1. Introduction

The increase in output power of diffraction-limited fiber laser systems has reached a plateau since the first observation of an effect that limits their average power scaling, known as transverse mode instability (TMI) [1,2]. TMI manifests itself as fluctuations of the output beam profile when a certain average power threshold is reached. These fluctuations are the result of a dynamic energy transfer between different transverse modes of the fiber. In the last decade, a strong effort has been made by scientists to understand the origin of TMI in fiber lasers. It has been found that TMI is caused by a thermally-induced refractive index grating (RIG) [3,4] which is written by a modal interference pattern (MIP) formed by the beating of the fundamental mode (FM) and a high-order mode (HOM), typically the LP$_{11}$, in the fiber core. Moreover, a phase-shift between the RIG and the MIP is required for TMI to happen [5].

Thus far, different methods have been proposed to increase the TMI threshold. According to [6,7], gain saturation plays a major role in TMI. In fact, the TMI threshold can be enhanced by increasing the gain saturation. This can be done, for instance, by decreasing the core diameter, changing the pump configuration, increasing the seed power, or shifting the pump and seed wavelenghts [2,6,8–10]. Also, in step-index fibers with a fixed geometry, the reduction of the NA and, thus, the V-parameter leads to higher TMI thresholds. Hereby, by coiling the fiber, high losses are introduced for the HOM and only the FM obtains significant net amplification [11,12]. In fibers with very large mode field diameters (MFD), which are typically used as straight rods, other methods for suppressing TMI have been presented. Some of these are based on washing out the thermally-induced RIG. This has been achieved, for example, by dynamic excitation [13] of the fiber or by pump modulation methods [14]. These techniques have proven to be effective, however, they require active control elements which increases the system complexity and hinder their widespread adoption. In fact, the laser community favors passive mitigation strategies. Most of the passive mitigation strategies presented to date are based on fiber designs such as SIF, distribute mode filter fibers (DMF) [15], large pitch fibers (LPF) [16,17] or fibers with pre-compensated refractive indexes [18]. Interestingly, none of the abovementioned fibers are polarization maintaining (PM). However, there is a fundamental interest in developing fibers that preserve a stable linear polarization state of the output beam. This interest is driven by applications where a stable polarization output is required, such as in glass material processing, nonlinear conversion and coherent polarization beam combining. In this context, a novel passive TMI mitigation technique for suppressing TMI in PM fibers was presented in [19], based on coupling light at a polarization input angle of around 45$^{\circ }$ in a PM fiber. Additionally, a different research group observed in a monolithic tapered PM fiber, tested under different angles with an offset splice, an increase in the TMI threshold when the angle was away from the main axes of the fiber [20]. However, due to the lack of detail in the guiding conditions of the fiber used and in the experimental setup, the results are difficult to interpret. In a previous publication from our group [21] we have demonstrated the enhancement of the TMI threshold when detuning the polarization from the slow-axis, but the fiber was efficiently single-mode in the fast-axis.To make the analysis of TMI in PM fibers more general, an experimental study with a fiber which is few-mode in both principal polarization axes is needed.

In this work, we present an experimental demonstration of the mitigation of TMI in PM fibers according to the technique proposed in [19]. Simulations of TMI are presented in a PM fiber under different linear polarization input angles to illustrate the mitigation method. They show that the RIG can be washed out when the polarization input angle is aligned around 45$^{\circ }$ with respect to the slow-axis of the fiber. Furthermore, a proof of principle experiment with a few-mode LMA PM fiber is presented, which verifies the numerical simulations. This fiber was tested in a counter-propagating amplifier scheme. The TMI threshold was measured systematically at different polarization input angles of the seed laser. We found that the maximum TMI threshold was reached at a polarization angle of 50$^{\circ }$ and it was a factor of two higher than the TMI threshold in the slow-axis and 60% higher in the fast-axis. Moreover, we measure the beam fluctuations as a function of the polarization input angle at a fixed output power, revealing the three different temporal regimes of TMI: stable, transition and chaotic. The beam fluctuations are the highest with the polarization aligned parallel to the slow-axis. This result in good agreement with the theoretical predictions. Finally we present the analysis of the beam fluctuations with a polarization mode resolved setup, which demonstrate that the beam fluctuations occur at two polarization axes simultaneously.

