In this paper we present a simple model to predict the behavior of the transversal mode instability threshold when different parameters of a fiber amplifier system are changed. The simulation model includes an estimation of the photodarkening losses which shows the strong influence that this effect has on the mode instability threshold and on its behavior. Comparison of the simulation results with experimental measurements reveal that the mode instability threshold in a fiber amplifier system is reached for a constant average heat load value in good approximation. Based on this model, the expected behavior of the mode instability threshold when changing the seed wavelength, the seed power and/or the fiber length will be presented and discussed. Additionally, guidelines for increasing the average power of fiber amplifier systems will be provided.
© 2015 Optical Society of America
The output average power of fiber laser and amplifier systems (both CW and pulsed) has seen an exponential increase over the last 20 years . Such an unprecedented power evolution rate demonstrated that active fibers have an outstanding potential for power scaling. This is due to the geometry of the fiber, which intrinsically offers a very high surface to active volume ratio that reduces the thermal problems which plagued most of the early solid state active media. However, this same geometry that is so beneficial for scaling the average output power of fiber laser systems becomes their Achilles’ heel when it comes to scaling the peak power in pulsed operation. The problem is that the tight confinement of the light in the fiber core over long lengths favors the onset of non-linear effects. Thus, until 2009-2010 it was widely thought that the scaling of the average power of fiber laser systems was mainly limited by non-linear effects, which onset threshold usually lays significantly lower than the thermal damage threshold of the material. Thus, with this mindset, the output average power that could be obtained from a single fiber aperture was estimated to be 36kW . However, in 2010 and early 2011 the first reports of a new phenomenon [3,4], transverse mode instabilities (TMI), started to challenge these predictions. In fact, TMI was a completely new and unexpected kind of non-linear effect in the sense that it did not limit the peak power of the laser radiation but its output average power instead. Since TMI shook the pillars that sustain the reputation of fiber laser systems, i.e. their average power scalability, its impact in the fiber laser community was deep and each new advancement in the topic has been received with interest and expectation.
Transverse mode instabilities are an effect that is characterized by the formerly stable high-quality output beam of a high-power fiber laser system becoming unstable once that a certain average power threshold has been reached . The beam instabilities are caused by the temporally unstable energy transfer between the fundamental transverse mode of the fiber and one or more higher-order modes . The temporal interplay between the different transverse modes involved in TMI give rise to three main regions of operation : a stable region (below the threshold), a transition region characterized by a quasi-periodic energy transfer between the modes (slightly above the threshold), and a chaotic region with seemingly chaotic energy transfer fluctuations between the different modes (well above the threshold). Shortly after the publication of the first experimental observations it was proposed that TMI are ultimately created by a modal interference pattern giving rise to an index grating in the active fiber either through the resonantly induced index change of doped fibers or through the thermo-optic effect [7,8]. Shortly afterwards A. Smith et al. showed that the measured beam fluctuation frequencies are compatible with a thermal origin of the effect .Since then intense research efforts have been devoted worldwide to gain a detailed theoretical understanding of the physical origin of TMI [9–11]. This global research effort has led to the development of very sophisticated simulation tools [10–16], which have allowed getting a clearer understanding of the behavior and dependencies of TMI. As a direct consequence of the progress in the theoretical understanding of TMI, the first mitigation strategies have been proposed [13–15,17–19] and even experimentally demonstrated [20–24]. All of these works, different in approach and scope as they are, share a common characteristic: identifying quantum defect as the main heat source in an active fiber and, therefore, pointing at it as being ultimately responsible for TMI.
In further experiments it has been observed that the TMI threshold degrades with operation time [25–27]. This, accepting the thermal origin of mode instabilities, points towards an additional heat source building up in the fiber with time. Additionally, it has been shown that this degradation of the TMI threshold can be (at least partially) reversed by thermally post-processing the active fiber [25,26]. Very recently we have reported and experimentally demonstrated that this degradation is a consequence of a photodarkening process in the active fiber . Even though the fact that photodarkening (PD) can increase the temperature in a fiber has been known for years [28,29], the strong impact that it actually has in the TMI threshold was surprising. In fact it has been shown in  that with only 7% loss of the output power due to PD, the heat load in the fiber can be doubled, thus leading to a strong degradation of the TMI threshold.
