1. Introduction

The design of high-power fiber laser systems has enabled an exponential evolution of their output power with a diffraction-limited beam quality [1,2]. This evolution, however, results in a high heat load inside the fiber, which triggers a thermally-induced change in the index profile through the thermo-optic effect [3–5]. Such change can have severe manifestations, such as the effect of transverse mode instability (TMI), which is currently hindering the power scaling in these systems [6]. The effect of TMI is characterized by a sudden, unstable, and bi-directional energy transfer between the fundamental mode (FM) and one or more higher-order modes (HOMs) in the active fiber. This energy transfer occurs in a millisecond time scale and results in fluctuations of the output beam profile [7,8]. TMI occurs upwards from a certain average output power level, known as the TMI threshold.

The origin of TMI is to be found in the thermally induced modification of the index profile of the fiber amplifier occurring at high average powers. The presence of HOMs in the fiber core, which interfere with the FM, results in a modal beating due to the difference in phase velocity of the propagating modes (i.e., FM and HOMs) along the fiber amplifier [9]. This beating leads to a periodic variation of the intensity profile of the amplified signal along the fiber, the so-called modal interference pattern (MIP) [10].The MIP, in turn, inscribes a thermally-induced refractive index grating (RIG) in the fiber via the thermo-optic effect. This RIG is perfectly matched (in terms of periodicity and symmetry) to the MIP and can potentially transfer energy between the FM and the HOMs. However, this modal coupling is only allowed if there is a phase shift between the MIP and the RIG [3,9,11]. Such a phase shift can be induced, for example, by perturbations of the system (noise and pump/seed fluctuations), because the RIG has a delayed response to the MIP changes due to the comparatively slow thermal diffusion time of the fiber [11–13].

Since TMI is one of the most pressing matters in high-power fiber laser systems, it has become more and more important to develop fast and reliable monitoring and characterization methods to gain insight into this effect. The standard method of TMI characterization and monitoring (from now on referred to as the 2 photodiodes or the 2PDs method) is still the most widely spread way of determining the TMI threshold and identifying the three characteristic regions of TMI (stable, transition and chaotic) [14]. This method uses two photodiodes to study the system stability. The first photodiode (power photodiode PD_p) measures the instantaneous power of the output beam (achieved after calibration with a thermal power meter). The second photodiode (stability photodiode PD_s) has a small detection area compared to the size of the incident beam, which allows the measurement of temporal changes in the beam intensity profile. Such temporal changes are employed to characterize the system stability at different power levels.

Even though this method (i.e., the 2PDs method) is simple and cheap, it only offers limited insight into TMI. Additionally, it is worth noting that this method is sensitive to power changes. This makes the characterization method unfit to analyze TMI in scenarios in which the output power fluctuates either as a consequence of the modal fluctuations or as a consequence of some mitigation strategy, for example, the pump modulation operating regime described in Ref. [15]. A popular alternative to the 2PDs method is the use of a high-speed camera (HSC). The HSC is more robust than the 2PDs method against other system dynamics and it offers a vastly increased amount of information of the TMI dynamics [7,16]. Unfortunately, HSCs are expensive devices, and this will prevent them from becoming ubiquitous for the characterization of TMI. In a compromise between characterization costs and detailed information, we offer in this work an alternative that is halfway between the 2PDs and the HSC. This novel approach marries many of the advantages of the above-mentioned characterization techniques (such as high speed, robustness against power fluctuations, detailed information of the beam dynamics, etc.) with an attractive price.

Our new method is based on the position-sensitive detection of the output beam of a high-power fiber laser system [17]. This method uses a 4-quadrant photodiode (4QPD) which records temporal traces that track the position of the beam in the X- and Y-axis. Thus, the temporal evolution of the center of gravity (CoG) of the fluctuating beam can be followed in space with a high temporal resolution. The use of the CoG is employed to measure the TMI threshold and to identify the three characteristic regions. Interestingly enough, this beam position measurement is inherently insensitive to, e.g., power fluctuations. Furthermore, the wealth of information contained in the trajectories described by the beam can provide a detailed insight into TMI.

