1. Introduction

The remarkable exponential evolution of the average output power of fiber laser systems seen over the last 20 years [1] came to a sudden halt around 2010 coinciding with the first observations of transverse mode instabilities (TMI) [2,3]. This effect, which is the first example of a thermally-induced non-linearity in optical fibers, refers to the sudden degradation of the beam quality of the light emitted by a fiber laser system observed once that a certain average power threshold has been reached. Above this average power level the output beam profile, which fluctuates on a millisecond time scale [4], clearly evidences the presence of higher-order modes [5]. Even though it is possible to operate well above the TMI threshold [6], one of the most sought-after characteristics of fiber laser systems, namely their power-independent nearly diffraction limited beam quality, is lost, which can be a severe restriction for many applications. Therefore, since the first reports on TMI, there has been an intense research effort worldwide focused on understanding and mitigating this effect. Such an effort has led to a fast progress in the understanding of the physics behind TMI [7–9]. According to the current understanding of TMI, the interference of two transverse modes in a few-mode fiber creates a modal interference pattern that gets inscribed in the inversion profile of the active fiber. This, in turn, generates a temperature profile with periodic features that, via the thermo-optic effect, gives rise to a long period grating that can potentially lead to the energy exchange between the interfering modes. This understanding of the underlying physics has crystallized in the development of several numerical [10–13] and semi-analytical [14,15] models that allow simulating TMI to gain a deeper insight into this effect. However, since some of these models are based on the beam propagation method (BPM), which cannot easily handle counter propagating signals, and others make some strong simplifications on the evolution of the signal amplitude along a fiber amplifier, the theoretical analysis of TMI has been limited to single-pass fiber amplifier systems so far. Despite this limitation, this analysis has already led to the proposal [16–18] and experimental demonstration [19–21] of several mitigation strategies for TMI.

Very recently we have revealed the unexpectedly high impact that photodarkening (PD) [22] has on the TMI threshold [23] of Yb-doped fiber laser systems. In this previous work it has been shown that the presence of PD not only results in a significant reduction of the TMI threshold, but it also modifies its behavior, which explains some of the observed discrepancies between the experimental measurements and the theoretical predictions, particularly those regarding the evolution of the TMI threshold with the signal wavelength [16,24,25]. Discovering this link between TMI and PD has allowed determining that the TMI threshold, in a very good approximation, is reached in a fiber amplifier system by a fixed average heat load. Based on this experimental observation we have found out that at the moment of reaching the TMI threshold the average heat load in a wide variety of fiber laser systems is around 34 W/m. This insight has allowed developing a simplified way of calculating the TMI threshold in fiber laser systems affected by PD (in alumino-silicate glass and once that the saturation of the PD losses has been reached) [26]. The predictions of this model, in spite of its simplicity, have shown a remarkable agreement with the experimental measurements and have revealed that even small amounts of PD-induced losses can double the heat-load in the active fiber.

In light of the new findings, the most straightforward approach to increase the average output power of laser systems based on Yb-doped silica fibers would be to use novel active materials that do not exhibit an increase in heat load due to PD. Even though this is a highly desirable goal, those materials are, to the best of our knowledge, not available yet and the progress in that front is steady but slow. Therefore, this approach can be considered as a mid- to long-term solution. In contrast, in the present work we provide some short-term guidelines to optimize the output average power of fiber laser systems in the presence of TMI and PD, without the need for a change in the composition of the active material. Guidelines for the design of high-power fiber amplifier systems will be given in the following. Note that some design guidelines, in particular those concerning the seed power and signal wavelength, have been presented elsewhere [23,26] and will not be further discussed herein. Therefore, in the following the dependence of the TMI threshold on parameters such as the core conformation, the ion concentration and the pump configuration (in terms of wavelength as well as propagation direction) will be studied. As already mentioned above, in order to carry out this study the simplified model presented in [26] will be used. This model has been calibrated for gain saturation levels typically found in the high power/low gain booster fiber amplifier stages found in high energy pulsed systems. Since the model does not include the impact of gain saturation on the transverse profile of the thermally-induced index grating, its predictions will lose in accuracy if the saturation level of the simulated amplifier significantly departs from the ones used in the calibration. Thus, the model has been extensively tested and calibrated with ~1m long large-pitch fibers (LPFs) [27], and even though it also seems to cast good results with other fiber designs and lengths [26], we will be mostly simulating fiber amplifying systems based on LPFs in this paper to minimize the impact of gain saturation in our results. In spite of this, the general guidelines and trends shown in the following for LPFs should be equally applicable to other fiber designs (although the resulting gain margin in terms of TMI threshold might be different to the one presented here).