Following the experimental demonstration of the link between PD and TMI , in this paper we present a fast and simple, yet accurate, model to simulate the response of the TMI threshold affected by PD to changes in different parameters of the system such as, e.g. the signal wavelength, the seed power, the fiber core size, the fiber length, etc. At this point it is worth mentioning that other authors have tried to model the impact of PD on the TMI threshold , but in those calculations the value of the PD losses was assumed, not calculated, and PD was rudimentarily modelled as an homogeneous linear absorption term. In contrast in this work we present a way to calculate the expected PD losses at the pump and signal wavelength in active operation in Yb-doped aluminosilicate fibers, for the first time to the best of our knowledge. In our model the PD losses are spatially resolved and their value (in saturation) is predicted from the physical parameters of the active fiber.
In this manuscript we describe the model used to predict the PD losses and the TMI threshold, discuss some of its physical implications, present its predictions and compare them with experimental measurements, discuss a new and intuitive interpretation of TMI, and provide some guidelines to increase the output average power of fiber laser systems without having to significantly alter the fiber core materials. Thus, the paper is organized as follows: in section 2 we present our heuristic model to predict the PD losses in an active fiber; in section 3 we introduce the method used to calculate the MIT threshold and discuss its validity; in section 4 we compare the predictions of the model with actual experimental measurements; in section 5 we provide a new thermal interpretation of active fibers and, finally, in section 6, we provide some guidelines to significantly increase the output average power of fiber amplifier systems in the near future.
2. Calculating the PD losses in high power Yb-doped aluminosilicate fiber systems
It is widely known that the PD-losses increase with the operation time of an active fiber until a certain saturation level has been reached. This temporal increase of the PD-losses (and, therefore, of the therewith associated heat-load) explains the observed degradation of the TMI threshold [25–27]. In this work we concentrate on the final (saturated) value of the TMI threshold ignoring the temporal evolution of the degradation of this parameter. Therefore, in order to do this, we require a means to predict the saturated level of the PD losses at the pump/signal wavelength during active operation. The heuristic formula employed for this task is based on a fit of the experimental data published in two papers (for aluminosilicate fibers) [31,32]. The first one shows that there is a nearly quadratic dependence of the maximum PD losses at 633nm with the total Yb3+ ion concentration in a fiber. On the other hand, the second paper describes that the actual PD losses at 633nm are linearly dependent on the (local) population density in the upper laser level of the active fiber. Therefore, by fitting these two graphs and merging them together in a single formula we get the following expression:
where PD633nm represent the PD-induced losses (in dB/m) at 633nm, N is the total Yb3+ ion concentration in the fiber (in ion/m3) and N2is the population density in the upper laser level (in ion/m3). Thus, the factor N2/N represents the (local) relative population density in the upper laser level, and it has been normalized to a value of 0.46 because the measurements in  where done for this relative population density. The factor 8.74e25 is used to transform the ion concentration from ion/m3 to wt%. This way Eq. (1) already allows predicting the saturated PD losses in aluminosilicate fibers but only at 633nm, which is where these losses are usually measured. However, for the formula to be useful for the purposes of evaluating the TMI threshold it has to predict the PD loss at the pump/signal wavelength. Note that, for simplicity, we will assume that the PD losses are constant across the complete Yb wavelength band, i.e. from 900nm to 1100nm, this might not be strictly true but we do not expect large deviations from this approximation. According to [33,34] there is a fixed ratio between the losses measured at 633nm and the losses at ~1µm. Therefore, it can be written:
Thus, in order to predict PD in the 1µm wavelength range we only need to estimate the value of γ. This can be done, for example, by using Eq. (2) in a laser simulation and adjusting the parameter γ to the value that best fits some experimental measurements. As it will be shown in the following, we have done this by using a Large-Pitch-Fiber (LPF) [35,36] and by fitting the evolution of the mode field diameter at the output of the fiber with the output average power.
