1. Introduction

Since its first observation in 2011 [1], transverse mode instability (TMI) has become the main limitation in power scaling of high average power fiber lasers with near diffraction-limited beam profiles [2]. TMI is caused by a thermally induced refractive index grating (RIG) and results in a threshold-like onset of a fluctuating modal energy transfer between the fundamental mode (FM) and higher order modes (HOM) [3–7]. The frequencies of TMI-fluctuations are limited by the thermal diffusion time in the fiber core and can reach tens of kHz for fiber core radii smaller than 15 µm [8,9]. Therefore, photodiodes and high-speed cameras (HSC) are usually employed to measure TMI. Over the last decade, the development of novel beam characterization [8,10,11] and measurement techniques [12] has enabled a better understanding of the fundamental properties of TMI. Most systematic TMI studies to date have been performed with non-polarization-maintaining (PM) fiber lasers. However, TMI in PM fibers show unique features. Simulations [13] and measurements [14–16] have shown that the TMI-threshold power depends on the linear input polarization angle. Tao et al. [17] observed a decrease of the polarization extinction ratio (PER) at the TMI-threshold and very recently, Palma-Vega et al. [15] measured a polarization static energy transfer from the fast axis FM to the slow axis HOM. Both effects are not fully explained so far. To understand how the individual modes interact in PM-fibers under the influence of TMI, novel polarization-resolved characterization methods in the sub-ms regime are required.

Polarization-resolved mode analysis is generally possible by different techniques, such as mode-matching to ring resonators [18], holographic methods compromising computer-generated holograms [19], spatial light modulators (SLM) [20,21] or digital mirror devices (DMD) [22] as well as with imaging polarimeters [23] in combination with suitable mode reconstruction algorithm [24,25]. The main limitation of these methods to be used for TMI measurements is their achievable measurement speed. In the case of the latest holographic techniques [21,22], measurement rates of 4 kHz [26,27] were demonstrated, but are currently limited to $\sim$ 20 kHz based on the DMD-refresh rates [28]. However, some camera-based imaging polarimeters can be further accelerated by the use of HSC. Divison-of-time Stokes polarimeter measure four to six [29] differently polarized intensity profiles sequentially in time. They usually rely on rotating polarizing elements, limiting the achievable measurement speed to a few hundred Hz [30,31]. Faster polarization measurements are enabled by simultaneous detection of these different intensity profiles, as in division-of-focal-plane- (DoFP) [32,33] or division-of-amplitude- (DoA) approaches [34]. The measurement speed of most imaging DoA- and DoFP Stokes polarimeter has been limited so far by the utilized cameras to video rates up to 100 Hz [35–37]. However, faster detectors such as HSC are available. Using pixelated polarizer arrays and a multichannel A/D converter, polarization acquisition from 32x4 pixels at ultra-high speeds up to 1.3 MHz was demonstrated [38] and subsequently commercialized [39]. The fastest DoFP-polarization camera [39] is optimized for measurements at 520 - 570 nm and limited to linear polarization imaging (1024x1024 pixels with 7 kHz). It is therefore currently not suitable for our application.

We have developed a novel high-speed Stokes polarimeter technique [40] that overcomes the speed limitation of current imaging polarimeters for wavelengths around 1 µm, which is presented in section 2. A simultaneous 4-beam mode reconstruction algorithm, which is described in subsection 2.3, enables the mode-resolved Stokes parameters to be calculated. We demonstrate the measurement capabilities of the polarimeter setup by characterizing a TMI-limited large-mode-area (LMA) Ytterbium- (Yb-) doped PM fiber amplifier in 35/200 µm geometry, which is introduced in section 3. The results are presented and discussed in section 4 and provide, for the first time, insights into modal polarization fluctuations at the onset of TMI.