The paper is structured as it follows: at first in section 2, we describe the operation principle and the simulations performed to illustrate the mitigation method. In section 3, we describe,in detail, the optical fiber and the guiding conditions used in the experiment, as well as the experimental setup. Thereafter, we present the experimental results and evaluate the performance of this mitigation method. Finally, we draw some conclusions in section 4.

2. Operation principle and simulations

It is widely known that PM fibers offer a significant degree of linear birefringence that gives rise to two main polarization axes (a fast and a slow one), which can preserve the polarization of a beam coupled into them. Crucially, as described in [22,19], each transverse mode of a PM fiber experiences a different birefringence. This, in turn, implies that the effective refractive index separation between two modes will be different for the slow and for the fast-axis of a PM fiber. As shown in Fig. 1 (upper and middle subplots), this leads to MIPs, e.g., between FM and the first HOM, that have slightly different periods in the slow-axis and in the fast-axis. This is the cornerstone of the mitigation strategy for TMI explored in this work. In fact, if the light is coupled in a PM fiber with a 45$^{\circ }$ polarization orientation with respect to the main axes, both MIPs of the slow and fast axis will co-exist and overlap in the fiber. As shown in the bottom part of Fig. 1, this results in a MIP that is regularly washed out (encircled areas in the bottom subplot). These regions of low visibility correspond, as schematically illustrated in Fig. 1 with the vertical dashed lines, with points at which the MIPs of the slow and fast axes are exactly out of phase. Note that the occurrence of these regions is not perfectly periodic along the fiber. This is due to the change of the birefringence seen by the different modes along the fiber, which is induced by the temperature gradient. A MIP with such washed out regions has deep repercussions for TMI since, as shown in Fig. 2 it will generate an inversion profile that mimics it and, therefore, also exhibits regions of low visibility (see middle subplot of Fig. 2). This, in turn, leads to a thermally-induced RIG that also has washed out regions (lower subplot of Fig. 2). Please note that only the radially anti-symmetric part of the RIG is shown in Fig. 2, since this is the one responsible for the coupling between the FM and the first HOM. In principle, such a RIG with washed out regions should have a significantly weakened coupling strength, which should, in turn, lead to an increase of the TMI threshold. In order to test the effectiveness of the proposed TMI mitigation strategy, we have carried out several simulations using the laser and TMI simulation tool described in [14]. The main modification that we have made to the original model is to incorporate a way to calculate the stress-induced birefringence in PM fibers. This model is an expanded and generalized version of the one proposed in [23]. The main feature of this simulation tool is that it incorporates, for the first time to the best of our knowledge, the impact of temperature on the modal birefringence of the fiber. This effect leads to the non-periodic distribution of the washed-out regions of the MIP shown in Fig. 1 and 2 as already discussed above.

 

Fig. 1. Modal interference intensity patterns created by beating between the FM and the first HOM of a PM fiber in the slow axis (upper subplot), in the fast axis (middle subplot) and when the polarization of the coupled light has a 45$^{\circ }$ orientation with respect to the main polarization axes of the fiber (lower subplot). The fiber parameters used to generate these plots and predict the TMI thresholds are described below.

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Fig. 2. MIP (upper subplot), inversion pattern (middle subplot) and radially anti-symmetric component of the thermally-induced refractive index change (lower subplot) in an Yb-doped, PM fiber seeded by linearly polarized light oriented at 45$^{\circ }$ with respect to the polarization axes.

