relatively broad transition area from the stable region (i.e. Pout ≤ Pthr) up to ~200 W at which mode instabilities were clearly observable on a low-speed camera. This transition region of the fiber laser system was characterized by periodic oscillations, which corresponded to discrete peaks in the Fourier spectrum, as shown in Fig. 4 (green and black spectra). This periodic behavior vanished for higher output powers, ultimately leading to a chaotic regime . Consequently, in the spectrum a noise-like pattern is observable at these higher output power levels (red and orange spectra). It is worth noting that the AOD was driven by a constant signal (i.e. constant deflection angle) to characterize the free running system. However, our very simple electronic circuits used to control the AOD showed clearly 50 Hz noise (and the corresponding higher harmonics thereof).
This noise led to a small variation of the signal beam’s position on the fiber core due to the deflection caused by the AOD. Thus, during these measurements the AOD influences MI creating the dominant peaks seen in the spectra (green and black graph). Moreover, these spectral peaks were even observable well below the MI threshold. However, if the AOD is not used, this perfectly spaced arrangement of spectral peaks vanish and only a few peaks appear (generally at different frequencies that those observable in Fig. 4) when the output power reaches the transition region . Moreover, the MI threshold is ~50 W higher in the system without the AOD. However, we expect that the use of more sophisticated electronics will avoid the unwanted influence of the AOD.
To complete the characterization of this fiber laser system, videos at different output power levels were recorded. We have used two types of cameras for our studies: a low- and a high-speed camera. The first one is able to capture ~25 images per second and, therefore, it was used to investigate the temporal behavior of mode instabilities over a time scale of several seconds. In contrast, the second one is a high speed camera which can capture 20.000 frames per second. Consequently, the temporal dynamics occurring on a millisecond time scale can be studied with this camera. In Figs. 5(a) and 5(b) excerpts from the low-speed (Media 1) and high-speed videos (Media 2) are shown, respectively.
The videos illustrate the changes of MI with increasing output power. At a power level of 182 W only small fluctuations of the mode profile could be seen due to low higher-order mode content. With increasing output power the higher-order mode content increased (195 W) and more modes got involved in the instabilities (229 W). A further increase in output power led to a more confined beam profile due to a stronger thermally induced refractive index change of the fiber core (424 W) . Additionally, it can be expected that an even larger amount of higher-order modes participate in the instabilities. To quantify these observations we exemplarily applied the mode reconstruction method based on the analysis of the intensity distribution of the output beam, recently published in , to the high-speed videos showing MI at 195 W and 229 W output power. The calculated evolution of the relative power contents of the involved modes is depicted in Fig. 6. The number of modes used to perform the reconstruction was conveniently chosen to obtain the best match between the reconstructed and the measured intensity distributions. For the sake of simplicity the LPF modes employed will be labeled after the similar LP modes of a step-index fiber. This way, for example, the fundamental mode (FM) of the LPF will be labeled LP01.
Media 3 and Media 4 show the evolution of the reconstructed intensity distribution for Pout = 195 W and 229 W respectively. In the case of 195 W output power, four different modes (LP01, LP11a/b and LP02) were used in the reconstruction. However at 229 W output power already eight strong different modes (additionally the LP21a/b, LP12a/b) were required. At 195 W the power content of the LP01 oscillated slowly between ~40 and ~80%. This contrasts with the case at 229 W where the power content of the involved modes can change between ~0 and ~50% within one millisecond.
5. Dynamic mode excitation scheme
The intention of our studies is to show that MI can be influenced and finally controlled by a dynamic mode-excitation scheme. In this proof-of-principle experiment we decided to keep the feedback algorithm and the corresponding electronic circuits as simple as possible. Thus, the logic comprised one microcontroller that can only generate predefined periodic functions (square-, sawtooth-, sinusoidal- and triangle-function) with variable amplitude, offset and frequency. During the iterations in the feedback-loop, the free parameters of a specific periodic function are adjusted to minimize the temporal fluctuations measured by the photodiode monitoring the output beam. The multi-dimensional optimization of the parameters is carried out with a minimization algorithm based on the direction set (Powell’s) methods . Besides the function characteristics, there are additional parameters that have to be adapted once for every specific experimental setup and that are not further iterated by the algorithm (e.g. length of the time window used to calculate the degree of stability of the photodiode signal or step sizes used to modify the function parameters).