2. High-speed imaging polarimeter

2.1 Polarimeter setup

To enable fully polarization- and temporally-resolved detection of TMI in Yb-doped fiber lasers, operating between 1.0 - 1.1 µm, we decided to develop a novel imaging polarimeter. We chose a classical DoA-approach with four parallel channels imaged onto a standard high-speed camera (HSC, model: FASTCAM NOVA S6 [41]). Our HSC-based Stokes polarimeter setup is shown schematically in Fig. 1. Similar to [42], the collimated incoming laser beam is divided into the four channels by non-polarizing beam splitters (BS). Linear polarizers with respective angles to the optical axis of $\theta = 0^\circ, 90^\circ$ and $-45^\circ$ are placed in three channels, while a quarter-wave ($\lambda /4$-) plate with the fast axis in the x-direction ($\theta = 90^\circ$), followed by a polarizer at $\theta = -45^\circ$, are inserted in the fourth arm. Zero phase shift mirrors (M) were utilized for the two $90^\circ$ beam deflections shown in Fig. 1. For a near-field imaging, the four differently polarized intensity profiles $I_{0^\circ }, I_{90^\circ }, I_{-45^\circ }$ and $I_\sigma$ are focused into a rectangular array on the HSC with plano-convex lenses of focal length $f' = 1.5$ m. To keep a breadboard-sized setup, the focused optical path was folded twice (not shown in Fig. 1). For our measurements, the detector area was reduced from the maximal resolution of 1024 x 1024 pixels (6400 frames per second, short: fps) to 512 x 386 pixels to achieve measurement speeds of 30000 fps [41]. A single pixel is 20 $\mu$m x 20 $\mu$m.

Assuming ideal components and a ray with a state of polarization (SoP) described by its Stokes vector $\vec {S}=(S_0,S_1,S_2,S_3)$, each pixel receives an intensity depending on the polarizer angle $\theta$ and waveplate-induced phase $\phi$ in front of it [29], according to

Vice versa, the parameters of $\vec {S}$ can be detemined by four appropriate angle/phase sets and thus intensity measurements. Inserting the polarizer angles $\theta$ and $\lambda /4$-plate phase of $\phi =\pi /2$ from our polarimeter’s beam paths in Eq. (1) and simple algebraic rearrangement yields

 

Fig. 1. Schematic setup of the 4-channel high-speed polarimeter. Abbreviations: BS - beam splitter, M - mirror, W - optical wedge, $\lambda /4$ & $\lambda /2$ - quarter- & half-wave plates.

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2.2 Calibration

Due to component- and alignment imperfections a calibration has to be performed to obtain the real system matrix $M_{cal}$. Since

Our calibration was performed prior to the measurements by adding a polarizer, a $\lambda /4$- and a half-wave ($\lambda /2$-) plate in front of the polarimeter, as shown in the dashed extension in Fig. 1. First, the polarizer was adjusted once to achieve a linear input polarization of a specific power level and was held fixed throughout the rest of the calibration process. Then, any desired input SoP, $\vec {S}_{in}$, could be set with the two waveplates. Utilizing the front-side reflex of an optical wedge (input angle $< 10^\circ$), $\vec {S}_{in}$ was controlled on a commercial, photodiode-based reference polarimeter (Thorlabs: PAX5710). For each $\vec {S}_{in}$, 100 frames were measured at 30 kfps. After dark current and background correction, rectangular sections around the center of mass of each beam were selected (see Fig. 2(b) top), averaged over all frames, and their gray values added. This way, the (spatially averaged) intensity array $\vec {I}_{out}$ was obtained. The calibration matrix $M_{cal}$ was then calculated according to Eq. (3) by solving the linear equation system.

 

Fig. 2. (a) HSC-based polarimeter measurements (blue) of random states of polarization (SoP) of a singlemode laser beam after calibration with 7 known Stokes values compared to the reference polarimeter (red) yields only small ellipse angle deviations $< 0.7{\% }$. (b) Schematic analysis from video frames to Stokes images.

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We observed better calibration results for an overdetermined linear system with 7-12 calibration vectors $\vec {S}_{in}$ (including the linear SoP: H, V, A, D, the circular SoP: R, L and random elliptic SoP). Exemplary measurements were performed utilizing a linear polarized diode laser at 1064 nm with a singlemode PM980-XP output fiber. After calibration with 7 Stokes vectors, random input SoP were set and measured with the PAX5710 and the HSC-based polarimeter. The results are compared in Fig. 2(a). The average azimuth- and elliptic angle differences were $\Delta \psi = (1.01 \pm 0.82)^\circ$ (azimuth) and $\Delta \chi = (0.61 \pm 0.42)^\circ$ (elliptic), respectively. Since the angles are defined from $0 \leq \psi \leq \pi$ and $-\pi /4 \leq \chi \leq \pi /4$, the maximum azimuth and elliptic angle difference could be $\Delta \psi _{max}=180^\circ$ and $\Delta \chi _{max} =90^\circ$. Therefore, the average angle differences between the reference polarimeter and HSC-based setup corresponded to relative deviations of $\Delta \psi /\Delta \psi _{max} = (0.56 \pm 0.46){\% }$ and $\Delta \chi /\Delta \chi _{max} = (0.68 \pm 0.47){\% }$. The PAX5710 angle accuracy is specified as $\pm 0.25^\circ$ [43], which is smaller than the measured angle differences to our setup. Nonetheless, this result is confirming that most component imperfections were corrected. Furthermore, the small deviations indicate that the remaining image noise (included in $\vec {I}$) has only a marginal effect on the calculated $\vec {S}$.