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We have simulated a 1 m long, rod-type, step index fiber with 80 $\mu$m core (V=7, $\Delta n=2.8\cdot 10^{-4}$) and 280 $\mu$m pump cladding diameter. The fiber is doped with an Yb$^{3+}$ ion concentration of 3.25 $10^{25}$ ion/m$^3$, it is seeded with 28.5 W in the FM (LP$_{01}$) mode and with 1.5 W in the first HOM (LP$_{11}$) at 1030 nm and it is pumped at 976 nm from the counter-propagating direction. The PM fiber is a panda-type (with a birefringence for the LP$_{01}$ mode of $\sim 2\cdot 10^{-4}$). With these parameters the model predicts a TMI threshold of $\sim$325 W when coupling light in the fast-axis of the fiber and of 310 W when coupling light into the slow-axis of the fiber. These TMI thresholds were obtained by increasing the power in 25 W steps until persistent oscillations in the mode content were observed in the simulations at a time scale of 8 ms. The TMI threshold was taken as the last power in which no oscillations were present. Please note that, while the simulated birefringence and fiber length are reasonably close to those of the fiber used in the experiments, the fiber core considered in the simulations is significantly larger. This has been done to reduce the computational cost of the simulations, since larger cores allow working with lower spatial resolutions in our model. Accordingly, the seed power has also been scaled up with respect to that used in the experimental setup, so that the seed intensity is roughly similar both in the experiment and in the simulations. These discrepancies in the parameters will not allow doing a one to one comparison of the absolute values of the TMI threshold between the experiment and the simulation. However, it is expected that the relative change in TMI predicted by the simulation should be also seen in the experiments.

The first numerical experiment that we have carried out is to operate the fiber at a constant output power of $\sim$405 W and rotate the orientation of the input polarization. At the same time, we have recorded the relative intensity noise (RIN) of the fluctuations (i.e., the rms value divided by the average value). Since we are operating above the TMI threshold of the system, this should give an indication of the ability of this technique to suppress TMI. Thus, in this simulation, a higher RIN value should correspond to a lower TMI threshold and viceversa. The results can be seen in Fig. 3. Three regions are observed: from 0$^{\circ }$ to 45$^{\circ }$, the beam is unstable and the fiber operates above the TMI thresholds. The oscillations are the highest with angles close to 0$^{\circ }$ and decrease when increasing the angle towards 45$^{\circ }$. Between 45$^{\circ }$ and 80$^{\circ }$ a plateau is reached that corresponds to the stable regime below the TMI threshold for these angles. Finally, a sharp increase in the beam fluctuations is obtained at 90$^{\circ }$. Thus, operating with orientations of the polarization that do not match the main polarization axes should lead to higher TMI thresholds. In particular, this simulation predicts a maximum increase of the TMI threshold for an orientation of the input polarization of $\sim$45$^{\circ }$ with respect to the main polarization axes. Also note that, as the polarization is rotated further (from 90$^{\circ }$ to 180$^{\circ }$), this pattern is expected to repeat itself, albeit mirrored. Another prediction of this model is that the slow axis (0$^{\circ }$) should have a somewhat lower TMI threshold than the fast axis (90$^{\circ }$), as already mentioned above. In order to confirm these predictions, we have looked at the temporal evolution of the beams at 405 W when the input polarization is oriented in the fast axis (Fig. 4(a)) and when it is oriented at 45$^{\circ }$ (Fig. 4(b)). As it can be seen, the system operates above the TMI threshold on the fast axis, whereas the output beam is stable when orienting the input polarization at 45$^{\circ }$. This confirms the mitigation of TMI when operating in the latter configuration. In fact, we have calculated the TMI threshold for the 45$^{\circ }$ input polarization and it is $\sim$475 W, which corresponds to, roughly, a $\sim$50% increase of the TMI threshold with respect to the configuration when the input polarization is oriented parallel to the fast axis. One thing that becomes apparent in Fig. 4(b) is that the evolution of the modal contents in the fast and slow axes is different. This is due to cross-talk between the polarization axes caused by the RIG induced by the MIP of one of the polarization causing modal energy transfer for the other polarization. This phenomenon leads to a quasi-static modal energy transfer, akin to that predicted in [24,25]. Further discussion about this phenomenon falls outside the scope of this paper. However, more information can be found in [26].

 

Fig. 3. Evolution of the RIN of the beam fluctuations at the output of the amplifier (when operated at a constant power of $\sim$405 W) as a function of the orientation of the input polarization.