2. Working principle

Considering that the TMI effect results in a temporal change of the output beam shape, a position-sensitive detector (i.e., 4QPD) can be used to track the CoG trajectories in the X-axis and Y-axis. If a high-resolution time measurement (i.e., for example with microseconds time resolution) of the X-axis data and the Y-axis data is plotted against each other over a certain time window (for example, 10 ms), the CoG trajectories can be reproduced and used to describe the temporal change of the beam stability. This concept is at the heart of the new technique (i.e., the 4QPD method). In other words, the crux of this technique is the use of the temporal change in CoG to measure the stability of the system. This approach not only brings more accuracy to the TMI characterization given its sensitivity to TMI-related dynamics, but it also offers high-speed detailed information about the temporal behavior of the beam. All these merits are achieved using a cheap and straightforward device instead of expensive high-speed cameras, as showcased in the following sections.

2.1. Setup

A seed signal of different input powers of 2 W or 5 W at 1030 nm is amplified with a 1 m long, rod-type fiber amplifier in the experimental setup, as shown in Fig. 1. This amplifier is a large-pitch fiber (LPF) with an active core diameter of around 63 µm and doped with trivalent ytterbium ions [18,19]. The fiber is pumped in the counter-propagating direction at 976 nm. The TMI threshold at input seed powers of 2 W or 5 W, as measured by the conventional 2 PDs method, is around 203 W or 224 W in the free-running system respectively [14].

 

Fig. 1. Setup in the free-running system and external pump modulation operating regime. DC: Dichroic mirror. BD: Beam dump. PD: Photodiode. LD: Laser diode. PDp: power photodiode. PDs: Stability photodiode. HSC: high-speed camera. 4QPD: 4-quadrant photodiode. PM: power meter.

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Furthermore, a position-sensitive detector (4QPD), Thorlabs PDQ80A, is placed at the position of the magnified near-field image of the output high-power signal to track the change of the CoG. This 4QPD contains four adjacent active areas spaced by a dead zone of around 1 mm width and has a detection area size of (∼ 8 mm x 8 mm). This device can measure shifts of the CoG both on the X- and in the Y-axis, as well as the total power of the output beam. In the same manner as in the 2PDs method, the output power of the 4QPD is calibrated against the readout of a power meter to have a fast measurement of the instantaneous power. The spot diameter of the incident beam is chosen to be around 4 mm to minimize the effect of the dead zones (i.e., incident power loss). Please note that the incident point of the beam is initially aligned (i.e., before the amplification process) to the sensor’s center. This point might slightly vary due to the thermal drift of the optics when the power increases. However, this drift will not affect the CoG measurement if the whole beam is confined to the detection area of the 4QPD.

An oscilloscope (Teledyne LeCroy HDO6104 12-bit) is used to display the three outputs of the 4QPD (i.e., the X-position information, the Y-position information, and the power) with 10⁵ samples per second. For more detailed measurements, i.e., to benchmark the 4QPD method, a high-speed camera (HSC) is placed in the near-field image position to record intensity frames (pixel size: 128 × 128) with the same temporal resolution of 10⁵ frames per second. In addition to the 4QPD method, the 2PDs method setup [14] is also implemented to evaluate the results of the 4QPD method. The power photodiode (PD_p) is a Thorlabs photodetector DET100A2, whereas the stability photodiode (PD_s) is a Thorlabs photodetector PDQ80A.

2.2 Center of gravity of the beam (CoG): trajectories

The beam fluctuations are mapped in the detection area of the 4QPD. This millimeter-scale movement of the beam on the 4QPD detection area (due to the magnification optics) can be converted to the real micrometer-scale movement of the beam at the fiber end-facet. Figure 2 shows the X- and Y-position information of the output beam at the fiber output facet recorded during 10 ms for three different output powers. It is worth noting that the diameter of the active core of this fiber amplifier is around 63 µm (i.e., the mode field area is ∼ 3100 µm²). Thus, the dimensions of the X- and Y- axis represent just the inner part of the active core dimensions. The thermal drift is corrected in such a way that the (0, 0) coordinates represent the average position of the center of gravity. In addition, the positive/negative coordinates in the X-axis and the Y-axis represent the right/left and the up/down 4QPD positions around the average CoG, respectively. Please note that in the measurements shown in Fig. 2, the TMI threshold has been determined (by the 2PDs method) to be at ∼ 224 W. Each plot, from left to right, shows the XY trajectory over 10 ms (blue line), which describes the temporal changes of the CoG at three different power levels: 155 W (stable region), 224 W (transition region), and 413 W (chaotic region). These trajectories cover a particular area in the XY plane of the active core that increases at higher powers.

