## Abstract

A new way of stabilizing the output beam of a fiber laser system operating above the mode instability threshold is described and the first proof-of-principle experimental results are presented. This technique, which relies on a modulation of the pump power, works by washing the thermally-induced refractive index grating out, which weakens the coupling efficiency between transverse modes. One of the main advantages of this simple, yet powerful, approach is that it can be easily incorporated in already existing fiber laser systems since it does not require any additional optical elements. Using this beam stabilization strategy, a significant pointing stability and beam quality improvement has been demonstrated up to an average power of ~600W, which is a factor of 2 above the mode instability threshold.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber laser systems have gained a solid reputation as a power-scalable solid-state laser concept largely devoid of the thermo-optical effects that plague traditional laser architectures and result in a progressive degradation of the beam quality with the average power. This widespread opinion is grounded on the exponential evolution of the output average power of fiber laser systems sustained over roughly two decades [1]. Such a development has led to fiber laser systems being the solid-state laser concept currently holding the record of the highest emitted average power with nearly diffraction-limited beam quality. Such a feat becomes even more impressive when taking into account that the thermal conductivity of the material fibers are made of (Silica) is a factor 5-10 worse than that of the crystalline host materials of competing laser technologies.

The exponential development of the average power of fiber laser systems lasted until 2010, when the first reports of a new effect, transverse mode instabilities (TMI), were published [2, 3]. TMI refer to the sudden loss of beam quality and stability in a fiber laser system once that a certain average power threshold has been reached. Such observation caused a great concern in the fiber laser community since one of the main advantages of this technology, namely its power-independent high beam quality, was under threat. This concern, together with the natural curiosity of understanding the physics behind a new phenomenon, sparked worldwide an intense research period in TMI that lasts until the present day.

Such research effort has led to the understanding of the physical origin of TMI. Thus, soon after the publication of the first observations, it was proposed that the effect could be related to a phase-matched energy transfer between two transverse modes of a few-mode fiber enabled by a thermally-induced index grating [4, 5]. Shortly afterwards, this theory was refined by pointing out that a phase shift between the modal interference pattern and the thermally-induced index grating is required for the energy transfer to actually take place [6]. Later on, new systematic measurements of the effect were published [7, 8], which helped in refining the models and in getting a deeper understanding of the effect. Thus, over the years, several theoretical models with different degrees of sophistication have been developed [9–13], which have greatly contributed to obtaining a deeper insight on the physics behind TMI.

Fascinating as the physics of TMI are, the effect is still very damaging for the reputation of fiber laser systems. Thus, understanding TMI is only the first step towards the main goal of mitigating it. Early in the process of investigating TMI, it became apparent that it is related to other thermo-optical effects, namely two-wave mixing [14] and, specially, stimulated Rayleigh scattering [15–18]. These effects still pose unsolved problems in bulk media and air. However, in this context, fibers are unique media due to their waveguide nature, which gives more degrees of freedom to devise mitigation strategies for TMI. Consequently, as a result of this and the understanding of the effect gained through the theoretical models, a few mitigation strategies have already been proposed [19–24] and some of them have been demonstrated.

Even though some of the mitigation strategies demonstrated to date are promising, most of them are not universally applicable since they require specialty optical fibers tailored to specific operation regimes [25–27] and/or the incorporation of new optical elements in the system [21]. This reduces the attractiveness of these techniques from the practical point of view. In contrast, a solution that could be easily incorporated in already deployed laser systems would be highly desirable.

In this work we present the proof of principle of a new way to stabilize the beam above the TMI threshold. This technique can be applied to already deployed fiber laser systems since it does not require any additional optical components. This approach is based on modulating the pump power of the fiber laser system with an appropriate frequency and amplitude to wash out the thermally-induced grating. As it will be shown, this weakens the modal coupling, resulting in a significant reduction of the beam fluctuations. Even though this technique has some similarities with the active stabilization of TMI through a dynamic excitation of the fiber [21] presented by the authors a few years ago, its physical operating principle is different. This confers this approach distinct advantages such as a stronger and more robust weakening of the thermally induced grating and a lower sensitivity to the modulation parameters. In a proof-of-principle experiment using the proposed technique, we have been able to stabilize the beam fluctuations in a high-power rod-type fiber amplifier up to a power of ~600 W which is the highest output average power reported so far from this kind of fibers with a nearly diffraction-limited beam quality.