The paper is divided into five sections. The first one details the contribution of the pump radiation to the thermally-induced index grating responsible for TMI. The second one is devoted to the impact of the core conformation (i.e. core size, area filling factor of the active ions, and doping concentration) on TMI. The third one analyzes the influence of the pump wavelength and the fourth one deals with the repercussion of the pump direction on the TMI threshold. At the end of the manuscript, the guidelines are summarized and conclusions are drawn.

2. Contribution of the PD-induced heat load to TMI

In our previous work [26], we have shown that the TMI threshold in a LPF amplifier significantly drops when the seed power is reduced. One possible explanation for this observation if the reduction of gain saturation [17], however the drop in TMI threshold seems stronger than the one expected exclusively from gain saturation effects. Interestingly, the simplified model used by us is able to accurately reproduce the measured evolution of the TMI threshold when taking into account the extra heat load caused by the PD-induced absorption of pump photons. Thus, it is intriguing to explore the theoretical possibility pointed out by our model of the pump radiation being able to contribute to TMI. This is the subject of this section.

2.1. Static contribution

As has been shown in [23] and [26], PD generates (through the absorption of pump and signal photons) a significant amount of heat that can be comparable to that induced by quantum defect. However, this extra heat load can only lead to a degradation of the TMI threshold if it is transversally inhomogeneous. Moreover, this heat load term has to be radially anti-symmetric to allow for the energy transfer between the FM and the first HOM (which is the usual case observed in the experiments). Seeing that the extra heat load term caused by the PD-induced absorption of signal photons exhibits such characteristics is straightforward since that symmetry is inherent to the modal interference pattern that ultimately gives rise to the thermally-induced index grating [9] responsible for TMI.

This symmetry condition, on the other hand, might seem difficult to achieve with a transversally homogeneous pump radiation. However, it can nonetheless be satisfied because the distribution of the PD losses (i.e. of the PD-induced pump absorption) mimics the three-dimensional profile of the population inversion (since the PD losses depend linearly on the local level of population inversion [26]). Additionally, as shown for example in [1], it is known that the distribution of population inversion, being created by the modal interference pattern, exhibits always the right symmetry (and periodicity) to allow for the energy transfer between the interfering fiber modes. Consequently, the distribution of PD losses in the fiber core (and, with it, the extra heat-load caused by the PD-induced absorption of pump photons) will also have the symmetry required for this energy transfer. This can be observed in Fig. 1. The figure illustrates the contribution of the PD-induced absorption of pump photons to the heat load in the fiber (in an 80µm core, 1.2m fiber similar to that used in [9]). As can be seen, and as it has been mentioned above, the profile of this contribution to the heat load mimics that of the inversion distribution.

 

Fig. 1 Normalized contribution to the heat-load caused by the PD-induced absorption of pump photons.

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However, even though PD is a dynamic process that reaches a well-defined equilibrium state after a certain time [28], the characteristic time constants of PD are usually orders of magnitude longer than those observed in TMI. This means that the grating contribution depicted in Fig. 1 is quasi-static and cannot follow the movement of the signal intensity profile during TMI. Such a quasi-static PD-loss distribution is generated by exciting different transverse modes with the same frequency content, as usually occurs in high-power fiber amplifier systems. Even though such a static index grating may influence the TMI threshold by causing a large instantaneous local energy transfer between the fiber modes (when a moving intensity pattern shifts through it), its average net contribution to the energy transfer is very low.

Therefore, if the pump power contributes to the TMI threshold as our experiments seem to indicate [26], a different mechanism able to generate a pump-induced moving asymmetric heat load pattern must be at work.