In order to illustrate the effect of the dynamic mode excitation on the output beam, an excerpt from Media 5 is shown in Fig. 7. In this example the AOD was driven by a square function at a constant frequency of 1 kHz as shown in green on the right-hand side. The spatial offset of the seed beam was approximately the core size of the LPF. However, the correlation between the voltage amplitude of the AOD signal and the spatial offset of the seed beam depends on the actual distance between the AOD and the focusing lens. In our setup typically a voltage of several 10 mV was need to deflect sufficiently the beam. The photodiode trace (in blue) of the output beam resulting from the dynamic excitation is also shown on the right-hand side. On the left-hand side, the high-speed video recording the evolution of the output beam in the near-field is shown. This video is overlapped by two colored circles. The blue one corresponds to the estimated spatial position of the photodiode and the green one corresponds to the estimated spatial position of the deflected input beam on the fiber core. The sizes of the circles are the estimated sizes of the photodiode and input beam, respectively.
A high AOD signal corresponded to an input beam that was deflected to the upper part of the fiber core. Similarly, the low-level AOD signal led to the input beam being deflected to the lower part of the fiber core. Both spatial positions of the input beam excited slightly different relative power contents of the LP01 and LP11, as well as, more importantly, different intermodal phases (ideally shifted by π). These two different modal excitations led to two different output beams after propagating through the fiber. At this point it should not be forgotten that the specific output beams are not only determined by the initial excitation, but they are also strongly influenced by the build-up and adaptation of the thermally induced index grating responsible for MI. Thus, as seen in the video in Media 5, the evolution of the output beam is slower than the change in the position of the input beam. This is because the thermally-induced grating needs a specific time (related to the finite thermal diffusion time) to react on the new excitation. Ultimately, as already mentioned, this produces a time delay between the change of the input beam position on the fiber core and the corresponding final output beam. If the position of the input beam is changed before a new steady-state of the thermally induced grating is reached, then MI can be driven (i.e. controlled) by the AOD as discussed in section 2.
6. Stabilizing mode instabilities
The aim when controlling MI is to stabilize the beam profile (i.e. to suppress the temporal fluctuations) and, simultaneously, to increase the relative power content of the preferred Gaussian-like FM. Media 6 and Fig. 8 shows how the stabilization scheme worked in our experiment at 220 W (i.e. two times the MI threshold). The free-running beam profile is displayed adjacent to the controlled one for comparison. In the first seconds of the video both beam profiles are quite similar and they show the typical appearance of MI. However, when the feedback loop is closed, the algorithm tries to find optimized parameters by coarsely sweeping different frequencies and periodic functions (using some constant pre-defined values for the amplitude and offset). During this sweep several beam profiles and different types of dominant modes are clearly observable at the output of the fiber. Most importantly, the preferred FM can also be seen for a short moment. After the coarse sweep is finished the best parameter set is automatically chosen. Afterwards the feedback loop performs a quick fine sweep of the frequency around the value found with the coarse sweep (~50Hz) in order to identify the optimum operation point. Once this fine sweep is over, the microcontroller enters in tracking mode where it simply tries to maintain this optimum state by iteratively performing just small parameter variations (of the frequency, amplitude and offset). Table 1 summaries the parameters used in the optimization algorithm. After the optimization process a rectangular function with a frequency of 600 Hz and deflection amplitude of 84 µm was found to produce the most stable output beam.
Stabilization in the transition region
We applied this stabilization scheme to different power levels above the threshold Pthr. In the power range from 100 to 200% Pthr (i.e. the transition region) it was possible to significantly improve the beam stability but not the power content of the FM. Thus, while for output powers below 180 W the output beam had a dominant FM content when the feedback-loop was switched off, we could only obtain a stable output beam with an intensity profile resembling the LP11 when switching the feedback loop on.
More interestingly, we were able to obtain a LP11-like output beam even at powers below the MI threshold. Excerpts from Media 7 and Media 8 are depicted in Fig. 9 and show the low- and high-speed videos of the excited beam profiles at 210 W of output power. Moreover, Fig. 10 illustrates the evolution of the relative power contents of the modes over 12 ms. As can be seen, the LP11 mode was dominant at the output of the fiber with an average relative power content of ~80%. On the other hand, for output power levels above ~200% Pthr it was indeed possible to obtain an output beam with dominant power content in the FM.