This paper is organized as follows: at first a brief description of the simulation tool used to study the response of the fiber laser system to the pump modulation is described. Those readers exlusively interested in the experimental results can directly jump to section 3, where the operating principle of the beam stabilization strategy is discussed and its performance evaluated. Afterwards, the first proof-of-principle experimental results are presented. Finally, some conclusions are drawn.

## 2. Simulation model

In order to simulate the behaviour of the thermally-induced index-grating under the influence of a pump modulation it is necessary to develop a dynamic model. Hereby the time axis will be divided in small steps. The main idea of the algorithm is, for each time step, to use the three-dimensional temperature profile of the fiber from the previous temporal step to obtain the local modes along the fiber (i.e. the evolution of the fiber eigenmodes along the fiber). These are then employed to determine the modal interference intensity pattern (under consideration of the modal coupling, usually between the LP_{01}-like and the LP_{11}-like modes, caused by the thermally-induced index grating). This is, in turn, used to calculate the inversion profile (together with the signal and pump power evolution) along the fiber taking into account the instantaneous pump power level in the current temporal step. Once that the inversion profile is known, it can be then employed to obtain the new thermal profile in the fiber. Then the simulation algorithm moves to the next temporal step. This process is repeated until the final simulation time has been reached.

According to the description above, the key feature of the algorithm is its ability to calculate the temporal evolution of the temperature and inversion profiles between time steps. This can be done using a semi-analytic model that will be detailed in the following. Based on our past experience with the stabilization of TMI using an acousto-optic deflector [21], we expect that the pump modulation frequency will be similar to the main frequency of the beam fluctuations. This, for the rod-type large-pitch fibers [27] that will be used in the experiments, results in an expected pump modulation frequency between 300 Hz and 2 kHz [7]. These modulation frequencies are too fast to consider that the system is able to adapt itself “instantaneously” to the new conditions (i.e. the steady-state solutions corresponding to the instantaneous pump conditions cannot be used to simulate the temporal evolution of the system). Additionally, the expected modulation frequencies are too slow to use a full temporally-resolved numerical method (because the calculation time would be usually too long). Thus, in order to simulate the response of the system to the pump modulation, we have developed a new semi-analytical model that is well-suited for modulation frequencies from the 10’s of Hz to some MHz.

The new simulation algorithm uses the full 3D-resolved steady-state rate equations of a fiber laser system presented by the authors elsewhere [28]. That model is able to solve the 3D rate equations, calculate the associated temperature and inversion profiles and, from them, obtain the evolution of the modes and their interference along the active fiber. However, as already mentioned, this is a steady-state model, which is not able to describe the transient response of the system to the pump modulation. In order to incorporate the temporal dynamics in this model, we have obtained some semi-analytic formulas that describe the temporal response of both the 3D inversion and temperature profiles to changes in the system.

Assuming that, after an instantaneous change, the parameters affecting laser operation remain constant with time, it can be demonstrated (from the temporally-resolved rate equations) that the temporal evolution of the population density of the upper laser level *N _{2}* over a time period

*Δt*is given by:

Where *N _{2o}* represents the initial state (before the change) of the population density of the upper laser level. Besides,

Here *h* stands for the Planck’s constant and *c* for the speed of light. Additionally, *τ* represents the average lifetime of the excited state, *λ _{pump}* is the wavelength of the pump radiation,

*λ*is the wavelength of the laser radiation, and

_{laser}*P*and

_{pump}(z)*P*are the pump and signal power along the active fiber (z-direction) in the co-propagating ( + superscript) and/or in the counter-propagating (- superscript) direction. Moreover,

_{signal}(z)*σ*,

_{apump}*σ*and

_{asignal}*σ*represent the absorption and emission cross-sections of the pump and signal radiation, respectively. Finally,

_{epump,}σ_{esignal}*Γ*(x,y) and

_{pump}*Γ*are the power filling distributions of the pump and signal beams, as defined elsewhere [4].