However, before the experimental fit can be presented and discussed there is an important topic that has to be addressed: the core conformation. As shown in Fig. 1, there are two conformations of the core widely used nowadays: homogeneous doping (as that obtained by MCVD process) and nano-structured doping (as obtained by the stack and draw process). Even though these two core conformations can lead to the same effective ion concentration (N), the relation between this parameter and the bulk ion concentration (Nbulk) will depend on the particular type of core in use. Thus, whereas N equals Nbulk in the homogeneous core, their value will usually differ in nano-structured cores. This is because these cores comprise a stack of Yb3+-doped glass rods (with a doping concentration equal to Nbulk) separated from one another by undoped glass. Therefore, by defining the Area Filling Factor (AFF) as the fractional area of the doped region actually covered by laser-active ions, the effective ion concentration can be calculated as .
Due to the non-linear dependence on the total (bulk) ion concentration, the different core compositions lead to different PD losses for the same effective ion concentration N. Thus, to take this into account we can modify the formulas above in the following way:
At this point it should be mentioned that Eq. (2) represents the PD losses that a mode with a 100% overlap with the doped region would see. Therefore, in order to calculate the actual PD losses that the pump/signal light experiences, the value predicted by Eq. (2) has to be multiplied by the overlap factor of the pump/signal light with the doped region.
Even though Eq. (2) focuses exclusively on the PD losses, a simulation tool using it has to consider not only the power loss but also the extra heat load that the absorbed photons (both signal and pump) generate in the fiber.
Finally, as mentioned above, in order to estimate the value of the parameter we need to compare the predictions of a simulation tool incorporating the PD loss with experimental measurements. In our case we used a 3D model that solves the spatially resolved rate equations coupled with the heat-transport equation similar to the one employed in . In this simulation tool the temperature profile modifies the index profile of the fiber which, in turn, changes the profile of the modes propagating through the fiber. Thus, this model, incorporating the extra heat load caused by PD, can be used to simulate the modal shrinking seen in some experiments, e.g. in . In this case we have used a 1.2m long LPF fiber with a core diameter of ~63µm (N = 3.5e25 ions/m3,AFF = 0.5 and pump cladding diameter ~200µm) and we have recorded the evolution of the mode field diameter (MFD) at the output end of the (counter-pumped) fiber as a function of the extracted average power. The experimental results are represented as blue dots in Fig. 2. For comparison the simulation results with PD (red squares) and without (green diamonds) considering PD (i.e. assuming only quantum defect heating) are shown. As can be seen, the results with PD fit better the experimental observations, even though it can be argued that the PD losses seem to be slightly overestimated. For this fit the value of the parameter is 24.5, which is the value that we will use in the rest of this work. At this point, however, it is important to remark that the fit value of the parameter actually depends on the simulation parameters chosen, such as e.g. the cross-sections (we have been using the ones published in ). This means that using different parameters might cast slightly different values of . In any case, the value that we have obtained falls within the range of 10-70 given by references  and .
Now it is worth discussing the general validity of the heuristic formula presented above. The formula presented in Eq. (2) predicts the saturation level of the PD losses in CW/quasi CW operation. This implies that other regimes of operation, e.g. Q-switching, might exhibit somewhat higher PD losses than the ones given by the formula. Additionally, the parameter has been calibrated for the generic aluminosilicate core composition of LPFs. Other core compositions with less PD losses will require a new calibration of the formula.
With this calibration of the parameter the expected maximum photodarkening losses at 633nm for a LPF fiber pumped at 976nm would be around 50dB/m. Additionally, in normal active operation the predicted PD losses at ~1µm for this fiber are ~1dB/m (which is relatively high due to the short length and, therefore, relatively high inversion levels that characterize LPFs). This level of PD losses leads to just a ~7% loss of amplification efficiency according to our simulations , which is in good agreement with our experimental observations that indicate an efficiency loss <10% in LPFs over time.