After calibration, polarization imaging is enabled, as shown by the exemplary Stokes-images $S_0(x,y)$ to $S_3(x,y)$ in Fig. 2(b). The spatial resolution is ideally given by the pixel size, but in reality it is limited by the accuracy of beam center determination and differences in beam sizes. Therefore, the original 180x180 pixels crops from the HSC-video were interpolated to 64x64 pixels. The beam profile, consisting of 90.5 % FM- and 9.5 % HOM-content, can clearly be differentiated from the surrounding noise. With this high-speed polarization imaging method, complete Stokes information about the incident beam profile are obtained. Therefore, it should be generally applicable for the characterization of more complex laser beams. However, additional calculations are required to obtain a mode-resolved SoP from the detected beam profiles.

2.3 Simultaneous 4-beam modal reconstruction algorithm

Since our experiments are performed with weakly-guiding LMA fibers ($n_{\text {core}} \sim n_{\text {clad}}$), we use the description of linearly-polarized (LP) modes as orthogonal mode set, as done by Flamm et al. [44]. The vector field $\vec {U}_{in}(\vec {r})$ guided in a fiber can be expressed as superposition of a finite number $n$ of LP-modes by

They contain the unknown phases $\varphi _k, \delta _k$ and amplitudes $A_k^{x}, A_k^{y}$ as variables and are used for the modal reconstruction. The transverse mode distributions $\psi _{k}$ are pre-calculated, assuming a straight step-index-fiber (SIF) with identical geometrical parameters as the real fiber. Additionally, they are scaled to match the recorded images. Before the reconstruction, the raw images of the four beams measured on the HSC ($I_{0^\circ }^m$ - $I_{\sigma }^m$) are first background-corrected. Then, equally-sized rectangular sections of i x j pixels are extracted around each beam’s center of mass, calculated as average over all frames. Additionally, an intensity calibration correcting component imperfections is performed using Eq. (3) and (4), obtaining $\vec {\tilde {I}}_{i,j}^m = M_{corr}\cdot \vec {I}_{i,j}^m$. Finally, a numerical minimization of the error function $\Delta (A_k^{x},A_k^{y},\varphi _k, \delta _k)$ is performed according to

The simultaneous minimization of $\Delta$ for all four beams ensures that the beams of high intensity have a higher impact on the error function and are therefore reconstructed with greater accuracy. Once all ellipse parameters $A_k^x , A_k^y$ and $\delta _k$ are known, the Stokes parameters of each mode k can be calculated [29]:

An example of the modal reconstruction is shown in Fig. 3. The measured multimode beam profiles in the four polarimeter channels (left column) are simultaneously reconstructed according to Eq. (6–9) (middle column). The differences between the original and reconstructed profiles are shown in the residual plots (right column). The relative residuals are in the order of $\sim$ 20 %, similar to the single-beam near-field modal reconstruction in [10]. Background cladding- and stray light, uncorrected intensity mismatch between the polarimeter channels, differences in detected beam sizes and the reconstruction with a limited LP-mode-set of a straight SIF are the main contributions to these residuals. With the modal fit parameters $A_k^x , A_k^y$ and $\delta _k$, for each mode a Stokes array $\vec {S_k}$ is calculated by Eq. (11). The corresponding normalized mode-resolved SoP, i.e. $\vec {S}_{norm.,k} = \frac {1}{S_{0,k}}\left (S_{1,k},S_{2,k},S_{3,k}\right )$, is illustrated in Fig. 3 on the Poincaré sphere. The implementation of and comparison to other fitting algorithm is beyond the scope of this paper, but may lead to improvements, especially for reconstructions with increasing mode numbers [25].

 

Fig. 3. Image of the simultaneous mode reconstruction of a TMI-limited laser beam (at $P_{out}=249$ W), obtaining all modal information. From these the mode-resolved Stokes parameters are calculated and shown on the Poincaré sphere.

