1. Introduction

Since the first demonstration of laser operation by Maiman in 1960 [1], the history of laser development has been characterized by the search of solutions to the different limitations that have arisen as the output power has been increased. Thus, the rod geometry of the first lasers was almost abandoned in favor of the slab [2], thin-disk [3] or fiber [4] geometries in an attempt to overcome the thermal limitations shown by these initial lasers as the output power was increased. Focusing the discussion in fiber lasers, the introduction of the double clad concept to overcome the limitation of the lack of high-power high-brightness pump sources, led to an unprecedented growth of the output power [5]. However, as the power was increased a new limitation in the form of non-linear effects [6] (in particular stimulated Brillouin scattering and stimulated Raman scattering) started to plague fiber laser systems. This led, over the years, to the development of several mitigation strategies. One of the most successful in the context of active fibers was the introduction of large-mode area (LMA) fibers [7], which mitigate non-linear effects by reducing the intensity of the light in the core. The constant strive towards higher peak powers has driven the development of this technology to the point that now a new generation of very large mode area (VLMA) fibers is readily available [8–10]. These VLMA fibers are characterized by core diameters >50µm and by relatively short lengths (which further mitigate non-linear effects). However, this impressive development in the core area, and the related increase in the output peak power and pulse energy of fiber laser systems, has come at the cost of multimode operation. This, however, has not implied a loss of beam quality since all new VLMA fiber designs [8–10] have in-built mechanisms that ensure the preferential amplification and/or excitation of the fundamental mode.

However, the multimode nature of LMA and VLMA fibers (unavoidable with today’s technology), combined with the large powers being extracted from these fibers, has led to the appearance of a new limitation: mode instabilities. This new limitation is different from conventional non-linear effects in the sense that it sets a limit to the maximum average power of the system and not to its peak power. The effect, which was reported for the first time three years ago [11], sees the stable high-quality beam of a fiber laser system becoming suddenly unstable once that a certain average power threshold has been reached. Shortly after the first observations of the effect were published, it was proposed that a thermally/inversion-induced index grating caused by modal interference along the fiber [12] could be responsible for the observed energy transfer between the fundamental mode and higher-order modes (HOM). Such a grating has automatically the right period to enable the coupling of the interfering modes. Later on, it was pointed out that the right period alone does not suffice for energy transfer to take place between the interfering modes, but a phase shift between the modal interference intensity pattern and the index grating is also required [13]. At this point two theories have been proposed to explain the origin of this phase shift: one of them relies on the fundamental mode and the higher order mode interfering with it having slightly different frequencies [13], whereas the other one is based on the initially quasi-static thermally induced waveguide changes becoming non-adiabatic [14]. Even though both theories agree that the energy transfer is caused by a thermally-induced grating, the physical processes triggering mode instabilities are quite different in them. Thus, the first theory is a periodic steady-state solution, whereas the second one requires the existence of a transient (e.g. increasing the pump). The physical feasibility of both processes has already been demonstrated with complex simulation models [15, 16].

In the relatively short time elapsed since the first reports on mode instabilities, this effect has become the most limiting phenomenon for a further scaling of the output average power of fiber laser systems. Thus there is an urgent need for mitigation strategies that allow for a significant increase of the mode instability threshold.

Mitigation strategies can be classified, according to Table 1, into passive and active. Passive mitigation strategies are those that do not require any actuation/control on the system during operation, whereas active mitigation strategies are those requiring such actuation/control. Additionally, passive mitigation strategies can be classified into intrinsic and extrinsic. Intrinsic mitigation strategies are those that involve changes in the fiber design to increase the threshold while leaving the rest of the system untouched. On the other hand, extrinsic mitigation strategies are those that, given a particular fiber, achieve the increase of the threshold by means of modifications of the rest of the system. Furthermore, as illustrated in Table 1, all active mitigation strategies are necessarily extrinsic ones. An example of an active mitigation strategy has recently been presented in [17]. Here the beam fluctuations characteristic of mode stabilities for average output power levels above the threshold are stabilized with a dynamic excitation of the modes at the input of the fiber (carried out with an acousto-optic deflector). With this technique the beam could be stabilized up to power levels three times higher than the mode instability threshold.

