1. Introduction

The evolution of the average output power emitted by fiber laser systems has stalled in the kilowatt level for ultra-short pulsed fiber lasers [1,2 ] and in the multi-kilowatt level for CW fiber laser [1,3–5 ] for approximately four years. The main reason for this stagnation is the onset of mode instabilities (MI), which degrade the beam quality emitted by high-power fiber laser systems once that a certain average power threshold has been reached. This sudden degradation of the output beam quality is accompanied by temporal fluctuations of the beam profile. In spite of this, it has been shown that mode instabilities do not necessarily pose a general limitation for the scaling of the output average power scaling of fiber laser systems if a certain amount of beam quality degradation can be tolerated by the application [6]. Nevertheless, even though tolerable in some cases, such beam degradations are always undesirable and, consequently, there is a growing interest and pressure in the scientific community to mitigate or even, ideally, to overcome this limitation. In this context the first experimental implementations of mitigation strategies have already been published. Some of these strategies employ advanced and sophisticated fiber designs in an attempt to ensure effective single mode operation [7–10 ], others employ multi-channel fibers [11] or some external means to stabilize the fluctuating beam [12]. Further approaches have been theoretically suggested including gain saturation/reduction of the pump absorption [13–15 ] or reduction of the quantum defect (QD) heating of the active medium [14]. In particular, the latter strategy directly addresses what is conventionally accepted to be the physical origin of MI [16–19 ]: the change of the refractive index profile of the fiber induced by the thermal load generated by the amplification process itself, i.e. predominantly by quantum defect. Figure 1 shows the predicted evolution of the MI threshold as a function of the signal wavelength when pumping at 976 nm and only quantum defect heating is taken into account [14]. This calculation predicts a 50% increase of the MI threshold by shifting the seed wavelength from 1030 nm to 1010 nm.

 

Fig. 1 Simulated evolution of the mode instabilities (MI) threshold as a function of wavelength based on [14]. The graph is normalized to the MI threshold at 1030 nm. It is assumed that the quantum defect (QD) is the only heat source in the active fiber.

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The approach of increasing the MI threshold by shifting the signal towards shorter wavelengths is very attractive since, in principle, it promises high improvements in terms of power and it is relatively easy to implement in practice. Therefore, it seems worth having a closer look at this strategy to determine its performance in a real system, which is what has been done in this paper. In order to do so a wavelength tunable system, covering discrete wavelengths from 1010 nm to 1060 nm, has been developed. As it will be seen below, it is experimentally shown that the wavelength dependence of the MI threshold contradicts the expected evolution given in Fig. 1. This unexpected result leads to the most likely conclusion that, besides QD, a second significant heat source is present in the fiber. Consequently, in an attempt to identify this additional heat source subsequent experimental investigations have been carried out. These strongly suggest photodarkening (PD) [20–22 ] as the second heat source, which up to now has been grossly underestimated and, therefore, neglected in most models and calculations. A corroboration of the large impact that PD can have on MI is given by simulations that estimate and take into account the additional PD-induced thermal load. The simulated results match accurately the experimental data leading to the unsettling conclusion that even a nearly negligible amount of PD-induced loss (~6%) can lead to a reduction of the MI threshold by a factor of roughly 2. At this point it should be mentioned that other authors [23] had previously briefly hinted at the strong impact that PD could have on MI. However, the calculations of PD in that work were very rudimentary and only considered as somewhat abstract theoretical experiment on the effect that any additional source of linear absorption would have on the MI threshold. Moreover, the wavelength dependence of the effect was left unexplored and the simulations were not contrasted with real measurements. On the other hand, there have been some experimental characterizations of the degradation of the MI threshold with time [8,24 ]where PD was briefly hinted as a potential cause for this effect but without providing any further conclusive evidence.

This paper is organized as follows: the first part describes the setup and experimental methods used in this work. In the following section the experimental results are presented. Additionally, in that same section the subsequent experiments aimed at finding evidence of a second heat source are described. The third part of the paper describes the model used, presents the results from the simulations and compares them with the experimental measurements. Finally, the article ends with a conclusion.