3. Calculating the TMI threshold
The calculation of the TMI threshold is based on the same hypothesis already proposed in : the TMI threshold is reached in one system when the grating strength grows to a certain fixed value. Under grating strength it is understood the product of the average value of the grating coupling constant (κ) times the fiber length (L). Additionally, in a grating the coupling constant is proportional to the grating amplitude .This way, for a constant device length the grating strength depends linearly on the grating amplitude. Therefore, taking into account the almost linear relationship between the grating amplitude and the average heat load in the range of average heat loads usually found in high-power active fibers (see Fig. 3), it is possible to approximate and simplify the hypothesis above as: the TMI threshold is found in one system when its average heat load reaches a certain fixed value.
It should be mentioned that Fig. 3 has been obtained by using the model already employed in  and described in , which basically couples the transversally-resolved rate equations  with the heat transport equation under consideration of mode interference. Thus, by simulating a fiber with different output average powers, the average grating amplitude in the fiber (evaluated as proposed in ) can be plotted against the average heat load, giving rise to Fig. 3.
The simplifications above allow calculating the TMI threshold directly from a model that solves the rate equations in 1D or 3D including the PD losses (and from its results estimates the heat load in the fiber). Thus, such a model only has to iterate looking for the output power that corresponds to the “threshold” average heat load (which will be determined in the next section).We have confirmed that both the 1D and the 3D approach cast similar results and, therefore, for simplicity and calculation speed, the results that will be presented in the following have been obtained with the 1D model. A further simplification that has been done is considering that the heat load profile in the fiber below the TMI threshold is dominated by the fundamental mode (LP01). This approximation implicitly assumes that the amount of HOM excited at the input of the fiber is of a few percent at most and its relative mode content does not grow significantly for output average powers below the TMI threshold (which implies that its contribution to the total heat load can be neglected in this power region).
In this section the steady-state rate and heat transport equations used for the 1D model are briefly presented. In obtaining the rate equations several assumptions were made: 1) the pump light is homogeneously distributed across the fiber cross-section and the Yb-ions are also homogeneously distributed across the doped region of the core, 2) polarization effects are ignored, and 3) monochromatic signals are considered throughout the calculations. Thus, taking into account these assumptions, the steady-state rate equations are:Eq. (3). Finally, Γpand Γs are the overlap factors of the pump and signal with the doped region, which can be expressed as follows:
In order to calculate the total heat load several contributions have to be taken into account: namely the quantum defect (QD), the PD and the propagation losses (PL)). Thus, the heat load induced by quantum defect is given by:Eq. (8) in the model for the sake of completeness, its contribution to the total heat load in our simulations can be neglected.
This way, the total heat load along the fiber is given by:
Please note that Eq. (9) provides a volumetric heat load (W/m3). In order to transform this to the linear heat load (in W/m) that will be used in the following, the result of Eq. (9) has to be multiplied by the area where the heat is generated (in this case, if the contribution of the propagation losses is neglected, this corresponds to the doped area A).
As mentioned above, the equations presented in this subsection describe the 1D model, however, they can be easily adapted to obtain a full 3D model as done in [7,8].
Due to the approximations used in the calculations there are some limitations of the model that need to be discussed. On the one hand, since only the fundamental mode is considered, the model is unable to predict the dependence of the TMI threshold on the coupling conditions. Thus, under this perspective the model gives more or less the maximum TMI threshold that could be expected from a given system. However, the actual measured TMI threshold could be lower (even significantly so) depending on the experimental conditions; but by optimizing the system (e.g. coupling conditions, coiling of the fiber, wavelength of the pump, etc.) it should be possible to approach the TMI threshold predicted by the model.
Additionally, simplifying the TMI threshold condition to just calculating an average heat load implicitly assumes that all the sections of the fiber contribute to the heat load/signal amplification. In practical terms this implies that the model is thought to be used for systems that have at least 1-2% of the pump power remaining at the output of the fiber. This condition, in practice, is valid for most high-power fiber amplifiers in use today. Should this condition not be met for a given system, then the average heat load should be calculated over an “effective” device length (approximately defined as the length in which 99% of the pump is absorbed) and not over the whole fiber length.