Table 1. Classification of mitigation strategies for mode instabilities

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In this paper we propose two passive mitigation strategies of mode instabilities: one extrinsic and one intrinsic. In order to theoretically evaluate the expected increase of the threshold, we introduce a semi-analytic formula. This simple formula, in contrast to others already published based on the coupled mode theory [18], relies on analyzing the strength of the thermally-induced waveguide changes.

The paper is organized as follows: in section 2 the semi-analytic formula used for predicting the behavior of the mode instability threshold is presented and the theoretical premises in which it is based are discussed. In section 3 an extrinsic mitigation strategy based on reducing the heat load in the active fiber is proposed. In section 4, an intrinsic mitigation strategy based on reducing the pump absorption is presented and discussed. Finally some conclusions are drawn.

2. Semi-analytic formula for the evaluation of the mode instability threshold

At this point there are already several numerical tools for modeling mode instabilities [14–16]. However, these models, even though very general in their scope, are very complex and computationally intensive. Therefore, using them to evaluate mitigation strategies for mode instabilities can be very inefficient due to the long computation times involved. Consequently, we believe that a simple and quick way to calculate the mode instability threshold is required, even at the price of sacrificing some accuracy.

In the semi-analytic formula that will be proposed in the following the physics of the amplification process are modeled in detail whereas there is a certain level of abstraction from the physics behind the origin of mode instabilities. On the contrary, other authors have chosen to model in detail the process of energy transfer between transverse modes while abstracting themselves from the laser physics [18]. However, the strength, shape and evolution of the thermally-induced grating below the threshold are mainly determined by the amplification process. Therefore, we believe that not taking into account the detailed physics of this process limits the ability of other formulas to analyze and find mitigation strategies for this effect.

2.1. Theoretical considerations

When injecting light into a LMA or VLMA active fiber it is almost unavoidable to excite, together with the desired fundamental mode, higher order modes (~1-5% higher order mode content being typical values achieved in our setups). The practical reasons for this typically range from the limited availability of focal lengths in commercial lenses, to the thermally induced changes of the fiber index profile at the input end during high-power operation. Thus, the presence of this unavoidable higher order mode (HOM) content leads to modal beating and to the generation of a thermally induced quasi-static grating as described in [19]. This grating is mostly static because both the fundamental mode and the HOMs, being excited by the same input beam, have identical spectra. The reason why we refer to this grating as quasi-static is because it can move during a transient due to local thermally-induced changes of the beat-length as, for example, the power is increased [14, 19]. Note that the existence of this static grating is compatible with the presence of a noise-generated moving grating suggested in [13, 18]. However, in case that this moving grating is present, it is expected to be much weaker than the quasi-static grating (at least below the threshold) and it will be running on top of it.

Having accepted the presence of the quasi-static grating as almost unavoidable in high-power fiber-laser systems, it is reasonable to think that it should be influencing the mode instability threshold. For example, as proposed in [14], the quasi-static grating can become so strong that it directly triggers mode instabilities by giving rise to non-adiabatic waveguide changes. On the other hand, even in the frame of the moving grating theory [13, 15], the quasi-static index grating should still be considered because it may significantly lower the threshold. The reason for this is that, as the moving part of the interference pattern (which drives the moving grating) becomes out of phase with the strong quasi-static grating, a strong instantaneous energy transfer can take place. This instantaneous energy transfer will be much stronger than the one that would be obtained without considering the presence of the quasi-static grating (which is what current models assume). Nevertheless, it is true that, in steady state, the net average energy transfer caused by the quasi-static grating will be approximately zero (because the energy transfer changes sign over a shift of the moving grating equal to one period of the quasi-static grating. This implies that for example over the first half of the period the energy would flow from the fundamental mode to the HOM and in the second part of the period the energy will flow from the HOM back to the fundamental mode). This is the reason why steady state models simply neglect the presence of the strong quasi-static grating. However, it is still reasonable to assume that the stronger instantaneous energy transfer (that will lead to an instantaneously/locally stronger moving grating) could lead to a lower mode instability threshold in the presence of transients (e.g. when increasing the pump power). However, this point still requires further investigation. Independently of this, it can be argued that below the threshold the strength of a possible moving grating will be proportional to the strength of the quasi-static grating (since their growth along the fiber is mostly determined by the amplification process).