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Fig. 4. Temporal evolution of the modal content (FM in blue and HOM in red) at the output of the amplifier when operated at 405 W. The solid lines represent the energy in the slow axis and the dashed lines represent energy in the fast axis. Two cases have been simulated: (a) the input polarization is oriented parallel to the fast axis and (b) the input polarization is oriented at 45$^{\circ }$ with respect to the main polarization axes of the fiber.

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This mitigation method has the drawback that if the linear polarization is not aligned parallel to the principal polarization axes, the incoming linear polarization will not be maintained, and elliptical polarization will result. However, elliptical polarization can straightforwardly be converted back to linear polarization at the output using a quarter-waveplate.

3. Experimental results

3.1 Fiber design

In order to test the theory and the numerical results, we have designed and drawn a LMA PM fiber. The fiber has a core diameter of 36 $\mu$m and a cladding diameter of 200 $\mu$m. The refractive index step ($\Delta n=n_{core}-n_{clad}$) in the fiber is $\Delta n\sim 4.5\cdot 10^{-4}$, resulting in a numerical aperture (NA) of 0.036. Thus the fiber has a V-parameter of 4. It also has two stress-applying parts (SAP), which break the symmetry of the fiber and, therefore, the degeneracy of the optical modes. Generally, in PM fibers the slow-axis is defined parallel to the line connecting the SAP, whilst the fast-axis perpendicular to it. Due to the stress induced by the SAP, the refractive index in the slow-axis is higher than in the fast-axis [27]. We have calculated the mechanical stresses and the supported modes in the fiber, following the approach described in [28]. The resulting refractive index profile in both the slow-axis and fast-axis is presented in Fig. 5. Thus, we obtain $V_{slow}\sim 4.3$ and $V_{fast}\sim 3.8$ for the slow and fast-axis, respectively. Hereby, it can be seen that the slow and fast-axis are both few-mode: the slow-axis supports the propagation of 6 modes, whilst the fast-axis supports only 3 modes. However, in this work, only the FM and first HOM are going to be considered, since they are the relevant modes at the TMI threshold. They are described here as linearly polarized (LP) modes: LP$_{01}^{\mathrm {slow}}$ (FM, slow-axis); LP$_{01}^{\mathrm {fast}}$ (FM, fast-axis); LP$_{11,\mathrm {e}}^{\mathrm {slow}}$ (first HOM, slow-axis, even); LP$_{11,\mathrm {o}}^{\mathrm {slow}}$ (first HOM, slow-axis, odd); LP$_{11,\mathrm {e}}^{\mathrm {fast}}$ (first HOM, fast-axis, even) ; LP$_{11,\mathrm {o}}^{\mathrm {fast}}$ (first HOM, fast-axis, odd). The FM birefringence of the fiber was calculated to be $B_{01}=1.56\cdot 10^{-4}$ and measured to be $B_{01}=1.47\cdot 10^{-4}$, which matches well. The slight difference between the expected and the measured values is due to geometry changes between the idealized simulations and the real fiber. The simulated HOM (even) birefringence was $B_{11,e}=1.76\cdot 10^{-4}$. Accordingly, the LP$_{01}$-LP$_{11,\mathrm {e}}$ beat length was calculated to be 7.0 mm and 6.1 mm for the slow and fast-axis, respectively. The fiber length was 1.2 m and was coiled to a diameter of 28 cm. With this coil diameter, losses are only expected for the two LP$_{11}^{\mathrm {fast}}$ modes, since the indexes of these modes are close to the one of the cladding. Therefore a higher TMI threshold is expected in the fast-axis of the fiber.

 

Fig. 5. Fiber design and geometry. Refractive index step along the vertical coordinate as a function of the radius. Slow-axis (blue), fast-axis (red). Fiber geometry (inset) with polarization aligned along the slow-axis (blue arrow) and fast-axis (red arrow), respectively. The dashed lines correspond to the $\mathrm {n_{eff}}$ of the LP$_{01}$ and LP$_{11}$ modes with their corresponding intensity profiles.