_{signal}(x,y)Finally, *N _{2f}(x,y,z)* in Eq. (1) represents the steady-state population density of the upper laser level after the parameter change and it is given by:

Where *N(x,y)* represents the cross-sectional distribution of the Yb-doping concentration in the active fiber.

In order to make the model work and capture the transient behaviour of the system, Eq. (1) to Eq. (3) are used to calculate the evolution of the population density of the upper laser level over a certain period of time *Δt.* Since the semi-analytic equations implicitly assume that the operation parameters do not change during *Δt*, this factor has to be chosen to be much shorter than the modulation period of the pump. For most of our simulations *Δt* ~15 µs is a reasonable choice. Once that the new value of *N _{2}* after

*Δt*has been calculated, it is inserted in the 3D steady-state rate equation model to obtain the new evolution of

*P*and

_{pump}(z)*P*along the active fiber. These are then used to recalculate

_{signal}(z)*N*using Eq. (1) to Eq. (3) and the process is repeated until convergence is reached. The assumption that allows using the 3D steady-state rate equation model to calculate the change in pump and laser power is that they react instantaneously to any variation of the inversion level. For the time scales and fiber lengths considered here this is a reasonable assumption.

_{2}Once that the new inversion and power levels have been calculated, it is time to calculate the new three-dimensional temperature profile. In order to do this, another semi-analytic formula has been obtained. Using a finite-difference approach, the heat-transport equation at each position along the fiber (neglecting longitudinal heat-flow) can be written in matrix form as follows:

Where *α* is the thermal diffusivity (~8.3∙10^{−7} m^{2}/s in Silica) and *κ* is the thermal conductivity (~1.38 W/(m∙K) in Silica). In Eq. (4) and the following equations the symbol *[ ]* represents a matrix. Thus, *[P]* is a matrix comprising the finite difference coefficients, *[T] _{z}* contains the 2D temperature profile in the fiber cross-section at position z and, finally,

*[Q]*describes the local 2D profile of the heat-load at position z. Additionally, by defining

_{z}*[A]*as the eigenvector matrix of

*[P]*,

*[Λ]*as its eigenvalue diagonal matrix and $[T]=[A]\cdot [{T}^{*}]$, Eq. (4) can be rewritten as:

With ${\left[X\right]}_{z}={\left[A\right]}^{-1}\cdot {\left[Q\right]}_{z}$.

This last equation can be solved as:

Where ${T}_{z\left(i,j\right)}^{*}(\Delta t)$ represents the elements of *[T*] _{z}* a time

*Δt*after the parameter change, ${T}_{z\left(i,j\right)o}^{*}$ stands for the elements of

*[T*]*before the parameter change (i.e. initial situation), ${X}_{z(i,j)}$ refers to the elements of

_{z}*[X]*and

_{z}*Λ*represents the eigenvalues of

_{i}*[P].*Functionally, Eq. (6) is identical to Eq. (1), so that the term ${X}_{z\left(i,j\right)}/({\Lambda}_{i}\kappa )$ can be interpreted as representing the elements of the matrix

*[T*]*once that the steady-state has been reached after the parameter change.

_{z}Once that the new temperature profile is known, it is used to obtain the thermally-induced refractive index change *Δn(x,y,z,t)* by multiplying it by the thermo-optical constant (*dn/dt*~1.2∙10^{−5} K^{−1} in Silica). This gives rise to the thermally-induced index grating responsible for TMI. Afterwards this index profile is employed to calculate the new modal profiles along the fiber. To simplify and speed up the model, it has been assumed that during *Δt* the temperature change is going to be small enough to have only a negligible impact on the modal profiles propagating through the fiber. Again, this implicitly assumes that *Δt* is small enough, but our step of ~15 µs seems to cast stable and accurate results.