2. Wavelength dependence of mode instabilities

Setup and measurement methods

The experimental setup consists of a seed source, a high-power amplification stage and some beam diagnostics. The seed source is driven by a fiber oscillator emitting nearly transform-limited 100 ps pulses at a repetition rate of ~19 MHz, which signal wavelength can be switched by a fiber-switch (Leoni FiberSwitch eol 1x8). The switch has eight ports that are spliced to passive single-mode fibers with inscribed fiber-Bragg gratings (FBG). Each FBG has a different center wavelength. Thus, the changing in wavelength of the laser oscillator is done by switching between the different FBGs. The resulting discrete tuning range covers the amplification band of Yb³⁺-doped silica fibers from 1010 nm to 1060 nm in eight steps. Subsequently, before delivering it to the high-power main amplification stage, the signal from the oscillator is boosted in three amplification stages. In order to get an efficient amplification of the seed signal (i.e. the amplified signal from the oscillator) even at the shortest/longest operation wavelengths its power level has to be appropriately chosen due to the different amounts of gain available at each wavelength [14]. Thus, the seed source has been configured to emit a maximum average output power of 35 W at each center wavelength. The ASE content is lower than 0.2%. After final amplification the ASE content slightly rises but do not exceed 1%. Note that a wavelength independent signal power is a necessary condition for comparable experiments due to the dependence of the MI threshold on this parameter [19,25 ].

Figure 2 shows a schematic of the entire setup. The signal from the seed system is launched into the main amplifier which is a 1.2 m long rod-type large-pitch fiber (d_core = 62 µm; α_976nm = 24 dB/m) mounted on heat a sink which is cooled by a water flow [7]. It should be noted that this particular fiber is already degraded at 1030 nm operation wavelength [1,8,24,26,27 ]. This means that the MI threshold has been measured several times until a saturation value is reached. A typical graph of such degradation measurements will be presented later. Finally, this main amplifier is pumped in the counter-propagating direction by a laser-diode operating at 976 nm (P_{out max} = 1.2 kW). The amplified output signal is analyzed by two photodiodes (P1 and P2) and a power meter (PM1). Photodiode P1 has an aperture that is larger than the imaged beam diameter to allow, after calibration, for a fast measurement of the average output power. The aperture of photodiode P2 is significantly smaller than the diameter of the imaged beam and it is used to measure the temporal stability of the beam as proposed in [28]. Examples of the typical time traces measured by both photodiodes (P1 and P2) are depicted in Fig. 3(a) . The pump diode operates always at a constant power to ensure that its emission wavelength does not drift (something that is essential in these experiments). In spite of this, the launched pump power can be varied very quickly by moving an aperture attached to a translation stage and placed right in front of the output port of the pump diode. Using this strategy a fast measurement of the evolution of the beam stability with the output power becomes possible, i.e. the power threshold of MI can be probed with a quick measurement. Probing the MI threshold this way at periodic intervals allows tracking its temporal evolution, i.e. its degradation. Furthermore, the whole process of probing the MI threshold and acquiring the data has been automatized to increase the accuracy and repeatability of the results. As can be seen in Fig. 3(a), when the pump power increases (blue line) the stability of the beam (black curve) decreases as indicated by the progressively stronger fluctuations, which become suddenly very pronounced after the onset of MI. In order to quantitatively determine the MI threshold, the beam stability (determined as the standard deviation of the fluctuations seen in the black plot of Fig. 3(a) is plotted against the output power, as shown in Fig. 3(b). From this plot the MI threshold can be calculated using the definition given in [28], which requires fitting the experimental data with an exponential curve (red line in Fig. 3(b). As can be seen in Fig. 3(b), the fitting curve adapts itself to the measured data both in the stable and transition regions but not in the chaos region. The MI threshold separates the stable and transition region and, therefore, it is important to match the fitting curve with the measured data in these regions.