Another limitation of the model as presented in this work is that, strictly speaking, it can only simulate fibers with similar V-parameters/modal characteristics. This is because when changing the V-parameter the relative overlap between the fundamental mode and the HOM change which, as shown in , affects the grating strength. In practice, though, this condition is relaxed since for V-parameters>4 the relative overlap between the fundamental mode and the first HOM (i.e. the LP11, which is the usual TMI partner) converges and, therefore, the impact of a change in the V-parameter on the TMI threshold becomes weaker and weaker. Consequently, the model should not be applied for fiber designs that work near the cut-off of the HOM that interferes with the fundamental mode to create the index grating (usually the LP11). Thus, in usual cases, for fibers with V-parameters<3.5 the model might become inaccurate. However, most of the fiber designs used for high power operation have a V-parameters of at least 4.
Finally, it is also worth mentioning that the model does not regard the change of the grating strength due to saturation (as reported in [17–19] and also indirectly seen in ). However, for a given fiber length this effect is relatively weak bringing only changes of 10-30% in the threshold.
In the following we will compare the predictions of the model with experimental measurements to gauge how accurate this simplified model can actually be.
4. Comparison of the simulation results with experimental measurements
The first experiment that has been carried out is the characterization of the evolution of the TMI threshold with the signal wavelength (blue dots in Fig. 4(a) and Fig. 4(b)). These experiments were done with a 63µm core, 1.2m long LPF (N = 3.5e25 ions/m3, AFF = 0.5, and pump cladding diameter ~200µm) pumped at 976nm and seeded with ~30W input signal . The experimental data of Fig. 4(a) can be used to adjust the value of the “threshold” average heat load by carrying out simulations with different values of this parameter. For the simulations shown in Fig. 4 we have used the value of the parameter given above (24.5) and neglected the heat generated due to linear propagation losses, since it is assumed to very small. As can be seen, the vast majority of our experimental results are comprised in the corridor formed by the simulation lines corresponding to “threshold” average heat loads of 29W/m and 34W/m (shaded area in Fig. 4(a)). The scattering of the experimental data is due to the change in the experimental conditions from one measurement to the next. This scattering creates a fork in the values of the “threshold” heat load, which opens up different possibilities for the choice of this parameter. However, when estimating the highest average power that a system can operate at we propose using a “threshold” average heat load of 34W/m, which fits the highest experimental MIT thresholds presented in Fig. 4(a) very well.
It should be mentioned at this point that the exact value of this maximum “threshold” average heat load might slightly change depending on the parameters of the simulation used to match the experimental results, such as e.g. the cross-sections. However, we do not foresee deviations larger than a couple of W/m with respect to the value that we have obtained. It is important to remark that once that the “threshold” average heat load has been determined and set, no other fits or adjustments are required to predict the maximum TMI threshold at any other signal wavelength (or when changing any other parameter of the system).
As can be observed in Fig. 4(b), the measured behavior of the TMI threshold is in strong disagreement with the prediction of models that only consider quantum defect heating (green line with diamonds in Fig. 4(b)). However, when including PD in our model, the change of the TMI threshold is faithfully reproduced (solid red line in Fig. 4(b)). The simulations of Fig. 4(b) have been done using a “threshold” heat load of 34W/m, and the experimental data correspond to the highest TMI thresholds found at each wavelength. Comparing these results with the predictions of the model considering only quantum-defect heating (green line with diamonds), it can be concluded that PD causes the TMI threshold to degrade by more than a factor of 2 in this system. This is consistent with our experimental observations .