2.2. Dynamic contribution

To identify such a mechanism we have performed a simulation of a 1cm long piece of air-clad fiber with 100µm pump core diameter using the beam propagation method (BPM). Distributed along the fiber in a periodic, radially asymmetric pattern there are different absorbing sections (deep blue regions in the imaginary part of the refractive index profile shown in Fig. 2). Note that at this point of the explanation we disregard the physical origin of such absorbing sections. This point is discussed later on. In order to illustrate the effect we are looking for in the clearest possible manner, we have set the absorbance of these regions to 30dB/mm. The pump (at 976nm) is modelled at the input facet of the fiber as having a flat electric field amplitude profile but a random spatial phase, which gives rise to speckle and leads to an almost even excitation of modes in the fiber. The result of the simulation can be seen as a movie online (Visualization 1). Figure 2 shows, in its central part, a longitudinal section of the distribution of absorbing elements and of the evolution of the pump intensity along the fiber. Additionally two frames of the movie showing the transverse distribution of absorption (left) and pump intensity (right) at two different positions along the fiber (one including an absorbing element –uppermost inset- and one without it –lowermost inset) are also presented.

 

Fig. 2 Generation of an asymmetric pump intensity profile in a fiber with transversally asymmetric absorption sections. The upper- and lowermost insets are two frames of a movie (Visualization 1) showing the imaginary part of the fiber cross-section (left) and the pump intensity distribution in the fiber (right) at two different positions along the fiber. The middle section of the image shows a longitudinal section of the imaginary part of the refractive index (up) and of the pump intensity distribution along the fiber. The dark blue areas of the imaginary part of the refractive index represent sections of high absorption. As can be seen, these get imprinted in the pump intensity distribution generating a transversal asymmetry in it (as highlighted by the yellow arrows).

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As can be observed, the absorbing sections of the fiber get imprinted in the transverse intensity profile of the pump (as indicated by the yellow arrows), thus giving rise to a transversal asymmetry in the local pump distribution. In a real active fiber, transversally asymmetric absorbing sections are generated by the transversally inhomogeneous distribution of inversion (created by the intensity interference pattern of the beating fiber modes - see e.g [9].), which, in analogy to the simulations presented herein, will give rise to a locally asymmetric (and quasi-periodic) pump intensity distribution along the fiber which resembles the negative of the inversion distribution.

At this point, we will consider that there are, additionally, constant losses present in the fiber (e.g. PD losses after reaching the equilibrium state). Note that in the following explanation the loss distribution will be considered constant in time. These losses will cause the absorption of pump and signal photons thus increasing the heat load in the fiber, as discussed elsewhere [26]. The most unfavorable case for the generation of a thermally induced index grating is when these losses are homogeneously distributed in the fiber core, i.e. when they can be considered as a kind of homogenous background losses. However, even in this most unfavorable case, the absorption of the transversally inhomogeneous pump intensity profile generated by the inversion distribution (see paragraph above) will give rise to a transversally asymmetric heat load pattern in the fiber.

Additionally, as several other works have shown [8,29], during TMI the signal interference pattern moves along the fiber. Moreover, since the inversion can be depleted/rebuilt in a µs time-scale in a high-power fiber amplifier, the inversion grating can follow the signal intensity pattern as it moves. Consequently, the pump asymmetry (caused by the instantaneous inversion distribution) will also follow the movement of the signal intensity pattern, thus giving rise to a moving transversally inhomogeneous heat load generated through the absorption of pump photons by e.g. the background PD-losses. This explains how this extra heat load caused by the pump radiation can indeed lead to a reduction of the TMI threshold.

It should be mentioned that the contribution of the pump to the total heat load through PD-induced absorption is, in most cases, expected to be small. However, it can grow significantly if the fiber amplifier operates in an inefficient regime or if the dimensions of the pump cladding are reduced, as it will be shown below. It is fair to say, though, that when this effect becomes relevant is usually when the gain saturation in the fiber is reduced. Since both effects (PD-absorption of pump photons and lower gain saturation) are expected to lead to a reduction of the TMI threshold, it is difficult to isolate their effect in a measurement. Therefore, for the time being we cannot say with certainty how strong the effect of the pump absorption is in comparison to the impact of gain saturation. In spite of this, we believe that it is important to raise awareness over a new effect that might contribute to a reduction of the TMI threshold.