 

Fig. 2 Schematic representation of the experimental setup. DC1 is a dichroic mirror which separates the pump and signal radiation. M1 to M5 are dielectric mirrors used to steer the IR signal; L1 to L4 are focusing lenses; L5 is a defocusing lens and W1 is a fused-silica wedge. P1 and P2 are photodiodes used to measure the average power and the stability of the beam, respectively, and PM1 is a power-meter. The moveable pinhole is used to adjust the pump power by cutting the beam caustic at different positions.

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Fig. 3 The graph on the left shows an example of the results from the automatic measurement of the evolution of the power and beam stability (signals from photodiodes P1 and P2, respectively). The graph on the right-hand side has been obtained from the left one by plotting the evolution of the beam stability (represented as the standard deviation of the fluctuations seen in the black photodiode trace in the left graph) against the evolution of the power. It shows the evolution of the beam stability with increasing power (blue dots), the fit function of the measured data (red solid line) and the calculated threshold (dashed black line). The stable, transition and chaotic regions have also been highlighted.

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Experiment

The first experiments are aimed at revealing the dependence of the MI threshold on the signal wavelength. Since, as explained above, the pump wavelength remains constant (something that has also been verified by tracking this parameter), a shift of the signal wavelength will be the only responsible for a change of the quantum defect (QD) induced thermal load (being reduced when tuning the signal to shorter wavelengths). Consequently, a significant increase of the MI threshold might be expected when blue-shifting the signal wavelength.

Using the setup described above, a measurement of the MI threshold as a function of the signal wavelength has been performed (Fig. 4 ). It should be noted that each dot in Fig. 4 represents the mean value of a run of twenty single MI threshold measurements where no characteristic drift of the MI threshold has been observed. As it becomes apparent in Fig. 4 both a blue- and a red-shift of the signal from its 1030 nm reference wavelength lead, against the expectations, to a substantial decrease of the MI threshold. In order to verify this trend the experiment has been repeated three more times (black dots in Fig. 4). As can be seen, in all the measurement runs the highest MI threshold has been measured at, or in the vicinity of, 1030 nm. However, after repeating the experiment several times, a second effect becomes observable, namely that the absolute values of the MI threshold degrade by about 10%. This observation can be related to the degradation of the MI threshold in pristine fibers [1,24,26,27 ] as previously mentioned and shown in Fig. 5(a) . It is interesting to note that this degradation occurred only after shifting the signal wavelength which imply a certain dependence of the degradation state on the seed wavelength. However, detailed investigations have to be carried out in the future.

 

Fig. 4 Evolution of the MI threshold power as a function of the signal wavelength when pumping at 976 nm (black, blue) and the simulated inversion level (green). (blue) Initial wavelength sweep and (black) subsequent wavelength sweep.

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Fig. 5 (a) Degradation of the MI threshold with the number of MI measurements using a pristine fiber. (b) The fiber under test is exposed to 100 W of 915 nm radiation for 6 h. The fiber initially degraded by the 915nm light shows a progressively higher MI threshold with time once that it amplifies the 1030 nm seed signal by pumping at 976 nm.

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The comparison of the experimental results (Fig. 4) with the theoretical prediction mentioned above (Fig. 1) shows a significant deviation at the shorter wavelengths. Instead of the expected increase of the MI threshold, a reduction has been observed. Such a gross mismatch between experiments and theory indicates an additional physical process at work not considered in the simulations. It should be remembered that theoretical models proposed up to date assume that MI are a thermal effect. Additionally they assume that the thermal load generated in the fiber during the amplification process is predominantly due to QD [13–15,17 ]. Consequently, such assumption must be flawed if the experimental observation does not follow the theoretical prediction. Pursuing this chain of thought leads to the conclusion that the flaw in the assumption can only be in considering QD as the main (and most of the time only) heat source in the fiber. Therefore, the conclusion from the experiments is that there must be an additional, up to now disregarded, source of heat load in the fiber.