 

Fig. 2. The trajectories of the center of gravity of the beam at different average output power levels at the fiber end facet recorded by the position-sensitive detector (i.e., 4QPD) within a time window of 10 milliseconds. The red ellipse estimates the area encircled by the trajectories. a) Stable region at 155 W. b) Transition region at 224 W. c) Chaotic region at 413 W. The (0, 0) coordinates represent the center of the stable beam in the fiber output facet.

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As indicated in Fig. 2, the trajectories are encircled by an elliptical fit (the red ellipse) which estimates the area covered by the experimental data (blue trajectories). To calculate this fit, the least squares method is employed to estimate the optimal elliptical fit to the experimental data [20]. The conic equation of a general ellipse is:

Using the least squares method, it is possible to estimate the coefficients of the conic equation. These coefficients will be employed to define the rest of the characteristics of the ellipse, such as its orientation, centering, and short and long axis. As shown in Fig. 2, the area enclosed by the ellipse grows with the output power (especially as the system reaches and surpasses the TMI threshold). In this manner, the area of this ellipse can be employed to measure the system stability.

Figure 2(a) shows the CoG trajectory which is confined in a small area of 0.39 µm² around the beam center at an average output power level of 155 W. This confinement indicates that the beam profile is stable at this power level. By increasing the power level to 224 W (i.e., around the TMI threshold), the CoG changes are more pronounced and, hence, its trajectory covers a larger area of around ∼ 4.54 µm², as confirmed by the elongated shape in Fig. 2(b). Based on the shape of the trajectory in Fig. 2(b), it is patent that the beam profile undergoes strong fluctuations primarily in one axis. This observation is consistent with a dynamic energy transfer between the FM (LP₀₁) and a radially anti-symmetric HOM (presumably the LP₁₁). At this power level (i.e., transition region level), it is possible to trace the energy exchange between the two orthogonal components of the radially anti-symmetric HOM by tracking the rotational movement of the trajectories at different temporal intervals. As shown in Fig. 2(c), the shape of the trajectory significantly changes at a power level of 413 W. This change confirms the appearance of more HOMs which are involved in the energy transfer and, as a result, the CoG moves farther and faster compared to the low power cases. Consequently, this movement results in a significant area increase of around ∼ 198.3 µm². At this point is worth mentioning that the area encircled by the trajectory is not the only parameter that can be used as an indicator of the system stability. Further parameters, such as the speed of CoG movement could also be employed, as it will be explained in Section 3.3.

3. Experimental results

3.1 New definition of the TMI threshold

In order to determine the TMI threshold by our new method [17], a pump power sweep is carried out over 20 s in the same manner as in the case of the 2PDs method [14]. Hereby the temporal traces of the beam trajectories are trimmed in 10 ms sections, and the encircled area in each case is calculated as previously explained in section 2.2. This process results in a plot of the evolution of the encircled area with the average output power as illustrated in Visualization 1. Figure 3 exhibits a frame taken from Visualization 1 which illustrates the production of the new stability curve. Figure 3(a) shows the trajectory at a power level of 396 W. It can be seen that at this power level the encircled area is ∼ 222 µm² (as estimated by the red fit). This value will be employed as one data point in the stability curve, as illustrated by the red markers in Fig. 3(b). This last plot shows that each power level has a corresponding area value. To this extent, the 4QPD method allows producing a new stability curve (area vs. power) to characterize the TMI dynamics, which replaces the stability curve (normalized standard deviation of the beam center fluctuation vs. power) of the 2PDs technique [14]. Moreover, this area can be normalized relative to the core or mode field area (i.e., the stability curve represents the normalized area vs. power) to make the plot independent of the fiber geometry.

 

Fig. 3. Frame taken from Visualization 1 showing a): the trajectory of the center of gravity of a fluctuating beam at an average output power level of 396 W during 10 ms, b): The stability curve which describes the evolution of the area covered by the CoG during 10 ms as a function of the output power. Each point represents the data of one trajectory plot.