Finally, in order to quantify the impact of the pump modulation on the thermally-induced index grating *Δn(x,y,z,t)* and its ability to couple energy between two transverse modes of the fiber, a coupled-mode equation has been obtained (in a similar way as done in [11]) and solved:

where *A _{i}* is the complex amplitude of the electric field of mode

*i*and

*β*represents its propagation constant. Additionally, the coupling coefficients

_{i}*κ*can be calculated as:

_{ij}In Eq. (8) *w* is the angular frequency of the signal light and $\overrightarrow{{\phi}_{i}}(x,y,z)$ is the normalized vector electric field of mode *i*. Finally, in Eq. (7) the modal gain coefficients *Κ _{LGij}* are defined as:

where ${N}_{1}\left(x,y,z,t\right)=N\left(x,y\right)-{N}_{2}\left(x,y,z,t\right)$ is the population density of the lower laser level.

Once that the new evolution of the modal amplitudes is obtained by solving Eq. (7), a new modal interference intensity pattern can be calculated. This will then be employed in the next temporal step to determine the distribution of the population density of the excited state in the fiber using Eq. (1).

Note that, even though it will not be fully analyzed in this work, the model as described above is able to reproduce the mode coupling and temporal dynamics of TMI.

To simplify the study and illustrate how the beam stabilization technique proposed herein works, we will use the integral of the coupling constant between the FM and the HOM along the fiber, defined by:

as a measure of the susceptibility of the system to TMI. Please note that the actual energy transfer between the transverse modes depends on the phase-shift between the thermally-induced index grating and the modal interference pattern [6], and this is a parameter which we are not considering in the equations above (but in the semi-analytic model described in Eq. (1) to Eq. (9) it is fully taken into account). Thus, using the absolute value of*κ*only allows getting a relative measure of the potential coupling strength of the thermally-induced index grating. Therefore, this approach will not lead to the calculation of an absolute value for the TMI threshold or the actual energy coupling between the modes, however, it should still serve as a measure for the pump-induced weakening of the thermally induced grating.

_{12}## 3. Operation principle

A pump power change in a high-power fiber laser system results in a variation of the heat-load and, in the steady-state, in a different overall temperature profile along the active fiber. This, in turn, leads to a modification of the local guiding properties of the fiber along its length and, therefore, to a change in the evolution of the modes [29]. In other words, the temperature profile results in a variation of the effective refractive index difference between the modes. Consequently, such a change is reflected as a modification of the periodicity of the modal interference pattern, as illustrated in Fig. 1. In this figure it can be seen that a higher pump power results in a modal interference pattern with a shorter period (in steady-state). This is because of the higher heat-load and, therefore, temperature which leads to a stronger thermally-induced modification of the refractive index profile of the fiber. Thus, if there is a slow periodic change of the pump power between the two levels shown in Fig. 1 (200 W and 400 W), the overall temperature profile will adapt, and the modal interference pattern will also change periodically between the two states shown in Fig. 1(a). On the other hand, if the pump modulation frequency is too high (much higher than the inverse of the thermalization time of the fiber core) then, even though the heat-load in the fiber will change (and with it the output power), the temperature profile will not be able to adapt and, therefore, no significant change of the modal interference pattern will be observed.

Finally, there is an intermediate region in the pump modulation frequency where the temperature profile has enough time to start adapting to the new pump power level, but cannot reach its final steady-state. In this situation the temperature profile is constantly in a transient state. This is an interesting regime, even though the change of the temperature profile will be smaller than the one expected from the two extreme pump levels in the steady-state situation (see e.g. Figure 1). However, if the modulation amplitude and frequency are adequate, the temperature profile (and, therefore, the thermally-induced index grating) can be, at least partially, washed out as shown in Visualization 1 (Fig. 2). Such a weakening of the thermally-induced grating will lead to a mitigation of TMI and, therefore, to a stabilization of the beam as was demonstrated, albeit with a different method, in [21].