Please note that in the following discussions we will be referring to the effect of the PD-induced pump absorption as a cause of sudden reductions of the TMI-threshold. Since, as mentioned above, when this effect becomes relevant the gain saturation is also reduced, our model is expected to lose accuracy in these cases. This means that in those situations we will only do a qualitative analysis of the simulation results. However, the general trend of the evolution of the TMI threshold predicted by our model should still be correct and the TMI threshold is expected to significantly drop in the situations indicated by the model.

3. Core conformation

The term core conformation encompasses all the parameters that describe the active core of an optical fiber which include the core size, the index step, the ion concentration and the way that the fiber core has been doped. In our previous work [26] we characterized this last property with the so-called area filling fraction (AFF), which is the fractional area of the doped region that has been covered with laser-active ions (see Fig. 3). The relevance of the AFF becomes apparent when taking into account the different ways in which an active core can be fabricated. Some classic methods such as the modified chemical vapor deposition (MCVD) [30] tend to result in a (usually homogeneous) distribution of the ions that completely fills the doped region, thus typically resulting in AFF = 1. On the other hand, fabrication techniques based on stack and draw [31] result in nano-structured cores in which many lumped doped regions are separated from each other by undoped glass. Therefore, in this kind of cores AFF is usually <1.

 

Fig. 3 The two most extended types of core conformation today in Yb-doped fibers: filled (left-hand side) and nano-structured (right-hand side). The fractional area of the doped region that has been covered with Yb-ions is called area-filling factor (AFF).

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In the examples presented in this work, we will always assume that the doped radius is 84% of the core radius and that the Yb-ions have been “homogeneously” distributed across this doped region, albeit with different AFFs. Since the doped area will not be changed, the impact of preferential gain on the TMI threshold will not be studied, but this is a topic that has been discussed in detail by other authors [13,14].

Unless otherwise stated, all the simulations will be done for the same test fiber which is a 1.2m long large-pitch fiber (LPF) [27] with 35µm pitch and pumped at 976nm in the counter-propagating direction. For simplicity this fiber is simulated as a 1.2m long step-index fiber (SIF) with a 63µm core diameter (~58µm mode field diameter -MFD- in the fundamental mode), ~50µm doped diameter, ~200µm pump cladding diameter, an AFF = 0.5 and a bulk ion concentration (i.e. the ion concentration of each one of the lumped doped regions) of 7*10²⁵ ions/m³. The fiber is seeded with 30W at 1030nm.

3.1. Mode-field diameter

One of the first decisions that must be made when designing a fiber is the size of its core. This greatly determines the ability of the fiber to amplify high energy pulses, but also, indirectly it may determine its length since fibers with large cores have to be coiled with large radii and, in extreme cases, they even have to be kept completely straight (e.g. as in rod-type fibers), thus limiting their practical lengths to just a few meters. Being this such an important parameter, it is extremely interesting to investigate the dependence of the TMI threshold on the core size or, equivalently, on the MFD of the fundamental mode guided in it. Figure 4 shows the predictions of our model when changing the fiber size (both core and cladding have been proportionally scaled to keep the pump absorption constant). The model assumes the same modal overlap of the fundamental mode with the doped region independently of the core size, which is equivalent to assuming a constant V-parameter. At first sight it might seem counter-intuitive to be able to scale the core size while keeping the guiding characteristics of the fiber constant (i.e. the V-parameter in SIF), but this is one of the remarkable features of LPFs [32]. As can be seen in Fig. 4, under these circumstances our model does not predict any significant dependence of the TMI threshold on the core size, something which is in good agreement with the results presented by other authors [14,33]. The residual change of TMI threshold with the MFD that can be observed in Fig. 4 is mostly due to the numerical accuracy of our model.

 

Fig. 4 Dependence of the TMI threshold on the MFD of the fundamental mode of a fiber which size is increased while keeping the V-parameter constant.