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The mode-resolved Stokes parameters are obtained under the assumption of fully polarized light and will therefore only yield results on the surface of the Poincaré sphere. This mode reconstruction algorithm is applicable for both PM and non-PM fibers in the absence of polarization changes of or between modes faster than the measurements speed (here: 30 kHz). Since the following TMI investigations are performed in a few-mode PM Yb-doped fiber, seeded with a singlemode, linearly polarized input beam, no faster random polarization changes are expected.

3. Experimental amplifier setup

For a conveniently low TMI threshold [2], which is easy to investigate experimentally, an Yb-doped LMA PM-fiber with a core/cladding diameter of 35/200 µm and a core NA of $\text {NA}_{\text {slow}} = 0.04$ was selected and tested in the free-space counter-pumped amplifier setup shown in Fig. 4. The effective mode field area of the fundamental mode was 760 µm$^2$ and the fiber’s birefringence was measured to be $B_{01} = 1.4\cdot 10^{-4}$. The 1.2-m long fiber ($\sim$20 dB pump absorption) was coiled to a diameter of 28 cm into a water basin, ensuring that the stressrods were parallel to the bending plane and avoiding twisting. A wavelength-stabilized 800-W pump diode centered at 976 nm was coupled into the YDF from the opposite direction than the seed laser utilizing dichroitic mirrors (DM). The linear polarized seed light from an external cavity diode laser (ECDL) at 1030 nm was spectrally broadened to 200 pm (3-dB width) by phase modulation with white noise and pre-amplified from 18 mW to a power level of 4.5 W. The beam passed through a polarizing isolator, and was therefore linearly polarized. A $\lambda /2$-plate in front of the fiber input allowed the linear input polarization of the seed laser to be adjusted. The output- and transmitted pump power were measured at PM1 and PM2, respectively. Reflexes of optical wedges were used to analyze the beam on the HSC-based polarimeter.

TMI-thresholds of PM fibers are known to be different in the fast and slow axis [14–16]. Due to stress-induced index changes, the fast axis has a lower V-parameter than the slow axis, which results in higher TMI-thresholds. For our fiber, the V-Parameters are $V_{\text {slow}} \sim 4.3$ and $V_{\text {fast }} \sim 3.8$ and the axes support six and three modes, respectively. For less complexity and better fitting results with a simple 3-mode-reconstruction (see Eq. (6–9)), we chose to operate the fiber near its fast axis. However, seeding the fiber with a linearly polarized seed parallel to one of their birefringence axes will lead to the smallest possible polarization changes. Therefore, the fiber was tested with a linear input polarization angle of $\alpha = 100^\circ$, that is slightly off the fast axis. We label the first three LP-modes ($\psi _k$ with $k = 1,2,3$ in Eq. (5)) considered in the result section $\psi _1 := \text {LP}_{01}^{\text {slow/fast}}$, $\psi _2 := \text {LP}_{11,e}^{\text {slow/fast}}$ and $\psi _3:=\text {LP}_{11,o}^{\text {slow/fast}}$, where e and o are abbreviations for the even and odd $\text {LP}_{11}$-mode. The distinction into the slow and fast axis is chosen to assign the modal amplitudes in x and y from Eq. (5) to the reference system of the fiber.

 

Fig. 4. Counter-pumped amplifier setup with abbreviations: DM - dichroitic mirror, PM - power meter, ISO - isolator, L - lens, ECDL - external cavity diode laser.

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4. Experimental results

4.1 Amplifier efficiency and TMI-threshold

The optical output power of the measured input polarization angle $\alpha = 100^\circ$ is shown in Fig. 5(a) over the launched pump power. The resulting amplification efficiency is $75.9\,{\% }$. To determine the TMI-threshold power, the RF spectral content in the TMI-relevant frequency region was evaluated. For this purpose, 20-ms long time traces in all four polarimeter arms ($I_{0^\circ }$ - $I_\sigma$) were measured with photodiodes, Fourier transformed, and then the standard deviation $\sigma$ was calculated from 0-15 kHz. The results are shown in Fig. 5(b) and the TMI-thresholds are identified by the rapid increase of the normalized standard deviation. The TMI-threshold was defined, where $\sigma$ reached three times it’s average stable value, $P_{\text {TMI}} = P(\sigma = 3\cdot \overline {\sigma }_{\text {stable}})$. The resulting TMI-threshold power averaged over all four channels is $P_{\text {TMI}}(100^\circ ) = (237\pm 1)\,W$. A simultaneous and rapid onset of TMI is observed in all four polarimeter channels.