2.2. Semi-analytic formula

According to the discussion above, in this paper we propose to analyze the strength of the quasi-static thermally-induced grating to predict the mode instability threshold and its behavior when different system parameters are changed. This approach to obtain a formula for the mode instability threshold, which significantly departs from the path chosen by other authors [18], is based on the important premise that below the threshold the energy transfer between the interfering modes is very low. This assumption, based on experimental observations [20], in turn implies that the strength of the quasi-static grating below the mode instability threshold is mainly dependent on the amplification process and, therefore, a steady-state rate equation model can be used to compute it.

All the theories about the origin of mode instabilities agree upon the fact that a thermally-induced long period index grating is responsible for the observed energy transfer between transverse modes. So we propose analyzing this grating in an analogous way as normal index gratings. In conventional long period gratings the coupling strength is, in the limit of weak coupling between the modes, proportional to the product $\kappa \cdot L$, where $\kappa$is the coupling constant and $L$is the length of the grating [21]. This, in turn, can be demonstrated to be proportional to $\Delta n\cdot L$, where $\Delta n$is the amplitude of the index perturbation (assuming an homogeneous grating).

In order to treat the thermally-induced long period grating in a similar way as its conventional counterparts, it is necessary to find a way to evaluate the amplitude of the thermal perturbation of the waveguide. The problem is that these thermal perturbations are three-dimensional entities that are not transversally homogeneous. Thus, in order to estimate the impact of the thermally-induced waveguide changes caused by modal interference, a parameter linking the two interfering modes with the waveguide characteristics is required. A natural choice for such a parameter is the modal beat length L_b. For example, the modal beat length between the fundamental mode LP₀₁ and a higher order mode HOM is defined as:

In Eq. (1) λ represents the wavelength and $n_{eff}{}_i$is the effective index of the mode i. Note that $L_b$is inversely proportional to the perturbation of the waveguide, whereas $\kappa$is directly proportional to it. Therefore, we propose using $1/L_b$as a measure of the strength of the thermally-induced waveguide changes.

Now, in analogy to conventional gratings, we propose employing the following ansatz to calculate the power threshold of mode instabilities $P_{th}$:

In analogy to conventional long period gratings, it can be said that the left-hand side of Eq. (2) is proportional to the overall coupling strength of the thermally-induced grating. At this point it is worth noting that here is where the abstraction on the origin of mode instabilities is made since Eq. (2) implicitly assumes, disregarding its cause, that there is a phase shift between the modal interference pattern and the index grating that allows the energy transfer to take place.

This way, Eq. (2) proposes that, for a given fiber design, mode instabilities are triggered once that a certain coupling strength is reached. The problem is determining the value of this certain coupling strength. In fact, with the very simple model proposed here (which ignores the actual mode coupling and the dynamic processes taking place during and before mode instabilities) it is only possible to use a free parameter γ to fit the calculated values to the actual mode instability thresholds. This reduces the ability of this formula to predict the mode instability threshold of new fibers. However, once that the factor γ has been fitted for a certain fiber, Eq. (2) should be able to predict the behavior of the threshold when changing different parameters of the system. Therefore, the ansatz presented in Eq. (2) is, in spite of its simplicity, still suited to investigate mitigation strategies for mode instabilities.

Equation (2) can be further simplified taking into account that $L\approx N\overline{L_b}$, where $N$is the total number of periods of the structure and $\overline{L_b}$is the average value of the beat length along the fiber. Thus we arrive at the following expression:

In order to evaluate Eq. (3), we simulate the three-dimensional steady-state thermally-induced waveguide changes that will appear in the active fiber under a certain excitation (i.e. relative power between the fundamental mode and the HOM) and pump conditions. This simulation is carried out using the thermally-coupled, transversally-resolved steady-state rate-equation model presented in [19]. Finally, once that the 3D thermally-modified index profile of the waveguide has been calculated, a mode solver is run at each point along the fiber to calculate the new local beat length.