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3.2 Area-filling factor

Another design parameter of interest, in particular for active fibers with nano-structured cores, is the AFF. The AFF is usually employed to help matching the active region of the core to the surrounding silica matrix (typically in large-core photonic crystal fibers), so it is important to study whether this parameter has a strong influence on the TMI threshold, as done in Fig. 5(a). It is worth noting that in these simulations the bulk ion concentration was left unchanged (7·10²⁵ ions/m³) but the pump cladding diameter was adjusted to obtain the same small-signal pump absorption in each case. As can be seen, the TMI threshold has only a weak dependence on the AFF decreasing by 15% between AFF values of 0.2 and 1. This is because even though the overlap of the signal with the region exhibiting PD is reduced linearly with lower AFF, the amount of inversion required to reach the same output power (i.e. the same gain) increases. This, in turn, results in higher PD losses (averaged over the fiber length). Thus, since the PD losses depend linearly on the product of the relative population in the upper laser level and the AFF [26], they increase only slightly for higher AFFs which results in the weak steady drop of the TMI threshold for AFF>0.3 (Fig. 5(a)).

 

Fig. 5 Dependence on the AFF of: a) the TMI threshold and PD-induced heat load for the signal (red) and pump (green), and b) pump power required to reach the TMI threshold in each case.

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This explanation above might seem in contradiction with the behavior of the TMI threshold observed for AFF<0.25, where a steep drop can be seen in spite of the PD losses becoming lower. In order to understand this effect it has to be considered that for lower AFFs the number of laser-active ions in the fiber is reduced. This ultimately limits the gain in the fiber, which has a negative impact in the amplification efficiency, since in order to get the same output power the pump power has to be progressively increased (to compensate for the lower number of active ions with a higher inversion level). As can be seen in Fig. 5(b) the increase of pump power for AFF<0.25 becomes quite dramatic, which significantly increases the extra heat-load caused by the PD-induced absorption of pump photons (black dotted line in Fig. 5(a)), as explained in section 2.

3.2 Active ion concentration

Finally, another parameter that defines the active core is the laser-active ion concentration. As we have demonstrated previously in [26], due to the quadratic dependence of the maximum PD losses on this parameter, its impact on the TMI threshold is expected to be very high. In Fig. 6 we analyze this dependence in more detail and in conjunction with the AFF. Figure 6 shows the predicted change of the TMI threshold as a function of the AFF and the bulk ion concentration (N_bulk), where this parameter represents the Yb-ion concentration in each one of the lumped doped regions of the core. In order to ensure the comparability of the results the diameter of the pump cladding has been adapted in each case to obtain the same small signal pump absorption. As can be seen, and in agreement with the results presented in [26], as a general trend lower ion concentrations result in higher TMI thresholds (at least for effective ion concentrations N>1·10²⁵ions/m³, with $N=N_{bulk}\cdot AFF$). For example, for AFF = 0.5 it can be observed that reducing N_bulk by a factor of three (i.e. from 10·10²⁵ions/m³ to 3·10²⁵ions/m³) results in roughly a twofold increase of the TMI threshold. This trend is maintained until the active medium becomes so diluted that the amplification efficiency drops due to the very low number of ions in the medium. This, as already discussed for the change in AFF in Fig. 5(a) (which illustrates a cross-section of Fig. 6 along the horizontal dashed line), leads to a reduction of the TMI threshold at low ion concentrations.

 

Fig. 6 Dependence of the TMI threshold on the bulk ion concentration (N_bulk) and the AFF. The isolines represent fiber designs with the same effective ion concentration ($N=N_{bulk}\cdot AFF$) and, therefore, with the same geometrical dimensions. The labels in each isoline provide the corresponding value $N/{10}^{25}$ . The horizontal dashed line represents the cross-section corresponding to Fig. 5(a).

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Additionally, it can be seen that higher AFFs result in lower TMI thresholds for N_bulk>4·10²⁵ions/m³. However, interestingly, that trend is inverted for low ion concentrations. This is because with higher AFFs the amplification efficiency does not drop so much for low N_bulk. Therefore, the benefits of reducing the PD losses with lower N_bulk can be stretched more with higher AFFs, thus resulting in higher TMI thresholds. As can be seen, in Fig. 6 we have also plotted some isolines. These represent fibers with the same effective ion concentration (N), which also implies fibers with the same geometrical dimensions. The labels of each one of the isolines correspond to the particular value of the effective ion concentration normalized to 10²⁵ ions/m³. Thus, the most interesting design guideline can be extracted from Fig. 6 when following these isolines. As can be seen, for a given fiber geometry/effective ion concentration it is always better to reduce the bulk ion concentration and increase the AFF than the other way around.