Visualization 1 has been generated using the model described in section 2. For this simulation 300 W average pump power at 976 nm with a 720Hz sinusoidal modulation and ± 77% amplitude change has been applied to a 1 m long rod-type fiber with 80 µm core diameter (MFD ~65 µm), 228 µm pump-cladding diameter, 1.2 mm outer fiber diameter, a V-parameter of 7 and doped with 3.25∙10^{25} Yb-ions/m^{3}. The fiber is seeded by 5.5 W of average power at 1030 nm. For this simulation ~8.4 ms (6 modulation periods) have been recorded starting from the steady-state situation with 300 W pump (which corresponds to the experimentally determined TMI threshold of the system at ~270 W output average power). Note that the pump modulation parameters used in this simulation correspond to the ones that will be employed to obtain our experimental results.

Visualization 1 (Fig. 2) shows the temporal evolution of three different parameters: the modal interference pattern (upper subplot), the population density distribution of the excited state, henceforth labelled as inversion pattern (middle subplot), and the radially anti-symmetric component of the thermally-induced refractive index change (bottom subplot). The reason for plotting only the radially anti-symmetric part of the refractive index change is because, as mentioned above, this is the one responsible for the energy transfer between the fundamental mode and the LP_{11} mode used in the simulation (and usually seen in experiments after the TMI threshold has been reached). Moreover, this radially anti-symmetric component of the refractive index change will also be used in the following for calculating *GS*. In Visualization 1 it can be clearly seen that, as time goes by, the thermally-induced index change is significantly weakened in average, even though there are some moments where it regains strength.

A careful analysis of the upper subplot of Visualization 1 also reveals dynamic changes of the modal content along the fiber, which are the result of an instantaneous modal energy transfer predicted by our model. This modal energy transfer is triggered by a phase shift between the modal intensity pattern and the index grating, which is induced by the pump modulation process. Discussing the nuances of such a controlled modal energy transfer goes beyond the scope of this paper since the focus of the present work is the impact of the pump modulation on the thermally-induced index grating. In any case, for the reader interested in this topic, the induced modal energy transfer has been measured and studied in detail in [30]. Suffice to say that, by a correct choice of the pump modulation parameters, such modal energy transfer becomes weak and is circumscribed to a brief portion of the modulation period. Thus, it becomes virtually undetectable for conventional cameras with acquisition rates of 20-60fps, such as the one used in this work. Therefore, since this manuscript presents and analyzes the observations done with these conventional cameras, this induced modal energy transfer processes will be no further considered in the following.

In order to get a better impression of the weakening of the thermally-induced grating, Fig. 3 shows a comparison between the profile of the grating coupling constant *κ _{12}(z)* corresponding to the highest

*χ*at t = 0 ms (in blue) and that corresponding to the lowest

_{ΤΜΙ}*χ*at t = 3.94 ms (in red) in the simulation (see Fig. 4). It should be noted that the highest value of

_{ΤΜΙ}*χ*is reached for the CW steady state situation at t = 0 ms, which is the starting condition for our simulations. Visualization 2 shows a video in which the temporal evolution of

_{ΤΜΙ}*κ*can be seen over the ~8.4 ms simulation window. As can be observed, confirming the impression obtained in Visualization 1, the pump modulation induces a significant weakening of the grating coupling constant.

_{12}(z)The data presented up to now in Figs. 2 and 3 (as well as the related movies in Visualization 1 and Visualization 2) only allow for a qualitative estimation of the impact of the pump modulation on the thermally-induced index grating. A more quantitative analysis can be done using the temporal evolution of *χ _{ΤΜΙ}* as shown in Fig. 4. The upper part of this figure shows the temporal modulation of the pump power and the lower one the evolution of the TMI susceptibility. The first thing that can be seen is the periodic behavior of

*χ*once that a certain initial adaptation period has elapsed (after ~2.5 ms). This transient is due to the evolution of the system from the initial CW steady-state situation to the pump modulation regime. The TMI susceptibility presents a dynamic behaviour exhibiting some pronounced maxima and minima that roughly follow the sinusoidal pump modulation. These are the result of the dynamically changing temperature conditions in the fiber. However, the most important feature revealed in Fig. 4 is that, during the pump modulation, both the maximum and average values of the TMI susceptibility become significantly lower than its CW steady-state value (at t = 0 ms). This weakening of the thermally-induced index grating is directly responsible for the stabilization of the beam that will be demonstrated in the next section.