3. Extrinsic mitigation strategy: reducing the heat load of active fibers

Up to now all the different theories explaining the origin of mode instabilities coincide in identifying heat load as the most likely cause for this effect. Thus, under the premise of quantum defect heating being the dominant heating mechanism in an Yb-doped fiber, the most straightforward mitigation mechanism of mode instabilities would be to reduce it. In turn, reducing quantum defect heating simply implies bringing the pump and signal wavelengths closer together. There are three possibilities for doing this: a) to shift the pump towards longer wavelengths, b) to shift the signal towards shorter wavelengths, or c) any combination of the previous ones. In the following the different characteristics, advantages and disadvantages of the first two approaches will be discussed.

Figure 1 illustrates the prediction of Eq. (3) for the change of the threshold when reducing the quantum defect. The calculations have been done for a 1.2m long fiber with 80µm core diameter (from which only 64µm are doped with $3.5\cdot {10}^{25}$m⁻³ concentration of Yb ions), 228µm pump cladding diameter and 1.2mm outer fiber diameter. The V-parameter of the fiber is 7 and the fitting parameter γ is $1.8\cdot {10}^{-2}$, which results in a predicted mode instability threshold of ~210W when pumping at 976nm and emitting at 1030nm. This threshold value is realistic for fibers of these dimensions. In the simulations the seed power in the fundamental mode is 50W and the LP₁₁ is excited with 0.5W. The reason for this high seed power is to allow reaching the mode instability threshold in spite of the lower gains that accompany the change in signal/pump wavelength. In Fig. 1(a) the pump wavelength is set to 976nm, whereas in Fig. 1(b) the signal wavelength is set to 1030nm and the pump is assumed to be monochromatic.

 

Fig. 1 Evolution of the mode instability threshold with the reduction of the quantum defect heating which was achieved a) by pumping at 976nm and by changing the signal wavelength or b) by emitting at 1030nm and by changing the pump wavelength. In both plots the black dashed line represents the expected evolution of the threshold due to the reduction of the quantum defect.

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As can be seen in Fig. 1 both approaches lead to a significant increase of the mode instability threshold. The reason why both evolutions deviate from the expected behavior proportional to the reduction of the quantum defect has mainly to do with the change in the pump absorption and/or temperature distribution along the fiber. This point, which will be discussed in more detail in the next section, implies that operating at a pump/signal wavelength combination different from 976/1030nm results in weaker and effectively longer gratings. The weakening of the grating happens due to two factors: a) the reduction of the quantum defect (not for signal wavelengths >1030nm) and/or b) a more linear amplification that results in a better distribution of the heat along the fiber (resulting in grating evolutions similar to those described in section 4).

The weakening of the grating and the increase of its effective length have opposite effects on the mode instability threshold: whereas a weaker grating results in a higher threshold, a longer one results in a lower threshold. The results in Fig. 1 indicate, however, that the weakening of the grating due to the reduction of the quantum defect is the dominant feature for signal wavelengths <1030nm and or for pump wavelengths >976nm; that is why, when the wavelength separation between pump and signal is reduced, the threshold increases, albeit at a lower rate than that dictated by the reduction of the quantum defect due to the simultaneous increase of the effective grating length.

At signal wavelengths >1030nm, Fig. 1(a) suggests that the weakening of the grating related to the more homogeneous heat distribution along the fiber is partially compensating the stronger thermal perturbations of the waveguide caused by the higher quantum defect. This explains why the threshold in Fig. 1(a) is higher than what would be expected from the quantum defect alone.