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3.2 Experimental setup

We have performed the systematic measurements of TMI using a counter-propagating fiber amplifier, similar to the one described in [21], which schematic setup is shown in Fig. 6. The fiber was seeded by a continuous wave (CW) 1030 nm signal, in which the spectrum was broadened to 120 pm to avoid Brillouin Scattering. The seed was amplified to 5 W in a first stage. The pre-amplifier was isolated and therefore the polarization state at its output was linear. The polarization input angle in the main amplifier was controlled with a half-waveplate (WP1, see Fig. 6). The fiber was mounted with the SAP parallel to the horizontal axis, it was coiled with a diameter of 28 cm without twist and it was water cooled. The fiber was pumped by a wavelength-stabilized diode centered at 976 nm, and with a maximum output power of 800 W. At the output, two wedge reflections were used to characterize TMI. One of the reflections was imaged onto a photodiode to record the fluctuations as described in [29]. The other reflection was analyzed with a high-speed full vectorial mode-resolved setup, described in [30]. This setup contains a high speed camera which has a maximum frame rate of 30,000 fps. Thus, it allows observing the beam fluctuations in the sub-ms regime, allowing to monitor the polarization of the individual modes under the influence of TMI.

 

Fig. 6. Schematic representation of the experimental setup. ISO: isolator; WP1: half-waveplate used to rotate the incoming polarization; M:mirror; DM: dichroic mirror; PM: powermeter; L: lens; W: wedge.

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3.3 Results

The TMI dependence on the linear input polarization angle was obtained by rotating the half-waveplate (WP1) in front of the fiber amplifier in 5$^{\circ }$ steps. Rotating the waveplate by an angle $\theta$ corresponds to a rotation of the linear polarization of $2\theta$. Thus, we systematically found the TMI threshold at different polarization input angles in 10$^{\circ }$ steps. At each fixed angle, the TMI threshold was measured. In order to do this, the pump power was increased until TMI was observed as strong fluctuations in the photodiode signal. Thereafter, the power was reduced and a fine scan of the power was performed to accurately determine the TMI threshold. The pump power was increased stepwise until the first fluctuations of the beam were observed in the photodiode, indicating an abrupt change in the beam stability. We recorded photodiode traces at powers below the TMI threshold, at the threshold and above it. We took 10 s traces with a resolution of 10 $\mu$s, which is much shorter than the temporal characteristics of TMI. This procedure was carefully repeated at each polarization input angle, from 0$^{\circ }$ to 180$^{\circ }$, covering a complete rotation of the linear polarization. As shown in Fig. 7, we measured a strong dependence of the TMI threshold on the polarization input angle. The lowest TMI threshold was found with the polarization aligned parallel to the slow-axis ($\sim$160 W). Then, the TMI threshold increased with larger angle up to 50$^{\circ }$, where we found the maximum TMI threshold ($\sim$350 W). For larger angles, the TMI threshold decreased to 90$^{\circ }$, corresponding to a polarization parallel to the fast-axis ($\sim$210 W). We have measured an increase in the TMI threshold of more than 100% with respect to the TMI threshold of the slow-axis and 60% with respect to that of the fast-axis. Please note that in this fiber, the TMI threshold of the fast-axis was measured to be higher than that of the slow-axis. The fast-axis has a lower V-parameter than the slow-axis, resulting in higher TMI threshold as it has been already explained in [21]. This represents a significant difference with respect to our simulation, where this TMI threshold difference was not so pronounced. The reason for this behaviour is that in our simulations the change in V-parameter between the slow and fast axis was only minor and a straight fiber was simulated. In the experiments, the fiber is coiled and the LP$_{11}$ in the fast-axis must experience bending losses. We covered a complete rotation from the slow-axis to the fast-axis. These results match fairly well the predictions presented in section 2. (at least for the fast-axis, as explained above). The curve from 0$^{\circ }$ to 90$^{\circ }$ is mirror symmetric with respect to 90$^{\circ }$, as it was expected and mentioned in section 2. A slight asymmetry can be identified in the curve, with small thresholds changes at some mirrored angles (for instance, 130$^{\circ }$ or 180$^{\circ }$). Since the change at 130$^{\circ }$ is of $\sim$20W, which is higher than the precision of the measurement of the power-meter (typically 2-3%), the reason of this threshold might be a slight change in the amplifier setup (for instance seed coupling) when the mirrored measurement was performed in a second run. The time evolution of these photodiode traces are presented in Fig. 8. Two traces are selected for each of the three polarization angles in Fig. 8, below and above the TMI threshold. The temporal evolution obtained with the photodiode in the three cases consist of a periodic fluctuation with a similar frequency. Typically, at powers around the TMI threshold, the beam oscillates in a periodic fashion with a well defined frequency [29]. In our case the frequency was of $\sim \,1.3\,$kHz for the three analyzed cases. The larger the MFD of a fiber, the longer the period of the oscillations. The three TMI fluctuations are measured at different polarization angles but with the same fiber and same MFD, therefore the same periodicity is expected.