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Hence, a new definition of the TMI threshold is required for the new stability curve. The current experiment aims at employing the empirical definition of the 2PDs method (i.e., the first derivative of the normalized standard deviation ($\frac{{d\,{\sigma _{norm}}}}{{d\,power}}$ = 0.01%/W)) [14] as a reference to deduce the new definition of the 4QPD method (i.e., both methods should cast similar values of the TMI threshold). This specific value was strictly chosen to mark the threshold beyond which the normalized standard deviation increases and spreads dramatically. This significant change in the normalized standard deviation indicates the instability of the beam profile at the corresponding average output power. To deduce the new definition for the 4QPD method, the amplifier system is operated as a free-running system while the corresponding stability curves of the 2PDs method (Fig. 4(a)) and of the 4QPD method (Fig. 4(b)) are produced simultaneously. The Y-axis in Fig. 4(a) and Fig. 4(b) represents the normalized standard deviation and the normalized value of the trajectory areas (normalized to the active core area, i.e., 3100 µm²), respectively. This measurement results in the TMI threshold being determined at a power level around 224 W by the 2PDs method, as illustrated by the orange vertical line in Fig. 4(a).

 

Fig. 4. Stability curves in a free-running system produced by a) the 2PDs method, b) the 4QPD method.

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Following that, the new stability curve (i.e., the 4QPD curve) is exponentially fitted as represented by the red exponential red curve in Fig. 4(b). Using the TMI threshold value measured by the 2PDs method, it can be determined that when the first derivative of the normalized area evolution with power is $\left( {\frac{{d\; ({normalized\; area} )}}{{d\; ({power} )}}\; = \; 0.008\; {\raise0.7ex\hbox{$\%$} \!\mathord{\left/ {\vphantom {\%W}} \right.}\!\lower0.7ex\hbox{$W$}}} \right)$ then the power level equals ∼222 W. This definition is considered as the new empirical definition of the TMI threshold measured by the 4QPD method. As shown in Fig. 4, it can be noticed that both methods can produce a smooth stability curve (blue curves) in the free-running system. The colored regions in Fig. 4 distinguish between the three stability regions of the system [14]. Therefore, both methods can discern the stable region (green area), the transition region (blue area), and the chaotic region (the red area). However, it is worth mentioning that the 4QPD method can produce the same results as the 2PDs method by using a single device (i.e., 4QPD) instead of using two (i.e., PD_p and PD_s). Furthermore, this method can be implemented in a complex operating regime, revealing the standard method's limitations, as explained in section 3.2.

This measurement procedure has been simultaneously repeated several times to test and guarantee the robustness and accuracy of the new technique. Additionally, it is known that the TMI threshold changes at different input seed powers due to the gain saturation effects. In other words, the TMI threshold with input seed power of 5 W is expected to be higher than the TMI threshold with input seed power of 2 W [21]. For this reason, the robustness of this definition (i.e., TMI threshold by 4QPD) can be tested at different seed powers as well. Figure 5 shows the results of these simultaneous measurements where both methods have shown similar TMI threshold with an average output power of 222 W and ∼ 203 W over consecutive measurements for different input seed powers of 5 W and 2 W, respectively. Please note that every measurement differs by less than 5 W from its consecutive measurement. This difference can be accepted because of the measurement accuracy. Moreover, by comparing the fluctuation of the blue and the red curves over consecutive measurements, it can be noticed that the 4QPD is more stable and accurate compared to the 2PDs method.

 

Fig. 5. Simultaneous and consecutive measurements for both the 2PDs and the 4QPD methods to determine the TMI threshold at different seed input powers.

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3.2 Impact of power fluctuations

In recent years, several ways to explore and mitigate TMI have been proposed [22]. Many of these experimental studies require complex operating regimes such as pulsed pump/pulsed seed operations [13], or/and artificially induced noise [12]. For this reason, it is of high importance to test the accuracy of both the 2PDs method and the 4QPD to characterize TMI dynamics in such complex operating regimes. This study should mainly illustrate the weaknesses and strengths of both methods.