_{ΤΜΙ}Even though the pump modulation technique as presented so far can successfully stabilize the beam fluctuations in a high-power fiber laser system, it has the drawback of producing a laser output power which is a strongly modulated. This might not be a deterrent for some applications (i.e. for those which processing time is longer than the modulation period, i.e. in some material processing and/or in directed-energy applications), but for others it might be a serious issue. One relatively straightforward solution might be to use a bidirectional pump configuration in which the pump modulations are synchronized and have a π-phase shift to one another. This will produce a constant output signal power while still allowing to wash out the thermally-induced index grating, since the point of maximum heat load will be continuously shifted in the fiber. A systematic study of this technique is the object of current investigation and a deeper discussion goes beyond the scope of this paper.

## 4. Experimental results

As already mentioned at the beginning of this paper, one of the key advantages of the pump modulation technique is that it can be readily incorporated into already existing setups since it does not require any additional optical components. This is illustrated in Fig. 5, where a schematic of the experimental setup is shown. As can be seen, the optical part corresponds to a conventional high-power fiber amplifier arrangement. The only modification to the system is the use of a pump diode driver that, in our system, can be modulated up to ~1 kHz with a function generator.

In order to characterize TMI, we used two weak reflexes of the output beam and directed them to two photodiodes (PD). The area of the first one (PD_{P}) was bigger than the beam itself, so it was used to measure the temporal evolution of the output power after calibrating it to the thermal power meter. On the other hand, the detection area of the second photodiode (PD_{S}) was significantly smaller than the beam at that point and, therefore, it was used to measure the beam stability [7]. Furthermore, we also used a reflex of the output beam to measure the improvement of the beam quality (M^{2} parameter) when switching the pump modulation on.

The active fiber was a ~1.1 m long Large-Pitch Fiber (LPF) [27] with an active core of ~65 µm. This fiber was seeded by a 5 W signal at 1030 nm and counter-pumped by a 976 nm laser diode able to deliver up to 2 kW. This diode was connected to a driver that can be modulated with up to several kHz, however, in the particular implementation of our system we were limited to a maximum modulation frequency of ~1 kHz. In such a system, without pump modulation, we have measured a TMI threshold of 266 W using the method detailed in [7]. In the following we will experimentally explore the limits of the beam stabilization technique.

We operated our system at 407 W output average power and recorded the emitted beam with a CCD camera (30 fps) when switching the pump modulation on and off with a period of 2 s. For this experiment the pump was modulated with a sinusoidal signal of 720 Hz and ± 77% modulation amplitude.

As can be seen in Fig. 6 and in Visualization 3, there is a very significant improvement in the beam stability when the pump modulation is switched on. Please note that the modulation of the output power is not visible in Visualization 3 because the frame rate of the camera is much lower than the pump modulation frequency.

We have noted that the system is fairly tolerant to the modulation frequency since we have managed to get good stabilization results also at ~1 kHz. Such a tolerance to this parameter is a clear advantage of this method over the active stabilization of TMI with a dynamic excitation of the fiber [21], which was characterized by a strong frequency sensitivity.

To quantitatively evaluate the improvement in beam quality obtained by the pump modulation technique, we have performed a M^{2} measurement both for the free-running system and for the stabilized system (using the same modulation parameters as before) at 407 W output average power. What we have obtained is that the M^{2} in the free-running system is ~1.6 and in the stabilized case it is <1.1. This further highlights the performance gain derived from employing the pump modulation technique in a high-power fiber laser system.