Besides through QD, the fiber may be heated due to, for example, bulk absorption of the fiber material (fused silica doped with Yb³⁺). However, the transmission of fused silica in the wavelength region of interest (from 1010 nm to 1060 nm) is largely wavelength-independent with less than ~0.01 dB/m [29]. This, on the one hand, does not clarify the origin of the wavelength dependence and, on the other hand, is too weak a heat source to compete with QD. Thus, we have to look elsewhere to explain the experimental observations.

Due to the wavelength dependence of the cross-sections of Yb³⁺ doped fused silica [30], the evolution of the undepleted inversion level in the fiber with the signal wavelength (Fig. 4.) exhibits a trend that is strikingly similar (albeit reversed) to that of the MI threshold shown in Fig. 4. This means that a higher undepleted inversion level remains in the fiber at the outermost wavelengths (1010 nm and 1060 nm) than around 1030 nm, where the minimum is located. From the literature it is well known that photodarkening (PD) increases with the amount of undepleted inversion [31]. However, typically PD is associated with a decrease of laser efficiency and not with an increase of the heat load. The reason for PD is the generation of color centers (CC) which are able to absorb IR radiation (both pump and signal) [32]. Nevertheless, nowadays with optimized doping compositions, PD decreases the amplification efficiency by just a few percent, which, over a long period of time, has misleadingly led to the conclusion that the impact of PD on MI should be negligible as well. However, even small amounts of PD-induced losses can have a strong thermal impact in an active fiber, as we will show in the section 3. In fact, under the assumption that nearly the whole absorbed photon energy is transformed into heat in the CC, even small amounts of photodarkening-induced losses should be enough to significantly increase the total heat load in an active Yb-doped fiber. Moreover, the CC responsible for PD will absorb not only the signal photons but some of the pump photons as well, which leads to an even higher overall extra heat load. Consequently, this absorption can heat the fiber and can be the origin of the additional thermal source we are looking for.

Experimental evidences of photodarkening

In order to explore the feasibility of photodarkening (PD) as the second thermal source in the active fiber, further experimental studies have been carried out.

Degradation of the MI threshold

The degradation of the mode instabilities (MI) threshold is by now a well-known feature of this effect [8,26,27 ]. In the following we will take a closer look at this degradation. This is done by using a pristine fiber as the main amplifier and by tracking the evolution of the MI threshold with time as explained above. Figure 5(a) shows the resulting typical evolution of the MI threshold as a function of multiple subsequent measurements. The degradation of the MI threshold observed in Fig. 5(a) has some parallels with the known evolution of PD with time in its temporal shape. The degradation evolves very quickly at the beginning to progressively decelerate as it approaches a saturation level [32,33 ]. So this experiment already hints at the link between PD and MI. However, the experiment has to be extended to provide conclusive evidence on the feasibility of PD as the effect responsible for the degradation of the MI threshold. This can be achieved by using a post-processed fiber instead of a pristine one. The idea is to exploit the fact that the level of PD-induced losses is directly related to the amount of undepleted inversion. Thus, the fiber under test is exposed to 100 W pump radiation at 915 nm for 6 h without amplifying any signal. Consequently, nearly 90% of the active ions are inverted, which should result in the strongest possible PD in this fiber.

After this treatment the evolution of the MI threshold is measured at an operation wavelength of 1030 nm. Figure 5(b) shows the obtained results. This measurement series starts with the lowest MI threshold and it progressively increases with each measurement.