An experiment has been carried out in which power fluctuations are externally applied to the system. For this purpose, the pump radiation is modulated with a sinusoidal signal at 1.5 kHz and ±15% modulation depth. Please note that this modulation process is similar to the one described in Ref. [23] to increase the TMI threshold via washing out the RIG. However, the modulation parameters in our case aim at inducting power fluctuations without substantially changing the TMI threshold (achieved thanks to the relatively low modulation depth). In fact, it was reported that the TMI threshold for this amplifier system changes if a modulation depth of ±75% is applied at frequencies less than 800 Hz. Moreover, the experimental observation by the low- and high-speed cameras in our experiments confirms that the TMI threshold does not change when a ± 15% modulation depth is applied at a modulation frequency of 1.5 kHz. In other words, the modulation parameters in this experiment are far away from the optimal ones required to change the TMI threshold.

Figure 6(a) and 6(b) show the simultaneously measured stability curves obtained with the 2PDs and the 4QPD methods, respectively. The stability curve of the 2PDs undergoes strong fluctuations, limiting the accuracy of the exponential fit, so the TMI threshold cannot be accurately measured without further signal processing [15]. These oscillations originate from the fact that the PD_s of the standard method is sensitive to all fluctuations (i.e., those from the power and those from the beam profile) which result in changes in the beam center (even if they are not TMI-related dynamics). For example, the pump modulation in our case results in a periodic heat load modulation along the fiber amplifier, resulting in a periodic change of the temperature profile. As a result of this change, the modal beat length undergoes a periodic change, resulting in a compression and stretching of the modal interference pattern [15,23]. This change in the MIP translates in small side-to-side fluctuations of the output beam. These external modulations, combined with the power fluctuations, lead to relatively strong intensity changes in the PD_s that cannot be fully compensated by the PD_p. Additionally, by comparing the 2PDs stability curves in the free-running system (Fig. 4(a)) and in the pump modulation operating regime (Fig. 6(a)), the power fluctuation becomes evident as a non-zero value of the normalized standard deviation value even at power levels below the TMI threshold.

 

Fig. 6. Stability curves in the pump modulation operating regime with a sinusoidal modulation of 1.5 kHz and ±15% modulation depth. a) The 2PDs method. b) The 4QPD method.

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On the contrary, the stability curve of the 4QPD (Fig. 6(b)) is smooth since this technique is based exclusively on recording the area covered by the movement of the CoG in a 10 ms time window as a measure of the system stability. This means that the small pointing fluctuations induced by the heat load modulation in the fiber (due to the pump power fluctuations) are transformed into a small offset in Fig. 6(b). This way the 4QPD method allows for a reliable exponential fit of the front part of the stability plot (Fig. 6(b)). Therefore, the TMI threshold could be accurately measured at around 229W, as illustrated by the orange vertical line in Fig. 5(b), which is very close to one of the free-running system.

The robustness of this method against power fluctuations will enable high accuracy and fast characterization in more dynamic systems, e.g. those already proposed as TMI mitigation techniques [22]. In addition to the simplicity and accuracy of the 4QPD method compared to the 2PDs method, it provides a wealth of information about the TMI dynamics, as it will be shown in the following sections.

3.3 Evolution of the center of gravity (CoG) of the beam in the fiber amplifier

The XY beam trajectory picture contains the temporal dynamics of the micrometer scale beam fluctuations. In other words, this information will enable the high-speed investigation of the spatial evolution of these fluctuations across the fiber core. In the following experiments, the input seed power is 2 W; therefore, the TMI threshold is decreased and measured at a value of ∼ 203 W. As already mentioned before, simultaneous to the 4QPD measurement, a HSC records the near-field intensity image with the same sample rate of 10⁵ frames per second.

Visualization 2 shows the spatial and temporal evolution of the CoG (over simultaneous 10 ms temporal measurement windows for the HSC and the 4QPD) at the average output power level of 228 W (i.e., in the transition region above the TMI threshold). Based on information in Visualization 2, it is clear that the 4QPD can replace the HSC to determine the spatial and temporal dynamics of CoG across the fiber’s core. Figure 7 shows the spatial and temporal evolution of the CoG across the fiber core, which is shown in Visualization 2. The XY trajectory of the beam’s CoG is tracked for 10 ms and is plotted by the blue curve. The corresponding intensity beam profiles- at different temporal points- are presented in the insets of Fig. 7, which are taken from the left-hand side video in Visualization 2. By comparing the intensity insets and the CoG trajectories, it is easy to understand the information delivered by the 4QPD. The red/white stars represent the CoG position recorded by 4QPD/HSC respectively at different temporal points (9, 9.5 and 10 ms). Moreover, a stable beam's initial CoG position point is measured as illustrated by the white star in the reference frame recorded by the HSC and the red star in the (0, 0) coordinates recorded by the 4QPD. This point could be determined thanks to a reference frame recorded at a power level of 115 W (i.e., when the beam is stable since the system operates below the TMI threshold). Interestingly, the measured CoG positions – at different temporal points – by the HSC can also be predicted and measured by the 4QPD.