Changing the pump wavelength might be in principle more attractive than changing the signal wavelength because the modification of the system can be reduced to the last amplification stage. However, changing the pump wavelength might become challenging very quickly unless the fiber design is properly modified as well. This is illustrated in Fig. 2, where the evolution of the amplification efficiency (${\eta }_{amp}=\left(P_{out}-P_{seed}\right)/P_{pump}$) as a function of the signal/pump wavelength is represented. As can be seen, in the case of changing the signal wavelength the amplification efficiency remains high (>0.7) up to a factor 2 reduction of the quantum defect. However, in our example, when changing the pump wavelength the amplification efficiency quickly drops to unacceptably low values (<0.3). More importantly, the dramatic loss of efficiency at the longer pump wavelength side means that in real high power pump diodes (with a typical spectral bandwidth of ~3nm) the short wavelength side of the pump radiation will dominate, which will lead to a lower MI threshold than expected from the central wavelength alone. Note, however, that increasing the seed power, the fiber length and/or the ion concentration will help increasing the amplification efficiency when changing the pump wavelength; but this is not possible or desirable in every application.

 

Fig. 2 Evolution of the amplification efficiency for a fixed seed power of 50W a) when changing the signal wavelength while pumping at 976nm and b) when changing the pump wavelength while emitting at 1030nm.

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Both approaches present substantial differences regarding the amplification process. It can be demonstrated that the maximum achievable inversion in an Yb-doped fiber amplifier is given by (when neglecting spontaneous emission):

In Eq. (5) h is the Planck constant, c is the speed of light, λ_l is the signal wavelength, A represents the doped area of the fiber core, L is the fiber length, E_stored is the maximum energy stored in the fiber (after discounting the quantum defect) and E_transparency is the transparency energy, which is not available for extraction. Finally, it is known that due to saturation the available energy E_G and the gain G are related by the following equation [22]:

Here Γ_l stands for the overlap of the signal beam with the doped region. Therefore, now using Eq. (7) it is possible to calculate the evolution of the available energy and gain in the fiber as either the signal or the pump wavelength is swept. The results of this calculation for a fixed pump power of 300W and for 50W seed power are shown in Fig. 3.

 

Fig. 3 Evolution of the available energy E_G and gain G for a fixed pump power of 300W a) when changing the signal wavelength while pumping at 976nm and b) when changing the pump wavelength while emitting at 1030nm.

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The red thick lines in Fig. 3(a) and Fig. 3(b) mark the situation with a similar mode instability threshold as predicted by Fig. 1. As can be seen, changing the signal wavelength offers, for a similar threshold, a significantly higher available energy. Thus, this approach might be attractive for pulsed systems pursuing high average powers and high pulse energies. Unfortunately changing the signal wavelength implies having to deal with a reduced gain (when compared to 1030nm) throughout the whole amplification chain. This, at the end, can imply the need of more pre-amplification stages, but the advantage is that no changes in the fiber designs are strictly required (at least up to a factor 2 reduction of the quantum defect).

On the other hand, choosing to change the pump wavelength as the way to reduce the heat load has, as mentioned before, the advantage that it can be done exclusively in the last amplification stage. Thus, the change is limited to the very final amplifier whereas the rest of the amplification chain remains intact. This approach, in order to counteract the potentially lower amplification efficiency, will require a redesign of the main amplifier fiber (regarding its length, total ion concentration and/or the ratio of the pump cladding to the doped region) and/or pump sources with higher brightness. These measures should also be ideally accompanied by higher seed powers.

With both approaches high gains in terms of the modal instability threshold are expected. However, as mentioned before, these predictions are based upon the commonly accepted assumption that the main heat source in the fiber is quantum defect heating. Should this not be the case, or should there be any other significant wavelength independent heat source in the fiber (e.g. photo-darkening-induced heating), then the expected increase of the mode instability threshold would be reduced and the plots in Fig. 1 would be flattened out.