As can be seen, Figs. 5(a) and 5(b) show a contrary behavior of the MI threshold evolution. So now the question is whether these two opposing behaviors can be explained by PD. In case of the pristine fiber the amplification process leaves a certain amount of undepleted inversion that results in the creation of CC with time, i.e. photodarkening, and subsequently in a progressive drop of the MI threshold with the number of measurements. However, the case of the postprocessed fiber is different. Here the amount of undepleted inversion during the measurement of the MI threshold is much smaller than the level of undepleted inversion generated by pre-irradiating the fiber with 915 nm light. This means that the total number of CC created by pre-irradiating the fiber at 915 nm is much higher than the number that would be obtained by amplifying a signal at 1030 nm. From the literature it is known that CC can be annihilated and generated during the amplification process with a certain probability depending on the amount of existing CC, undepleted inversion and intensity of signal and pump [31]. The final amount of PD in a fiber is the result of reaching an equilibrium state between the generation and annihilation of CC [31]. This implies that the probability for annihilation of CC is higher than the generation due to the large total number of existing CC previously generated by the 915 nm radiation. This reduction of the total number of CC leads to a reduction of PD and, consequently, to a smaller heat load, resulting in an increase of the MI threshold. The final state of the MI threshold depends on the amount of undepleted inversion during the amplification process and on the remaining CC that were not bleached [31].

A further consequence of the above is that the final MI threshold after irradiation at 915nm should be different depending on the signal wavelength. In order to prove this, the fiber under test has been repeatedly post-processed by exposing it to 915 nm for 30 min and, after each time, the evolution of the MI threshold has been measured for different operation wavelength). Figure 6 shows the saturation of the MI threshold after a certain number of measurements for all three signal wavelengths (1015 nm, 1030 nm and 1060 nm). The final values of the MI threshold are indeed different for each wavelength. The highest MI threshold is reached for an operation wavelength of 1030 nm, followed by 1015 nm and the lowest one is for 1060 nm. As already mentioned, this trend matches the amount of undepleted inversion which reaches its lowest point around 1030 nm and it becomes higher at 1015 nm and 1060 nm. It is interesting to point out that the level of undepleted inversion at 1015 nm and at 1060 nm is roughly equal. However, in all measurements the MI threshold at 1060 nm is consistently lower than at 1015 nm. The most likely cause for this behavior is the fact that QD heating at 1060 nm is higher than at 1015 nm. This implies that the thermal loads caused by QD and PD must be of the same order of magnitude.

 

Fig. 6 The fiber under test is exposed to 915 nm radiation for 30 min. The graphs show the evolution of the MI thresholds with the number of measurements at three different signal wavelengths.

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Transmission of a 633 nm probe signal

In the context of PD it is common practice to measure the additional propagation losses induced by CC by using a probe signal. The wavelength of this signal is chosen so that it falls outside the amplification window of the Yb-doped fiber. Typically, a probe beam operating around 600 nm is used. In order to do so the setup from Fig. 2 is extended by launching the radiation emitted by a HeNe laser at 633 nm into the amplifier fiber. Hereby, the mirrors M1 and M3 are replaced by dichroic mirrors able to transmit the probe beam. This allows measuring the power of the probe beam before and after its propagation through the fiber under test even during the pumping and amplification of the seed signal. However, it is important to note that this transmission measurement significantly differs from other PD-loss measurements in that the fiber under test is a multi-mode fiber surrounded by a high NA air-cladding structure. Thus, a significant amount of the launched probe signal is mainly confined in the structure outside of the signal core. The consequence is a reduced overlap of the probe signal with the active core, which leads to a reduced sensitivity to the PD-induced losses. Therefore, it can be expected that the uncertainty associated with this measurement is larger than comparable measurements in strictly single mode fibers.

The relative transmitted power of the probe signal at 633nm for three different operation wavelengths (1015 nm, 1030 nm and 1060 nm) are depicted in Fig. 7 . Moreover, for the sake of comparability, the amplified output power is for all three seed wavelengths fixed at ~200 W. The results in Fig. 7 show that the highest transmission of the 633 nm signal corresponds to 1030 nm, the second to 1015 nm and the lowest to 1060 nm. Simulations reveal that the inversion levels show the same trend as the measured transmission losses. The highest inversion and, therefore, the highest PD losses corresponds to 1060 nm; the lowest inversion and, therefore, the lowest PD losses corresponds to 1030 nm. In case of 1015 nm, both, the inversion level as well as the PD losses are in between. This mirrors the trend seen in the MI threshold and establishes a correlation between the MI threshold and the generation of additional propagation losses at 633 nm due to PD [33].