4.2 High-speed polarimeter measurements at the TMI-threshold

With the HSC-based polarimeter we analyze the TMI onset in greater detail. Slow motion videos of the output beam ($S_0$) are recorded for the output power levels of $P_{out} = 236$ W (stable, Visualization 1), 240 W (TMI onset, Visualization 2) and 249 W (stronger TMI, Visualization 3). Selected video frames at t = 1.1 ms are shown in Fig. 6. Polarization ellipses illustrate the local state of polarization at positions of $S_0 > 0.15$ and are plotted as arrows for elliptic angles $|\chi | < 10^\circ$ for better visibility. The ellipse axes are additionally scaled by the degree of polarization (DOP), given by $\text {DOP} = \sqrt {S_1^2+S_2^2+S_3^2}/S_0$.

 

Fig. 5. (a) Output power over launched pump power, and (b) standard deviation $\sigma$ from 0-15 kHz extracted from FFT of photodiode time traces measured in the four polarimeter arms, normalized to its average value in the stable region, for a linear input polarization of $100^\circ$.

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Fig. 6. High-speed video frames at t = 1.1 ms showing the amplifier’s output beam ($S_0$) at output powers of (a) $P_{out} = 236$ W (stable beam), (b) $240$ W (TMI-onset) and (c) $249$ W (stronger TMI). The local polarization state is illustrated by polarization ellipses in areas where $S_0 > 0.15$. From Visualization 1, Visualization 2 and Visualization 3.

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Fig. 6(a) shows a weak negative ellipticity, but the polarisation remains constant throughout the 10 ms video (see Visualization 1). The maximal DOP is 95.9 %. At the TMI-threshold and above, a well defined oscillating energy transfer between the FM and even HOM occurs and is accompanied by slight changes of the polarization angle and ellipticity (see Visualization 2 and Visualization 3). Fig. 6(b,c) show slightly inhomogeneous transverse polarization profiles, which are the results of the superposition of differently polarized modes. To determine the SoP of each mode, the modal reconstruction described in section 2.3 is applied. We fit the modal amplitudes and phases with Eq. (6–9) and Eq. (10).

Firstly, we present the relative mode contents $|A_k^{x,y}|^2/\sum (|A_k^{x,y}|^2)$ with $k = 1-3$ in Fig. 7, separated into the x- (= slow axis, bottom) and y- (= fast axis, top) components. The presented information without the intramodal phase differences could be extracted from the simultaneous measurement of only two channels, i.e. $I_{0^\circ }$ and $I_{90^\circ }$. The reader should note that for the powers shown, the $LP_{11o}$ mode content is negligible (on average < 0.95 %) even above the TMI-threshold. This is also indicated by the beam fluctuation along the horizontal axis from Visualization 2,Visualization 3. The energy exchange takes place at the TMI-threshold only between the $LP_{01}$ and $LP_{11e}$ modes, since the guiding losses of the $LP_{11o}$ are larger than for the $LP_{11e}$. The reason for the stronger guiding of the $LP_{11e}$ are the index-reduced stress-inducing rods parallel to its intensity lobes. These confine the $LP_{11e}$ mode better into the fiber core and thus reduce its bend loss. Therefore, the $LP_{11o}$ mode will not be included into further analysis.

 

Fig. 7. Relative modal content separated into the fast-axis (top) and slow-axis components (bottom) for (a) $P_{out} = 236$ W (stable beam), (b) $240$ W (TMI-onset) and (c) $249$ W (stronger TMI). All charts in a row/column have the same y/x axis scaling.