Using the photodiode PD_{S}, we have measured the stability of the beam with and without pump modulation at different output average powers. This analysis, however, requires a certain amount of signal processing to render meaningful results. The reason is that, as illustrated in Fig. 7, the raw data of the photodiode (black curve) shows the sinusoidal power modulation of the output signal. This modulation of the output power is controlled by the pump modulation parameters and is stable in time. On the other hand, the onset of TMI manifests itself as beam fluctuations which are registered by the photodiode as amplitude noise in the sinusoidal modulation of the output signal. Therefore, in order to analyze the temporal stability of the beam, it is necessary to isolate this amplitude noise from the raw photodiode data. This can be done by filtering the modulation frequency and its harmonics (since the modulation is not perfectly sinusoidal) from the raw data, which results in the blue line in Fig. 7. Then, in order to quantify the magnitude of the beam fluctuations, the standard deviation of the blue line is calculated.

Figure 8 shows a comparison of the evolution of the standard deviation of the beam fluctuations for the free-running system (red crosses) and for the stabilized system (blue dots) using optimized modulation parameters at each output power. As mentioned above, in order to obtain a clear evolution of the standard deviation in the stabilized case, we have filtered out the spectral components corresponding to the modulation frequency and its harmonics from the PD_{S} trace.

A strong suppression of the TMI-induced beam fluctuations becomes apparent in Fig. 8 when comparing the evolution of the standard deviation of the free-running and the stabilized system. According to our measurement, the TMI threshold of the free-running system is ~266 W but we can significantly improve the stability of the beam up to ~600 W of output average power. Furthermore, the stabilized system shows extremely low values of beam fluctuations below ~400 W of output average power. In this region the fluctuations of the stabilized beam are comparable to the ones of the free-running system below the TMI threshold. Above 400 W the beam fluctuations become progressively stronger. Based on our past experience with the active stabilization of TMI using a dynamic excitation of the fiber [21], we believe that the stabilization can be improved if a feedback loop is incorporated in the system (which would comprise a photodiode used to monitor the beam stability and a control unit used to decide the best modulation parameters).

Additionally, in the upper part of Fig. 8 it is possible to get an impression of the improvement of the beam quality obtained when using the pump modulation technique all the way up to an output average power of 600 W.

In order to evaluate the performance limits of this technique, we operated our system at 563 W output average power, i.e. two times above the TMI threshold of the free-running system. For this, the pump power was modulated with a frequency of 694 Hz and a modulation amplitude of ± 90%. Using these parameters we recorded another video while switching the pump modulation on and off every 2 s. The result of this experiment is shown in Fig. 9 and in Visualization 4. As can be seen, the pump modulation can still achieve a remarkable improvement in stability and quality of the output beam, even though in this case some light high-frequency trembling can be observed in the stabilized beam. In spite of this, the M^{2} could still be improved from 2.5 in the free-running system to 1.4 in the stabilized case.

## 5. Conclusions

In this work we have introduced a novel technique to stabilize the TMI-induced beam fluctuations by modulating the pump power. This beam-stabilization technique can be applied to any kind of fiber. Furthermore, this technique can be readily incorporated in already existing setups since it does not require the modification of the optical system.

We have presented both a theoretical study of the operating principle of the pump modulation technique and an experimental demonstration thereof. The theoretical analysis, based on a new simulation model, provides a theoretical frame to understand the experimental results. Thus, using this technique in a rod-type fiber amplifier, it has been possible to obtain a robust stabilization of the beam up to ~407 W of average power. This power level is 1.5x above the TMI threshold of the free-running system. Furthermore, we have also shown that a significant improvement of the beam quality and stability is still possible at an average power of ~600 W, i.e. roughly 2x above the TMI threshold of the free-running system. This is the highest output average power reported so far from a rod-type fiber with a stabilized beam.

In our opinion, this encouraging first experimental results can even be further improved by optimizing the modulation function and by incorporating a feedback control loop in the system. Thus, the further development of this technique can lead to significant gains in the average power of fiber laser systems with nearly diffraction-limited beam quality in the near future.

## Funding

This work has been supported by the Fraunhofer and Max Planck cooperation program within the project “PowerQuant”, by the German Federal Ministry of Education and Research (BMBF), project no. (PT-VDI, TEHFA II), by the European Research Council under the ERC grant agreement no. [617173] “ACOPS” and by the German Research Foundation (DFG) within the “International Research Training Group 2101”.

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