At this point it is worth mentioning that our colleagues from the Fraunhofer IOF in Jena have recently drawn a fiber following the guidelines for the core conformation provided above and they have obtained a pump-limited narrowband output average power of 3kW out of a 30m long step-index fiber with 24.5µm core diameter [34]. This result represents a significant increase (>50%) of the TMI threshold with respect to previous attempts with non-optimized fiber designs.

4. Pump wavelength

There have been some reports showing that detuning the pump wavelength from 976nm to lower wavelengths can result in higher TMI thresholds [35,36]. Even very recently more than 3kW output average power have been obtained by pumping a monolithic fiber laser system at 915nm [37]. In this section the impact of detuning the pump wavelength will be considered in detail in an attempt to learn when such a strategy can be advantageous.

In order to study the impact of the pump wavelength on the TMI threshold we have performed several simulations. In the first one, represented in Fig. 7, to isolate the effect of the pump wavelength, the fiber length has been fixed at 1.2m but the pump cladding diameter (red line in Fig. 7) has been modified for each pump wavelength to ensure the same small-signal pump absorption. If no PD would be considered, the TMI threshold would roughly increase with a reduction of the quantum defect (since we have a constant signal wavelength of 1030nm). However, when PD is present this behavior is modified and a maximum of the TMI threshold is reached around 976nm. This is mainly because, in order to compensate for the lower absorption cross-sections, the pump cladding diameter has to be made smaller for pump wavelengths detuned from 976nm. This, in turn, increases the overlap of the pump with the doped region leading to significantly increased PD-induced losses at the pump wavelength. This, consequently, leads to a higher contribution of the pump to the PD-induced extra heat load and to a reduction of the TMI threshold. On top of that, for pump wavelengths much shorter than 976nm the quantum defect increases significantly, which further contributes to lowering the TMI threshold.

 

Fig. 7 Dependence of the TMI threshold on the pump wavelength for a fixed fiber length of 1.2m with PD (blue solid line). The pump cladding diameter (red line) has been changed to achieve always the same small-signal pump absorption.

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The existence of a maximum of the TMI threshold for a pump wavelength of 976nm seems to contradict the experimental evidence presented by other authors. In particular in [36,37] they report an increase of the TMI threshold when shifting the pump wavelength from 976nm down to 970-972nm without changing the active fiber. However, when analyzing these results in detail it can be seen that, in both cases, the effective length (i.e. the length where the amplification takes places, roughly defined as the length in which 99% of the pump has been absorbed) for the 976nm was much shorter than the actual fiber length. In other words, the fibers were too long for the 976nm pump. This implies that, for the amplification process, the effective fiber lengths in those experiments were different for the various pump wavelengths in use. In order to simulate this situation we have fixed the dimensions of the core and cladding and we have adjusted the fiber length (red line in Fig. 8) to obtain the same small-signal pump absorption in all cases. Note that herein we assume that the threshold condition of an average heat load of 34W/m is independent of the fiber length. In fact, as discussed in [26], even though our model casts reasonable results for long fibers (~15-20m), a progressive loss of accuracy cannot be ruled out in those cases. Therefore, the results presented in Fig. 8 will be only interpreted from a qualitative point of view. In this case, as can be seen, in agreement with the experimental evidence, there is a minimum of the TMI threshold around 976nm (blue solid line in Fig. 8). Thus, under these circumstances our simulations predict that detuning the pump wavelength should lead to higher TMI thresholds. However, this increase of the TMI threshold has more to do with the increased length over which the amplification takes place than with a reduction of the PD-induced heat load. This is not a surprise since it has been shown before [16,17,26] that increasing the fiber length leads to higher TMI thresholds. The results in Fig. 8 are in good qualitative agreement with those presented in [33].

 

Fig. 8 Dependence of the TMI threshold on the pump wavelength for a fiber with 63µm/200µm core/clad diameter when considering PD (blue solid line). The fiber length (red line) has been changed for each pump wavelength to achieve always the same small-signal pump absorption.