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Below the TMI-threshold (Fig. 7(a)), the output beam is stable with a $LP_{01}$-content of 90.0 % and a smaller $LP_{11e}$ mode content of 9.3 %. The dominant $LP_{01}^{\text {fast}}$ and $LP_{11e}^{\text {fast}}$ fast-axis mode contents are 79.0 % and 7.9 %. Increasing the power level over the TMI-threshold to $P_{out} = 240$ W and $249$ W, mode content fluctuations occur and become stronger, with TMI-freqencies of 1.22 kHz and 1.43 kHz. The average modal content of the $LP_{01}^{\text {fast}}$ decreases to 72.9 % and 63.7 %, while the $LP_{11e}^{\text {fast}}$ mode content increases to 13.2 % and 26.3 %, as shown in Fig. 7(b) and 7(c), respectively. Similar trends are observed for the slow-axis components. The average $LP_{11e}^{\text {slow}}$ mode content increases from stable 1.1 % to 2.0 % and 3.3 % (Fig. 7(a) to 7(c)). The $LP_{01}^{\text {slow}}$ mode content shows more interesting features. From 236 W to 240 W, the average mode content initially remained constant at 11.1 %, although it began to fluctuate, and then also decreased to 6.1 % at 249 W. Furthermore, for TMI we expect an anti-correlation between the FM- and HOM-content due to the mutual energy exchange. This is the case for the fast axis modes $LP_{01}^{\text {fast}}$ and $LP_{11e}^{\text {fast}}$ shown in Fig. 7(b), resulting in a linear correlation coefficient of -0.99. However, in the slow axis, the maxima positions of the $LP_{01}^{\text {slow}}$ and $LP_{11e}^{\text {slow}}$ are shifted by $\sim$ 170 µs, (see dashed lines in Fig. 7(b), bottom). The resulting linear correlation coefficient of 0.04 indicates an uncorrelated behaviour and could imply, that the mode content fluctuation in the slow axis do not solely originate from mutual energy transfer between both slow-axis modes. Additionally, the mode contents of the $LP_{01}^{\text {slow}}$ and $LP_{01}^{\text {fast}}$ modes are unexpectedly negatively correlated with -0.53, indicating a rotation of the FM polarization state. This can also be identified by the shifted maxima positions (dashed lines in Fig. 7(b), top) at 2.43 ms and 2.17 ms for the fast- and slow axis component, respectively. At 249 W, the correlation coefficient between the FM-content in the fast and slow axes is suddenly positive with 0.88. This points towards different or competing underlying physical effects. A slight offset of $\sim$ 100 µs between the maxima of the $LP_{11e}^{\text {slow}}$ and $LP_{11e}^{\text {fast}}$ modes can be identified in Fig. 7(c) as well. The polarization fluctuations at $P_{out} = 240$ W will be discussed in subsection 4.3. We would like to emphasize that already the mode content analysis of two simultaneously measured orthogonal polarization states provides much more information than a single beam HSC analysis.

Including the intramodal phase information $\delta _k$ in the analysis, the calculation of the mode-resolved Stokes parameters (Eq. (11)) is possible. The Stokes parameters $S_{0,k}$ are plotted in a 4-ms interval in Fig. 8 (top row). Additionally, the normalized mode-resolved SoP is depicted on the backside of the Poincaré sphere for the three previously discussed power levels (bottom row). In the Poincaré plots the markersizes were scaled according to $S_{0,k}$, to help the reader focus on the relevant and reliable data points. In Fig. 8(b), with the onset of TMI, the modal SoP begin to oscillate. To better illustrate the modal SoP changing periodically over time at the TMI-threshold, a video is provided (Visualization 4). The trajectory of the SoP of the $LP_{11e}$ (red) and $LP_{01}$ (blue) are similar to a lower semicircle and an ellipse. Fitting the depicted $LP_{11e}$-trajectory as a circle in the azimuth-elliptic angle representation (2D) yields a circle radius of $r_{\text {circ},\,240 W} = (17.9 \pm 0.2)^\circ$. For 249 W depicted in Fig. 8(c), the reliable red data points approximate a circle as well (here: $r_{\text {circ},\,249 W} = (20.0 \pm 0.5)^\circ$). Mathematically, this is a result of the periodically fluctuating mode contents in slow and fast axis in combination with an oscillating intramodal phase difference $\delta _k$ (see $S_{2,k}$, $S_{3,k}$ in Eq. (11)). To investigate its physical origin, the temporal evolution of the Stokes parameters is analyzed in greater detail at $P_{out} = 240\,$W.

 

Fig. 8. Stokes Parameters $S_{0,k}$ (top) and mode-resolved SoP on the Poincaré sphere (bottom, markersize = $S_{0,k}$) for linear input polarization of $100^\circ$ at (a) $P_{out} = 236$ W (stable beam), (b) $240$ W (TMI-onset, see Visualization 4), (c) $249$ W (stronger TMI).