 

Fig. 7. Dependence of the TMI threshold on the linear polarization input angle. TMI threshold measured at each polarization input angle.

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Fig. 8. Temporal evolution of the beam fluctuations for operation in the slow-axis (blue), fast-axis (red) and at 50$^{\circ }$ (purple). Selected traces from data in Fig. 7 normalized to the mean value in 10 ms.

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In order to compare the simulations and experimental results, we operated the fiber at a constant power of $\sim$285 W, which is above the TMI threshold in both the slow and fast-axis of the fiber. Thus we can reproduce the results shown in Fig. 3. We analyzed the time traces by calculating the standard deviation (STD) of the beam fluctuations in 0.5 ms intervals to determine the spreading of the STD points. The results corresponding to the beam fluctuations dependence on the polarization input angle at a fixed power are presented in Fig. 9. The blue points in Fig. 9 (a, left subplot) are the STD in different analyzed time intervals of 0.5 ms, whilst the white circles represent the average STD of all time intervals. Analyzing the evolution of the STD of the photodiode traces and their corresponding spectra reveals different stability regions characteristic of TMI. At 0$^{\circ }$(slow-axis) and 10$^{\circ }$, the STD shows the largest spreading of points and the spectrum reveals a continuum of frequencies, which is characteristic of the chaotic regime of TMI [29]. This can also be seen in the trace of Fig. 9 (b, bottom plot). As the angle is increased towards 50$^{\circ }$, the fluctuations starts to become periodic and the spreading of the points decreases. The spectra shows the presence of discrete frequencies between 15$^{\circ }$ and 45$^{\circ }$, which correspond to the transition regime of TMI. From 50$^{\circ }$ to 60$^{\circ }$, we reached the expected plateau which has been theoretically predicted by the numerical simulations. This correspond to a stable regime where TMI is mitigated, which can be also seen in the time trace at 50$^{\circ }$, Fig. 9 (b, middle subplot). In our experiment the stable regime is reached at 50$^{\circ }$ and not at 45$^{\circ }$ as predicted by the numerical simulations. The reason for this slight discrepancy might be due to the differences between the simulated and experimental fiber. As the angle is increased further, an abrupt increase in the STD is obtained and as the polarization becomes close to 90$^{\circ }$ (fast-axis) the beam starts to fluctuate again and the transition TMI regime is reached. Here the fluctuations are again periodic, as it can be seen in the spectra as discrete frequencies and in the periodic time trace in Fig. 9 (b, upper subplot). It must be emphasized, that in contrast to 0$^{\circ }$ (slow-axis), at 90$^{\circ }$ (fast-axis) the fluctuations are periodic (inset of Fig. 9).

 

Fig. 9. Evolution of STD deviation of the beam fluctuations at the output of the amplifier (operated at a constant power of 285 W) as a function of the polarization input angle, and corresponding spectra (a). Photodiode traces at selected angles 0$^{\circ }$ (lower subplot), 50$^{\circ }$ (middle subplot) and 90$^{\circ }$ (upper subplot) (b).