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5. Pump direction

When designing a fiber laser system it is necessary to ponder over the trade-off associated with choosing the pump direction. For example, for practical reasons many monolithic all-fiber laser systems opt for co-propagating pump configurations in the last stage. However, this configuration increases the impact of non-linear effects. In this context, it is necessary to analyze whether the pump direction also influences the TMI threshold. Note that a similar study has also been done in [38], but in that work the authors did not consider the impact of PD, which is the main focus of the simulations presented herein. We have carried out this study for the test LPF described above and the results are presented in Fig. 9, which shows the evolution of power along the fiber amplifier for the case of co-propagating pump (red lines), bi-directional pump (green lines) and counter-propagating pump (blue lines) configurations with (solid lines) and without (dashed lines) PD. As can be seen, our model predicts that the impact of the pump configuration is nearly negligible in the absence of PD. This result is expected since the model does not take into account the effect of a change in gain saturation on the TMI threshold. It has been shown in [39] that gain saturation breaks the symmetry between co- and counter-propagating pump configurations, thus resulting in slightly different thresholds. Nevertheless, this effect usually amounts to ~10-30% difference in the TMI thresholds. However, not taking the impact of gain saturation into account allows us isolating the raw effect of PD. Thus, as can be seen in Fig. 9, the TMI threshold in the co-propagating pump case is the lowest when PD is considered, whereas the TMI threshold for the counter-propagating case becomes the highest (with the one corresponding to the bi-directional pump configuration being somewhere in-between). The reason for this behavior has to do with the fact that in the co-propagating pump configuration the highest pump power is located where the lowest signal power is, thus resulting in very high local inversion levels and, therefore, PD losses which, in turn, lead to a higher PD-induced heat loads.

 

Fig. 9 Dependence of the TMI threshold on the pump direction. The plot represents the evolution of the signal power along the test LPF for co-propagating pump (red lines), bi-directional pump (green lines) and counter-propagating pump (blue lines) configurations with (solid lines) and without (dashed lines) PD.

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Even though the TMI threshold difference between the co- and counter-propagating pump configurations seems small in Fig. 9, this difference can become very significant in other fibers. Table 1 provides the simulated and measured results obtained for an un-published bendable version of the LPF (~34µm core diameter, ~25µm doped diameter, ~370µm pump cladding diameter, AFF = 0.6, N_bulk = 7·10²⁵ ion/m³) with two different fiber lengths (5.5m and 6.5m) and seeded with 15W at 1038nm. As can be seen, our simulation predicts the TMI threshold for the 5.5m long fiber in the co-propagating pump configuration quite accurately. The model also predicts a very significant increase of the TMI threshold when switching the pump configuration from co-propagating to counter-propagating. Unfortunately, our 6.5m long fiber was repeatedly destroyed by fiber fuse at a power of 860W in the counter-propagating direction and, therefore, we could not really measure the TMI threshold in this configuration. However, until that output average power no signs of TMI were observed in the system. Anyways, what these measurements indicate is that in this system the TMI threshold with a counter-propagating pump is significantly higher than with a co-propagating one, as predicted by the model.

Table 1. Measured and expected TMI thresholds for the bendable LPF

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6. Conclusions

In this paper we have studied the dependence of the TMI threshold on different parameters and configurations of an alumino-silicate Yb-doped fiber amplifier system in the presence of PD (after saturation of the PD losses). The dependence of the TMI threshold on structural fiber parameters such as the core size, the AFF and the ion concentration has been analyzed and discussed. Additionally, the impact of the pump wavelength and the pump direction in fiber amplifiers has been presented and compared with experimental data.

From all these results, several guidelines for the optimization of the output average power of fiber amplifiers can be derived. First of all, the best strategy to increase the TMI threshold is to reduce the PD losses by decreasing the ion concentration in the fiber core. However, in order to reach sufficiently low ion concentrations in nano-structured fibers the AFF has to be increased (which may impact on the ability to match the index of refraction between the doped and undoped sections of the core). Additionally, detuning the pump wavelength from 976nm (especially towards longer wavelengths) can lead to higher TMI thresholds if the fiber length is increased. Besides, it has been shown that a counter-propagating pump configuration leads to higher TMI thresholds.

We believe that these guidelines should be useful to significantly increase the output average power of fiber amplifier systems in the near to medium future.