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4.3 Discussion of polarization changes at the TMI-onset

The four Stokes-parameters are plotted over time in Fig. 9. The SoP of the $LP_{01}$ (blue) and $LP_{11e}$ (red) is approaching and separating with a defined period of 0.82 ms. This corresponds to the same frequency of f = 1.22 kHz as the mode content fluctuations from Fig. 7. We have highlighted in green where $S_1/S_0$ to $S_3/S_0$ of the two modes are closest. Interestingly, at these positions the $LP_{01}$-mode content starts to couple to the $LP_{11e}$ mode (see top chart, $S_0$). When the SoP is closest the interference contrast of the mode interference pattern (MIP) is maximal, which enables the formation of strong refractive index gratings (RIG) [7]. From the respective Jones vectors, it can be calculated [45], that the interference contrast in the Stokes formalism is proportional to (1+$\vec {S}_{norm., LP01} \cdot \vec {S}_{norm., LP11e}$)/2, with the scalar product of the two normalized modal Stokes vectors $\vec {S}_{norm., LP01} \cdot \vec {S}_{norm., LP11e}$. The summand and factor of 1/2 are necessary because orthogonal polarization states are on opposite sides of the Poincaré sphere, resulting in 1 for identical and 0 for orthogonal SoP. This measure of the interference contrast is shown together with the FM content ($S_{0,LP01}$) in Fig. 10 in a 2-ms interval. At this time, we can only qualitatively discuss the observed effects based on our experimental data. Further numerical and theoretical investigations are required to fully explain the phenomena presented, which are beyond the scope of this paper.

At t = 0.78 ms, the maximum FM-content of 93.8 % is reached (#1 in Fig. 10). Here, the polarization states are very close with a high interference contrast of 90 %. As described before, this enables high-contrast MIP and strong index modifications (a strong RIG) to be written. Given a negative phase shift $\psi$ between the MIP and the RIG [3,46], i.e. $-\pi <\psi <0$, this is known to enable the negative modal energy transfer from the FM to the HOM. The origin of the phase shift is still the subject of scientific debate and its discussion beyond the scope of this paper. The rate of the negative energy transfer to the $LP_{11e}$ increases, as the modal SoP approach each other. If the phase shift were to remain constant, this could be attributed solely to the steeper index change of the RIG written by the MIP. The interference contrast reaches its maximum of 99.3 % at t = 0.93 ms, where the FM-content is slightly reduced to 89.1 % (#2 in Fig. 10). Between #2 and #3, further $LP_{01}$-content is coupled to the $LP_{11e}$ and the modal SoP start separating, which is causing the interference contrast to decrease. As a result, the strength of the RIG decreases again, which should attribute to the decreasing rate of negative energy transfer to the HOM. At t = 1.28 ms (#3) the FM-content has reached its minimum of 73.1 % and the interference contrast reduced to 67.8 %. Thereafter, the power begins to transfer back to the FM, which indicates that a change in the sign of the phase shift ($0<\psi <\pi$) between the MIP and the RIG occurred. The rate of positive energy back-transfer is increasing, as the interference contrast is approaching it’s minimum of 62.2 % at t = 1.39 ms (#4). As more $LP_{11e}$ content is coupled back to the $LP_{01}$, their modal polarization states converge again, increasing the interference contrast, slowing the rate of energy back-transfer, and starting the cycle all over again.

 

Fig. 9. Temporal evolution of the Stokes parameters of the $LP_{01}$ and $LP_{11e}$-mode at the TMI onset power level of $P_{out}=240$ W. The modal polarization state is closest in the areas highlighted in green.

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Fig. 10. A measure proportional to the modal interference contrast (left axis) and the total $LP_{01}$-mode content (right axis) versus time at the TMI-onset at $P_{out} = 240$ W. The highlighted positions (#1)-(#4) are located at the maximal $LP_{01}$-content (#1), the maximal contrast (#2), the minimal $LP_{01}$-content (#3) and the minimal contrast (#4).