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Thus, changing the polarization input angle allows to cover the different regions characteristic of TMI: stable, transition and chaotic [29]. And this, at the same average power level, in contrast to the work in [29], where the change of regimes is due to an increase in the average power. As it has been discussed in [31], at average powers that exceed the TMI threshold the RIG has a strength that make it sensitive to small phase shifts between the MIP and the RIG, caused by perturbations of the system such as signal or pump noise. In our experiments, at 0$^{\circ }$ (slow-axis) the system is operated at a power which exceeds considerably the TMI threshold. Therefore the grating strength is high and the RIG is very sensitive to small phase shifts, leading to chaotic fluctuations (chaotic regime). As the angle is increased, i.e., the linear polarization is detuned from the slow-axis, the contrast of the MIP starts to decrease and the RIG strength is reduced. Thus, the system starts to get more robust against phase shifts. As a result, the fluctuations start to decrease and become periodic (transition regime) until 50$^{\circ }$ where the system operates with a stable output beam (stable regime). In this explanation is assumed, that we do not change the phase shift between the MIP and the RIG by rotating the linear polarization of the seed. As the angle is increased further and it approaches the fast-axis the RIG strength increases again leading to the unstable regime with periodic fluctuations (transition regime). These experimental results prove that the strength of the RIG can be reduced by rotating the polarization input angle in a PM fiber and confirms the theory and numerical predictions.

Finally, we have performed a detailed characterization with the high-speed polarization mode-resolved setup for input polarization parallel to the the slow-axis and at 50$^{\circ }$. With this setup, we can characterize TMI at both principal polarization axes of the fiber, independently, and determine if both polarization axes fluctuates simultaneously at the onset of TMI. From the polarization high-speed measurements, we can extract the full information about the polarization of each mode, in each polarization axis of the fiber with sub-ms resolution. In Fig. 10(a) and (b) we present the normalized mode contents of the slow-axis and fast-axis of the fiber, respectively, for an input polarization parallel to the slow-axis. Please note that here, the power of the slow-axis was much larger than in the fast-axis (PER$\sim$13 dB). As it can be seen, both output beams corresponding to the slow and fast-axis of the fiber fluctuates above the TMI threshold. The RIG generated in the slow-axis induces fluctuation in the beam profile of the fast-axis, as well. And this, even though its power is of few watts (<10 W). For 50$^{\circ }$ and at 371 W, which is above the TMI threshold at this angle, we can see again that both axes fluctuate with a defined frequency. It is evident from Fig. 10 that the main HOM present is the LP$_{11,\mathrm {e}}$ in both cases. This mode has lower losses than the odd one. The even LP$_{11}$ mode has its lobes aligned with the SAPS and they act as a barrier from tunnelling of the mode due to bending of the fiber. It can be therefore concluded, that TMI triggers the fluctuations of the output beam profile in both polarization axes of the fiber.

 

Fig. 10. High-speed mode content in the different polarization axes of the fiber under 0$^{\circ }$ and 50$^{\circ }$ operation. Slow-axis (a) and fast-axis (b) output for slow-axis operation. Slow-axis (c) and fast-axis (d) output for 50$^{\circ }$ operation.

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4. Conclusion

In this work we have presented a novel and simple way of increasing the TMI threshold by rotating the linear polarization input angle in a PM fiber amplifier. We have presented a theoretical explanation of the method, which was supported by numerical simulations. The RIG is partly washed out by operating the fiber with an input angle around $\sim$45$^{\circ }$ (between the slow-axis and the fast-axis). This occurs due to modal birefringence in PM fibers, which leads to different beat lengths in the slow and fast-axis of the fiber. The difference of beat length leads to a MIP which presents regions with low-visibility. This in turn, reduces the strength of the RIG that triggers TMI. To confirm the theoretical predictions, an in-house LMA PM fiber was designed, fabricated and tested in a counter-propagating amplifier scheme. We have demonstrated, with photodiode measurements, that the TMI threshold can be doubled when rotating the polarization input angle from the slow-axis to 50$^{\circ }$. Furthermore, at fixed average power above the TMI threshold of both polarization axes, the beam fluctuations can be reduced by operating the fiber between the slow and fast-axis, confirming the numerical results. Finally, we have presented a high-speed mode-resolved polarization analysis of the output beam. This has revealed that TMI triggers the fluctuation of the output beam profile in the two polarization axes of the fiber. These experimental results, which are in good agreement with the theoretical predictions, provide further insights in the understanding of TMI and paves the way to stabilize high-average power PM fiber lasers.