 

Fig. 7 Relative transmission of a 633 nm probe-signal. The transmission is measured while the seed signal is amplified to ~200 W. For comparison, the corresponding relative change of the saturated MI threshold with the signal wavelength is also depicted (black curve).

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3. Simulation of the influence of photodarkening on mode instabilities

In order to further establish the viability of PD as the physical mechanism responsible for the degradation of the MI threshold [1,24 ] and for its unexpected dependence with the signal wavelength, which significantly departs from previous theoretical predictions [14], several simulations have been carried out. However, the first step has consisted on incorporating the expected photodarkening-induced power loss and extra heat load in the simulation models. The estimation of the expected PD power loss has been done using experimental graphs published in the literature [31,33 ]. Thus, for example, in [33] it is shown that the maximum PD-induced loss depends on the square of the total ion-density. Additionally, the results published in [31] show that the actual PD-loss, as opposed to its maximum possible value, is roughly linearly proportional to the relative population density in the upper laser level in the fiber. Thus, using this information and correcting the PD losses of those graphs (which were measured at 633nm) for 1030nm [32], it is possible to calculate the expected PD loss as a function of the local inversion level along the fiber and in dependence of the total ion concentration. This way a spatially resolved estimation of the PD loss can be incorporated in the simulation models. Furthermore, under the assumption that all the energy of a photon absorbed by the color centers (CC) that induce PD is transformed into heat, it is possible to associate a spatially-resolved heat load to the estimated PD losses. Besides, it is worth noting, that the CC can potentially absorb both photons from the signal or from the pump. This, in turn means, that not only the signal but also the pump contributes to the extra heat load.

In order to evaluate the impact of photodarkening, the fiber laser system used in the experiments has been simulated by solving the rate equations. Thus, an Yb-doped fiber with 80 µm core diameter (from which only 64 µm are doped with 3.5*10²⁵ ions/m³), 177 µm pump core diameter, 1.2 m length and doped has been considered. The fiber is seeded by 35 W at 1030 nm and pumped in the counter-propagating direction at 976 nm. For this configuration of the fiber laser system the calculations provide an expected average photodarkening loss of 1 dB/m. In spite of this, as shown in Fig. 8(a) , the fiber has enough gain to partially compensate for this PD-induced loss, which results in an overall reduction in power of just ~6% when taking PD into account (blue curve). Such a small degradation of the output power is consistent with the typical experience with these fibers in which power losses lower than 10% have been observed. Furthermore, from the practical point of view such a fiber would be considered nearly PD-free. However, the situation becomes significantly more dramatic when looking at the impact of photodarkening from the perspective of the heat load, as done in Fig. 8(b). As can be seen, the extra heat load caused by the PD-induced absorption of the signal photons at 1030 nm (orange line) is comparable to (actually even slightly higher than) the heat load created by the QD (red line) in this fiber. Moreover, as mentioned above, the absorption of pump photons creates an extra heat load (green line). Therefore, when adding all these contributions to the heat load together (blue line) it can be seen that this parameter has been more than doubled with respect to the total heat load found in a fiber pumped with the same pump power but in which no PD is present (red line).

 

Fig. 8 a) Evolution of the signal power along the fiber amplifier when considering (blue line) or not (red line) photodarkening. b) Evolution of the heat load induced by the quantum defect (red line), the absorption of signal photons (orange line) and the absorption of pump photons (green line). Additionally, for comparison purposes, the total heat load (blue line) is shown.