Stabilization in the chaotic region
In Fig. 11 an excerpt of the low speed videos showing both the stabilized and the corresponding free-running near-field output beam profiles for different output powers ranging from 200% Pthr (220 W) up to 430% Pthr (472 W) is depicted. Table 2 summaries the parameters of the microcontroller used to stabilize the beam profile. However, these parameters should only be understood as exemplar, since their values strongly depend on the specific laser system to stabilize. Moreover, it should also be mentioned that, for most powers, there were different combinations of function, frequency and amplitude that produced similar results. However in Table 2 we are only giving the instantaneous solution that the microcontroller held for the optimum at that moment.
From Media 9 it becomes apparent that both the pointing stability and the beam quality of the output beam are significantly improved with the dynamic excitation in comparison to the free- running case. In order to quantify this impression, the mode reconstruction technique was applied to the high-speed videos. The graphs depicted in Fig. 12 show the temporal evolution of the relative power content of the FM (in blue) and of the strongest HOM (in grey) for different output powers. Graph a) in Fig. 12 shows the result of the stabilization at a power level that lies in the upper part of the transition regime of MI (~217 W). At this power level it was still possible to obtain a strong LP11 -like output beam, but it was also possible to obtain a FM-like one. This way the parameter set was chosen to obtain the FM. With increasing output power from 200% Pthr (Fig. 12(a)) to 240% Pthr (Fig. 12(b)) the FM became clearly the most dominant mode and, consequently, the beam quality was improved. At power levels as high as 300% Pthr (Fig. 12(d)) the FM was still clearly the dominant mode. However, the FM power content became more and more instable. By increasing the output power to 430% Pthr (Fig. 12(f)), it could be seen that the influence of the dynamic mode excitation became weaker. In consequence the FM power content decreased and the temporal instabilities increased.
The data from Fig. 12 were used to calculate the time-averaged relative power content of the FM (Fig. 13) and its corresponding standard deviation (indicated by the error bars). For comparison, the same values have been calculated for the free-running system and they have also been depicted in Fig. 13.
It can be seen that the average relative power content was increased by a factor of at least 3 for output power levels as high as 300% Pthr. However, at output power levels higher than 300% Pthr the average power content of the FM began to drop below 50%, indicating a reduced influence of the stabilization scheme on MI. A comparison with the free running system at these high power levels was unfortunately not possible due to the ambiguity of the mode reconstruction method when too many different modes are involved in the instabilities . However, it can be expected that the relative power content of the FM in the free-running system would still be lower than that of the stabilized case. Thus, it can be assumed that even at the highest power of 430% Pthr this simple stabilization scheme is able to increase the beam stability and the relative power content of the FM. This assumption is partially supported by the results shown in Media 9.
In section 2 we have presented the working principle of this stabilization technique. If the simplified vision of the physical processes taking place in the fiber given in that section were perfectly accurate, it should be expected that the stabilization scheme would be able to produce a FM-like output beam right from Pthr. However, what we saw in the experiments was that when switching on the feedback loop it was only possible to obtain a stabilized output beam with a dominant FM content for powers >200 Pthr. Below this value the feedback loop was only able to obtain a stabilized output beam resembling an LP11 mode. However, at this point it is important to discuss the practical relevance of this fact. Looking at Fig. 5 (Media 1) it becomes apparent that in the free running case below ~180 W the FM is the dominant mode on the output beam. The beam quality and stability up to this power level was fairly stable and sufficient for most practical applications. Hence, no real stabilization would be required. That leaves just a <30 W gap in which we were not able to obtain a FM-like output beam.