It is worth mentioning that the combination of the parameter and the “threshold” average heat load is unique and the one given here exhibits the best overall agreement for the measurements shown in Fig. 4(b) (which correspond to the highest experimental TMI thresholds presented in Fig. 4(a)). If the parameter would be higher, the value of the TMI threshold at 1030nm could still be matched using a lower “threshold” average heat load, but this would result in an overall prediction curve with a maximum shifted towards shorter wavelengths (around 1020nm). Should, on the other hand, the parameter be lower, then the TMI threshold at 1030nm could still be matched using a higher “threshold” average heat load, but this would result in a prediction curve with a maximum shifted towards the longer wavelengths and with a short wavelength edge (between 1010nm and 1030nm) that is significantly steeper than the one observed in the measurements.
At this point it should be pointed out that Brar et al.  have also done an experimental study on the dependence of the TMI threshold on the signal wavelength similar to the one presented above. However, they concentrated in the long wavelength range (λ>1050nm) which, as can be seen in Fig. 4(b), does not reveal the discrepancy caused by PD between the measurements (blue dotted line) and the expected behavior of the TMI threshold due to QD-induced heating (green line).
The hypothesis of a constant “threshold” heat load can be further tested when changing the seed power of a fiber amplifier. Figure 5 shows the measured (blue dots) and predicted (red solid line) relative change of the extracted power at the TMI threshold as a function of the seed power. The extracted power is defined as the output average power at the TMI threshold minus the seed power. Note that in Fig. 5 the extracted power has been normalized to its maximum value in the seed power range considered. The measurements in Fig. 5 were done with two different LPFs. For the low seed power experiment in Fig. 5(a) we used a 63µm core, 1m long LPF seeded at 1042nm, whereas for the high seed power experiment in Fig. 5(b) we used a 90µm core, 1.15m long LPF seeded at 1033nm. In order to best fit the experimental results a constant “threshold” heat load of 30W/m has been employed (which falls within the experimental ambiguity range determined in Fig. 4(a)).
As can be seen in Fig. 5(b), the extracted power at the TMI threshold is reduced with higher seed powers, a trend that is well reproduced by our model. More dramatic is the strong non-linear dependence of the TMI threshold with low seed powers observed in Fig. 5(a). As can be seen, for seed powers below 5W the TMI threshold decreased rapidly in this experiment. Such a non-linear dependence is adequately reproduced by our model. According to the simulation results, this behavior is caused by the loss of amplification efficiency for low seed powers in these large core short fibers. As a result the pump power has to be increased significantly to reach the TMI threshold, which leads to a strong increase of the spontaneous emission and of the PD-induced heat load originated by the absorption of pump photons. However, it is worth mentioning that, according to our model, this non-linear dependence seems to be a characteristic of short fibers, since this feature does not appear in the simulation of long (several meters) fibers.
Finally, we have applied our simulation tool to different fiber systems that have been tested by us either at the institute of applied physics (IAP) or at the Fraunhofer institute for applied optics and precision engineering (IOF) in Jena, Germany. As can be seen in Table 1, the fibers in use range from different variants of the LPF design (including a new unpublished flexible one, referred to as Bendable LPF in the table), to nano-structured step-index fibers (see Fig. 1. This fiber is referred to as ns-SIF in the table) or to a commercially available Nufern 20/400 step-index fiber (SIF). The characteristics of these fibers (listed in the table) are widely different in terms of core size, length, doping concentration, core conformation, emission wavelength and pump direction. In spite of this, applying the simulation model presented in this paper with the maximum “threshold” average heat-load proposed above (i.e. 34 W/m) casts reasonably accurate predictions for all the systems. This experimental observation is quite intriguing since, as mentioned before, it would be expected that the TMI threshold is reached when the grating strength (which is proportional to the product of the average grating amplitude times the length) is constant. However, the results in Table 1 apparently suggest that longer fibers require stronger gratings to reach the TMI threshold. Even though this point has to be analyzed in detail in a future work, we believe that what happens is that the average grating amplitude actually depends non-linearly on the fiber length due to saturation effects. Thus, the longer the fiber the weaker the grating becomes for the same output power. This effect, which has already been discussed in [17,18], would explain the dependence of the TMI threshold with the fiber length or, in other words, why the same grating strength might be reached for different output average powers depending on the fiber length.