4. Intrinsic mitigation strategy: reducing the pump absorption

As mentioned above, intrinsic mitigation strategies are those involving modifications in the fiber design. According to Eq. (3), an intrinsic approach to reduce mode instabilities is based on the reduction of the pump absorption level (understood as the amount of pump power absorbed by the fiber in absence of seed per unit length and for low powers). A reduction of this parameter leads to a more linear evolution of the signal power along the fiber as illustrated, for example, in Fig. 4(a). This change of the amplification profile is similar to what happens in the fiber when choosing a pump/signal wavelength configuration different than 976/1030nm, as done in the previous section. In Fig. 4(a) there is a comparison between two 1.2m long, 80µm core fibers (the same doping parameters as in the previous section have been used) pumped at 976nm and seeded with 50W of LP₀₁ mode and 0.5W of LP₁₁ mode at 1030nm. The only difference between the fibers is that in one case the pump core diameter is 228µm and in the other case it is 395µm. The pump power has been appropriately chosen in each case to reach the mode instability threshold, i.e. ~210W for the fiber with 228µm pump core diameter and ~242W for the fiber with 395µm pump core diameter. As mentioned above, it can be appreciated that the power evolution of the signal in the fiber with the larger pump core is considerably more linear than in the other case. This, in turn, leads to a more homogeneous distribution of the heat along the fiber and, consequently to lower absolute temperatures, to lower temperature gradients and, more importantly, as shown in Fig. 4(b), to a weaker but effectively longer grating/thermal perturbation. Note that the grating/thermal perturbation is characterized in Fig. 4(b) by the normalized amplitude of the changes of the inverse beat length along the fiber, as proposed in Eq. (2).

 

Fig. 4 a) Evolution of the signal power along the amplifier when considering two fibers with different pump absorption and b) measure of the evolution of the grating strength along the active fibers simulated in a).

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Figure 5 shows the evaluation of the MI threshold for several fibers with different lengths and pump core sizes (i.e. pump absorptions levels). The pump core diameters have been chosen so that the same total amount of pump absorption is maintained along the black line (the pump core area has been increased by a factor of 2 in the red line and by a factor of 3 in the green line). As before, the fibers were pumped at 976nm and were seeded with 50W of LP₀₁ and 0.5W of LP₁₁ at 1030nm. Moreover, the outer diameter of all fibers was kept constant at a value of 1.2mm. Figure 5 clearly shows that a lower level of pump absorption per unit length is beneficial for MI. This is consistent with anecdotal evidence suggesting that longer fibers exhibit higher MI thresholds. However, length alone cannot explain these observations. In fact, it can be seen in Fig. 5 that for a fixed pump core diameter (i.e. pump absorption level) the MI threshold can be almost independent of the fiber length over a relatively wide range of lengths. Only for lengths shorter than about half the optimum pump absorption length for each fiber does the MI threshold start to sink significantly. Thus, according to Fig. 5, the higher mode instability thresholds typically reported for long fibers seem to be mainly the result of the lower pump absorption levels characteristic of those fibers.

 

Fig. 5 Comparison of the mode instability threshold for fibers with different pump absorption levels. All the fibers have an 80µm core, have been seeded with 50W at 1030nm and are pumped at 976nm.

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5. Conclusion

Passive mitigation strategies for mode instabilities can be divided in two general categories: intrinsic and extrinsic. Intrinsic mitigation strategies involve modifications of the fiber design, whereas extrinsic mitigation strategies involve acting upon system parameters other than the fiber design.

In order to evaluate the relative performance of different mitigation strategies a means to calculate the mode instability threshold is required. Even though this can be done with complex BPM models, this would be extremely time-consuming. From the practical point of view a simple expression that can be quickly evaluated would be more advantageous, even at the price of sacrificing some accuracy. In this paper we propose a new approach to obtain such an expression, which is based on analyzing the strength of the thermally-induced grating. Since the formula obtained this way contains a free parameter γ (that has to be fitted once per fiber type to experimental data), the ability of this expression to predict the threshold of new fiber designs is rather limited. However, once the γ parameter has been fitted, the formula can evaluate the expected change in the MI threshold associated to different mitigation strategies (that involve using the same fiber type). The main differential feature of this formula is that the detailed physics of the amplification process are taken into account.

Thus, using this expression for the MI threshold, two passive mitigation strategies, one intrinsic and one extrinsic, have been presented and analyzed. The extrinsic mitigation strategy is based on reducing the quantum defect in fiber amplifiers by either changing the signal or the pump wavelength. The advantages and disadvantages of these two approaches from the practical point of view have been discussed. The intrinsic mitigation strategy is based on reducing the pump absorption level. Both strategies promise significant increases of the MI threshold with only moderate changes in current fiber amplifier systems.