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At this point we have to discuss, what may cause the SoP of both modes to separate. We believe, that two effects play a role here. Firstly, Andermahr et al. [47] have observed, that the polarization of the modes in a few-mode fiber evolve in such a way that the polarization distance between them is maximized. This was qualitatively explained and later numerically verified [48] as a consequence of local gain saturation and transverse spatial hole burning (TSHB), leading to preferential gain for orthogonal polarization components. At the TMI-threshold in our experiment, the dominant $LP_{01}^{\text {fast}}$ mode content is partially coupled to the $LP_{11e}^{\text {fast}}$ mode. With the increasing power share of the latter, both modes should therefore experience preferential gain for their orthogonal (slow-axis) components, trying to maximize their distance on the Poincaré sphere from one another. This would explain, why both the $LP_{01}^{\text {slow}}$ and the $LP_{11e}^{\text {slow}}$ mode content increase positively correlated to the $LP_{11e}^{\text {fast}}$ mode content in Fig. 7. Secondly, Palma-Vega et al. [15] have observed a static energy transfer from the $LP_{01}^{\text {fast}}$ to the $LP_{11e}^{\text {slow}}$ mode, likely due to similar modal effective indices under the presence of small birefringence fluctuations. For our fiber, the effective indices of these two modes are very close at room temperature, i.e. $n_{\text {eff}}(LP_{01}^{\text {fast}}) = 1.45033$ and $n_{\text {eff}}(LP_{11e}^{\text {slow}}) = 1.45035$. With an increasing thermal load the effective indices approach each other. Note, that the effective index of the $LP_{11o}$ mode, $n_{\text {eff}}(LP_{11o}^{\text {slow}})=1.45032$, is similar as well, but the index difference to the $LP_{01}^{\text {fast}}$ mode increases under operation. This makes dynamic coupling between $LP_{01}^{\text {fast}}$ and $LP_{11e}^{\text {slow}}$ more likely during TMI in the presence of small pertubations. It would lead to $LP_{01}^{\text {fast}}$ content being coupled directly to the $LP_{11e}^{\text {slow}}$ mode. Thereby, the SoP-movement of the $LP_{11e}$ mode would be amplified, contributing to the trajectory of this movement being much larger than for the $LP_{01}$ mode (see Fig. 8).

Equally important is the discussion of what might cause the reversal of mode transfer under weak interference contrast (#3 in Fig. 10). In the theory of Jauregui et al. [4], the transfer from the HOM to the FM is induced by the increasing effective index separation of the two modes along the longitudinal exponential temperature gradient. The MIP period is compressed towards the amplifier end and a positive phase shift with respect to the RIG induced. In our amplifier, the increased amount of incoherently superimposed light leads to a blurring of the MIP, as nicely illustrated in [13,48] and subsequently a reduction in TSHB. This results in more adiabatic refractive index changes, which reduce the negative energy transfer and finally allow the RIP to catch up with the MIP. Therefore, the effect of the compressing grating period [4] should become more dominant with decreasing interference contrast and ultimately reverse the energy transfer. Another consequence of reducing the TSHB is a more homogeneously saturated gain profile, which reduces the gain for orthogonally polarized components [47], allowing the modal SoP to converge again.

It is worth to mention, that beyond $P_{out}=249$ W the movement of the modal polarization states becomes increasingly chaotic and the average $LP_{11o}$ mode content increases. Similar to the SoP-movement of the $LP_{01}$ mode at $P_{out}=249$ W (see Fig. 8(c)), characteristic trajectories could no longer be identified, which was presented in [40]. The chaotic TMI-regime [8] was not reached up to our last measurement value of $P_{out}=281$ W. However, due to the close relation between the polarization oscillations and the mode content fluctuations discussed above it is to be expected that the polarization dynamics become chaotic as well.

5. Conclusion

We developed a novel high-speed Stokes polarimeter technique based on the simultaneous detection of four paralleled channels on a high-speed camera that is overcoming speed limitations of current polarimeters. This method enables full-Stokes polarization imaging with measurement speeds from a few kHz to several hundreds of kHz. Additionally, an algorithm has been introduced, that allows the calculation of mode-resolved Stokes parameters by a simultaneous 4-beam mode reconstruction under the assumption of fully-polarized modes. The measurement capabilities of the high-speed polarimeter setup are demonstrated by the detailed characterization of TMI in an Yb-doped PM fiber amplifier in 35/200 µm geometry. At the TMI-threshold of $P_{\text {TMI}} = 237\,W$, a simultaneous onset of mode content and polarization fluctuations was observed for the first time. Detailed measurements with the high-speed polarimeter revealed an oscillating modal energy transfer between the $LP_{01}$ and $LP_{11e}$ modes, that was accompanied by oscillating modal polarization states on circular and elliptic trajectories. Interestingly, the modal polarization states approached and separated with the same characteristic TMI-frequency as the mode content fluctuations occur. We characterized the temporal evolution of these modal polarization fluctuations and the resulting oscillating modal interference contrast and provided a first qualitative explanation. These results provide new insights in the physics behind TMI, especially in PM fibers. Further numerical investigations and theoretical considerations are now required to fully understand this behaviour, which are beyond the scope of this paper. We believe that our Stokes-polarimeter approach is beneficial since HSC are commonly utilized for visualization and modal reconstruction of TMI-limited laser beams.