Table 1 also suggests that the “threshold” average heat load value of 34W/m is a “constant” adequate for widely different high power fiber laser systems. However, at this point some discussion about the “constant” nature of this parameter is necessary. For example, quite likely the “threshold” average heat load will depend on the emission wavelength (possibly as a function ofλ2, as this mimics the scaling law required for index perturbations to cause the same impact on the guided modes of a fiber at different wavelengths ). Supporting this claim is the fact that thulium-doped fiber MOPA systems have already demonstrated operation with average heat loads in excess of 80W/m without reaching the TMI threshold . Additionally, as discussed above, possibly the “threshold” average heat load depends on the V-parameter if this is lower than ~3.5, since it has been shown that in this range there is a pronounced dependence of the TMI threshold on this parameter .
In Table 1 it has been assumed that all the fibers present roughly the same PD-losses. However, some experimental evidence points out that the Nufern fiber used in the experiments might show slightly lower PD losses than the LPF. Assuming that the final PD-losses in this fiber would be ¾ of those predicted by our model (this value seems consistent with independent experimental evidence), then the predicted TMI threshold using a “threshold” average heat load of 34W/m would increase from ~2.4kW to2.8kW.In this case, in order to get the same expected TMI threshold as that given in Table 1 a “threshold” average heat load of 29W/m would be required. This deviation, which falls within the experimental ambiguity range given in Fig. 4(a), might well be caused by the experimental conditions (such as non-optimum coiling or excitation) or it might indicate a slight loss in accuracy of the model for long fibers. Anyway, this still represents a relatively small change in the value of the “threshold” average heat load taking into account that the fiber length has changed by a factor of 15 with respect to the experiment with which the model was calibrated. Thus, a weak dependence of the “threshold” average heat load with the fiber length cannot be fully discarded, but this point is still highly speculative and needs future systematic experiments to be completely clarified. In any case, as it is, the proposed “threshold” average heat load of 34W/m seems to cast reasonably accurate predictions for the TMI threshold in widely different high power fiber laser systems.
5. Thermal interpretation of active fibers
Using the results of Fig. 4 and representing the evolution of the average heat load (quantum defect and total) with the signal wavelength, as done in Fig. 6, allows for a new interpretation of active fibers and the phenomenon of TMI. Under the perspective given by Fig. 6 an active fiber can be understood as a heat-load bucket with a certain capacity (34W/m in this case). This heat-load bucket can be filled with either “productive heat load” (i.e. heat load that produces signal photons, or in other words, quantum defect heat load) or with “non-productive heat load” (i.e. heat load that does not produce signal photons, e.g. PD-induced heat load). Independently of how the heat-load bucket is filled, once that it is full to its rim the TMI threshold is reached. Thus, increasing the output power beyond this point would result in an overflow of the heat-load bucket (yellow arrows) and TMI occur.
From the interpretation of TMI derived from Fig. 6 follows that the heat-load bucket can, in principle, be completely filled with “non-productive heat load”, e.g. that due to propagation losses or photodarkening. This leads to the conclusion that TMI should not be a phenomenon exclusive of active fibers, but few-mode passive fibers should exhibit TMI as well provided that the seed power is sufficiently high. This establishes a limit to the amount of average power that can be sent though a passive fiber. However, with current low loss silica glasses, it is likely that this limit is higher than the threshold of other non-linear effects.
It can be seen in Fig. 6 that the heat-load bucket (corresponding to the experiments presented in Fig. 4) is filled to at least half of its capacity with “non-productive heat load”. Therefore, in order to increase the TMI threshold a way of filling the bucket predominantly with “productive heat load” has to be found. This is something that will be in the following section.