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In order to estimate the impact of this dramatic increase of the heat load on the MI threshold, a simple model has been used. The model uses assumptions similar to those presented in [14], i.e. that the MI threshold is reached once that a certain average grating strength is reached in the active fiber. Since the average grating strength at which the MI threshold is reached has to be fitted to one experimental measurement, this kind of model is not able to predict the MI threshold of a new fiber design. However, it can predict the dependence of the MI threshold to changes of the system, once that the condition for reaching the threshold has been found. To simplify matters even more the amplitude of the thermally-induced waveguide perturbations is proportional to the heat load. Therefore, in good approximation, calculating the evolution of the MI threshold in a given fiber becomes a matter of evaluating when a certain constant average heat load has been reached in the system (regardless of the emission wavelength). In order to do this, only the fundamental mode is considered since it is assumed that below the threshold the higher-order mode content is fairly low (a few % at most) and, therefore, only a negligible amount of the heat load will be generated by it in the active fiber. This very simple approach disregards the actual shape of the thermally-induced grating and, therefore, its predictions will slightly differ from those obtained with the more complex model presented in [14]. Nevertheless, the advantage of this simplified model is that it calculates the MI threshold in a matter of minutes. For a detailed description of the model please refer to [34]. Furthermore, in spite of its simplicity it is shown in the following that this approach is able to reproduce the experimental results with a high degree of accuracy.

The simulation model is used to reproduce the experimental measurements presented above. In order to do this, the average heat load that leads to the onset of MI is determined by fitting this parameter in a way that the simulation threshold at 1030 nm corresponds with that seen in the measurements. This leads to a “threshold” average heat load value of 34W/m in this case. From that moment on, there is no free parameter in the simulations and the evolution of the threshold is simply calculated by changing the wavelength of the signal in the model. The simulation results (in blue) are presented in Fig. 9(a) together with the experimental measurements (in green). As can be seen there is a very good agreement between simulation and experimental results at all wavelengths, which validates the approach used in the simulation model.

 

Fig. 9 (a) Comparison between simulation (blue line) and experimental (green line) results. b) Dependence of the mode instability threshold with the signal wavelength in an Yb-doped fiber pumped at 976nm when considering (blue line) and not considering photodarkening (red line).

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As shown in Fig. 9(b), the wavelength dependence predicted by the model when considering PD (blue line, which are the same results already presented in Fig. 9(a)) is very different from the one expected when taking only QD heating into account (red line). In particular, in accordance with [14], the model predicts that when only QD heating is present (red curve) the MI threshold should increase when the signal is shifted towards shorter wavelengths, i.e. when the QD is reduced. In contrast, when taking into account the extra heat load created by PD (blue curve), it can be seen that the wavelength dependence of the threshold is strongly modified and the highest threshold is reached at ~1030 nm, as also observed in the experiments (see Fig. 9(a)). The reason for this is that the average undepleted inversion along the fiber, and with it the PD losses, reaches its lowest point around 1030 nm. Thus, as the signal wavelength moves away from this central region around 1030 nm the PD losses become higher and the extra heat load increases, resulting in lower MI thresholds. Additionally, it can be seen that the threshold at 1060 nm is lower than at 1010 nm. This, as already discussed in a previous section, has to do with the fact that the QD heating at 1010 nm is significantly lower than at 1060 nm (see red curve). Another interesting point of Fig. 9(b) is that the MI threshold with PD (~312 W) is more or less a factor of 2 lower than that for the case without PD (~750 W). This factor of 2 is the level of MI degradation that is expected in this fiber, and it is consistent with experimental observations. This implies that even the small amount of PD present in this fiber, which leads to a power loss of just 6%, is strong enough to halve the MI threshold. Consequently, any measure that will lead to a reduction of PD, even if it is only a modest one, can a have strong impact on the MI threshold.

4. Conclusion

Experiments designed to prove that changing the seed wavelength is an effective mitigation strategy for mode instabilities (MI) have led to the unexpected result that the highest MI threshold is located at a seed wavelength around 1030 nm. The subsequent experimental investigations have revealed that there is an additional heat source (apart from the quantum defect), which depends on the amount of undepleted inversion and shows a time dependent saturation behavior. An effect that combines all of these features is photodarkening (PD). The simulation of the influence of PD-induced propagation losses on the MI threshold matches accurately the experimental results, which further strengthens the viability of PD as the second heat source. The simulations also show that even if the reduction of output power due to PD losses is negligible, PD can double the heat load in the fiber. Thus, it is expected that even a small reduction of the PD-induced losses will increase significantly the MI threshold.