In any case, the fact that a stable output beam resembling an LP11 could be obtained with the stabilization setup for powers ranging from ~Pthr all the way up to ~200% Pthr depicts a more complex picture than that given in section 2. Indeed, we believe that what we see in this power range when using the dynamic excitation is the result of seeding the travelling wave as predicted by Smith et al. . These authors published simulations in which they investigated the influence of a modulated seed signal on the threshold of mode instabilities. The simulations were done considering a sinusoidal amplitude modulation of the seed signal in the kHz band with 30 parts per million peak-to-peak amplitude. The results show an increase in the relative power content of the considered LP11 and a decrease in the FM content, respectively. Consequently the MI threshold was significantly reduced. The authors explained this behavior by considering that a modulated seed signal has additional side-bands in the kHz range around the center frequency in the power spectrum. These extra frequencies, present for both the FM and the LP11, seed the traveling interference wave leading to an energy transfer from the FM to the LP11 . The experiments described in this paper are for the first time very close to the scenario described by these simulations. One main difference, though, is that the entire launched seed signal is in good approximation constant in power and only the intermodal power distribution is varied periodically with time. Due to the spatial symmetry of the modes (the FM has one intensity peak and the LP11 two) our dynamic excitation causes a modulation of the LP11 with twice the frequency of that corresponding to the FM. Additionally, our initial frequency-shifted power in the LP11 is many orders of magnitude larger than that considered in . Despite the differences, our dynamic excitation also produces frequency side bands in the kHz range in the power spectrum. Thus, following Smith’s argumentation the dynamic excitation should lead to an almost complete energy transfer in the LP11 and to a very significant drop of the MI threshold. Even though the drop in the MI threshold is only moderate, i.e. much smaller than what could be expected from , we did observe it. Furthermore, a significant (albeit not complete) energy transfer in the LP11 was also observed. Thus, we believe that these measurements represent the first experimental evidence of the feasibility of the MI mechanism proposed by Smith et al. However, it is important to stress that the behavior of the free-running system is quite different from the behavior of the system under dynamic excitation. In fact, as mentioned in the introduction, the experimental evidence of the free-running system (at least for our short very large mode area fibers) does not seem match the predictions of Smith et al. This is why we believe that, as demonstrated in this paper, while the origin of MI proposed by Smith et al. is entirely possible, it is not the dominant mechanism at work in our experiments.
As far as we are aware, Smith’s theory [6, 17] does not predict any high-power regime in which the FM becomes again the dominant mode while modulating the seed signal. However, as shown above, this is what we see with our dynamic excitation scheme for powers >200% Pthr. The way that we interpret this is that with the dynamic excitation we are indeed seeding a moving grating which propagates on top of a strong quasi-static one. By increasing the output power the homogenization of the quasi-static grating described in section 2 starts to become more relevant and we find a broad power range over which the beam could be stabilized.
In conclusion, we have shown that MI can be controlled externally by a dynamic excitation of the fiber modes using an acousto-optic deflector. In this first proof-of-principle experiment both the beam quality and the pointing stability of the output beam were significantly improved over a wide range of powers above the threshold. In particular the fundamental mode content could be increased by more than a factor of three at three times the typical threshold of the free-running system. Moreover, even at output power levels larger than four times Pthr we showed that MI could be influenced and it may even be possible to control them with a more sophisticated approach.
The presented technique can be further improved and refined. For example, the excitation may be improved with advanced algorithms, which allow using arbitrary functions for the beam stabilization. Moreover, a much faster electronic logic might be able to calculate the instantaneous transfer function and apply the corresponding inverted function to completely compensate the influence of the instantaneous induced grating within a certain time window. Besides the presented dynamic excitation scheme, alternative techniques to control MI are feasible, for instance a modulation of either the pump (preferably) or the seed power in the kHz range should also lead to a stabilization of MI. This latter point is subject of current investigation.
Finally, it can be expected that a further evolution of this technique will be able to stabilize the beam profile at even higher powers.
The research leading to these results has received funding from the German Federal Ministry of Education and Research (BMBF), the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no.  “PECS,” and the Thuringian Ministry for Economy, Labour, and Technology (TMWAT, Project no. 2011 FGR 0103) with a European Social Fund (ESF) grant and the Thuringian Ministry of Education, Science and Culture TMBWK) under contract B514-10061 (Green Photonics). Additionally, F.J. acknowledges financial support by the Abbe-School of Photonics Jena. We thank Manuel Krebs for fruitful discussions regarding the transfer function.
References and links
1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]
2. F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012). [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]
7. 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 20(12), 12912–12925 (2012). [CrossRef] [PubMed]
8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19, 23965–23980 (2011).
9. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. Part 1: Adiabaticity criteria,” Optoelectronics, IEE J . 138(5), 343–354 (1991).
10. 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]
13. 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]
14. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef] [PubMed]
15. N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, A. Tünnermann, and O. De Vries, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012). [CrossRef] [PubMed]
16. S. Breitkopf, A. Klenke, T. Gottschall, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “58 mJ burst comprising ultrashort pulses with homogenous energy level from an Yb-doped fiber amplifier,” Opt. Lett. 37(24), 5169–5171 (2012). [CrossRef] [PubMed]
18. 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]
19. 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]
20. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012). [CrossRef] [PubMed]
21. 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]
22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).