6. Guidelines to increase the TMI threshold
Finally, the model can be used to get some guidelines to increase the output average power of fiber laser systems in the near future. Figure 7 shows the predicted evolution of the TMI threshold with the length of a SIF fiber (with AFF = 1). In these simulations the core of the fiber (30µm in diameter) has been left unchanged and the pump cladding has been adapted to always maintain the same total small-signal pump absorption in the fiber (in dB) for each fiber length. The ion concentration in the core is assumed to be N = 6e25ion/m3. Furthermore, the emission wavelength is changed with the fiber length to track the change in the spectral position of the gain maximum. Thus, for example, the solid blue line, which roughly corresponds to state-of-the-art fiber laser systems today, starts with a cladding diameter of ~125µm and an emission wavelength of 1033nm at 1m fiber length and ends with a diameter of ~500µm and an emission wavelength of 1074nm for a 17m long fiber. According to Fig. 7, using a material without PD-degradation (dashed blue line) should result in a factor ~2 increase in the TMI threshold, thus allowing to reach ~6kW diffraction-limited output average power with 17m fiber. However, the simulation shows that better results can be obtained even when considering PD degradation. The way to achieve this is to dilute the ion concentration in a way that the total number of ions in the fiber remains constant regardless of the fiber length (red and green solid lines). Please note that in this case the total small-signal pump absorption in the fiber is lower than in the previous cases because for 1m long fiber with ~125µm pump cladding the ion concentration is just N = 3.5e25ion/m3 (instead of N = 6e25 ion/m3 as it was before). By diluting the ion concentration in the proposed way, the emission wavelength does not change with the fiber length (since there is not a higher re-absorption) and, consequently, the quantum defect heating does not increase with the fiber length (as it did in the blue lines causing their non-linear shape). Additionally, since with this approach the PD-induced heating is not dominant anymore, the TMI-threshold can be further increased by blue-shifting the emission wavelength of the signal (green line) .Thus, following this strategy~10kW diffraction-limited average power can be expected from a 17m long fiber. The price that has to be paid with this approach is that the seed power has to be increased with the fiber length (in the simulations it grows from 10W for 1m to 170W for 17m) and that the dimensions of the cladding remain constant at ~125µm regardless of the fiber length/output average power. Therefore, in order to achieve the expected performance levels high-brightness pump diodes or tandem pumping  schemes become necessary.
In this work we have shown that the behavior of the TMI threshold can be adequately simulated when considering that it is reached for a constant value of the average heat load. Comparison with experimental measurements indicates that this is a good approximation that holds for most high-power fiber laser systems. Furthermore, we have shown that in order to faithfully model the behavior of the TMI threshold it is necessary to incorporate the impact of PD in the simulation tools. Thus, for the first time to the best of our knowledge, we present a way to calculate the expected PD losses at the pump and signal wavelength in active operation in Yb-doped aluminosilicate fibers. In our model the PD losses are spatially resolved and their value (in saturation) is predicted from the physical parameters of the active fiber. The heuristic approach presented in this work has been compared with experimental measurements resulting in a good match between the predicted and the measured results.
A comparison of the experimental data with our simulations shows that a constant “threshold” heat load condition of 34W/m adequately predicts the TMI threshold in widely different systems. This approach not only allows developing simple and fast simulation tools to predict the TMI threshold, but also leads to a novel interpretation of TMI. In this interpretation, a fiber can be understood as a heat-load bucket with a capacity of 34W/m. This heat-load bucket can be filled either with “productive” or with “non-productive” heat load, but once that it is full TMI occur. Therefore, in order to increase the TMI threshold ways to fill the heat-load bucket predominantly with “productive” heat load have to be found. This implies reducing the PD-losses (or any absorption term) in an active fiber. One way of doing this is using core active materials with intrinsically lower PD losses. Another more immediate way is proposed in this work and is based on significantly diluting the ion concentrations used in active fibers. However, only the development of high brightness pump diodes or tandem-pumping schemes will allow accessing the expected performance gains.
According to our simulation model output average powers in excess of 10kW with nearly diffraction limited beam quality could be achieved with a new generation of optical fibers in the near future.
This work has been supported by the German Federal Ministry of Education and Research (BMBF), project no. 13N11972 (PT-VDI, TEHFA) and by the European Research Council under the ERC grant agreement no.  “ACOPS”. The authors also want to thank the Fraunhofer institute Jena (IOF) for providing some of the experimental data.
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