 

Fig. 7. XY trajectory of the CoG across the fiber core at 228 W in the transition region (free-running system) recorded over 10 ms (blue curve). The red stars represent the CoG position at different temporal points. The insets are taken from Visualization 2 which shows the corresponding HSC intensity frames recorded simultaneously at the same temporal points. The reference frame represents the intensity profile of a stable beam below the TMI threshold. The white stars represent the CoG position in the frames.

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By comparing the distance travelled by the CoG between 9 ms and 9.5 ms, and the distance travelled between 9.5 and 10 ms in Fig. 7, it is patent that the CoG moves at different speeds. The rich information recorded by the 4QPD allows, for example, calculating the instantaneous speed of the beam CoG movement (in units of µm/µs). Figure 8 shows the trajectory at different power levels of 206 W (Fig. 8(a)) and 228 W (Fig. 8(b)) with different colors, which encode the instantaneous speed of the CoG. Based on this information, the acceleration and deceleration of the CoG across the fiber’s active core can be analyzed. The color map represents the instantaneous speed of CoG resultant from the X- and Y-axis speeds. As shown in Fig. 8(a), it can be seen that the CoG speed in the X-axis is faster than in the Y-axis, which leads to a beam rotational movement when the TMI dynamics are triggered. The maximum CoG speed is reached when the beam is close to the core’s center, whereas this speed is decreased to almost zero when the beam CoG reaches its maximum excursion at the spatial point’s ± 10 µm and/or ±5 µm in the X-axis and Y-axis, respectively. This speed will evolve with power since the RIG strength (and, therefore, the beam fluctuations) gets stronger. Figure 8(b) shows the speed map for the trajectories at a power level of 228 W, where it can be noticed that the maximum speed is reached not only around the core center but also in some side parts of the excursion which indicates the onset of new HOMs compared to the case in Fig. 8(a).

 

Fig. 8. XY trajectory of the CoG across the fiber core at power levels of a) 206 W and b) 228 W in the transition region recorded over 10 ms. The color code represents the instantaneous speed of the CoG movement across the amplifier core (in µm/µs).

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Along these lines, and as mentioned above, the CoG instantaneous speed across the core diameter in both the X- and Y-axis can be separately calculated, as exemplary depicted in Figs. 9. The trajectories in Fig. 8(a) over a time window of 10 ms can be divided into short trajectories over time windows of 100 µs. Figure 9(a) and Fig. 9(b) illustrates the difference between the travelling speeds of this short trajectory in the X- and Y-axis at the power level of 206 W respectively. Based on this information, the evolution of the average travelling speed over 10 ms can be plotted as a function of the output average power as shown in Fig. 9(c). At the TMI threshold (i.e., ∼203 W measured by the 2PDs method), the speed suddenly increases, indicating the onset of TMI. At the TMI threshold, the movement starts mostly in the X-axis indicating energy transfer to just one of the two LP₁₁ modes of the fiber. Consequently, a side-to-side movement of the beam is induced, as shown previously in Fig. 2(b). Below the TMI level, the speed value is very small (in comparison to its value above the TMI threshold). Based on this detailed speed report, the CoG speed parameter can also be employed as a measure of the stability of the system and, therefore, it could be used to determine the TMI threshold.

 

Fig. 9. Instantaneous CoG speed across the fiber core at a power level of 206W on a) the X-axis and b) the Y-axis. c) Average CoG speed over 10 ms time window as a function of the output average power.

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3.4 Spectral information

In Ref. [7] a HSC was used to extract the spectral information of the TMI dynamics. It is worth noting that this information can also be extracted by the 4QPD method, including details about the rotational dynamics of the beam movement (i.e., different speeds in the X- and Y-axis). Figure 10(a) shows the XY trajectory over a time window 10 ms at a power level of 206 W. Please note that this power level is in the transition region where the energy transfer happens predominantly between the FM (i.e., LP₀₁) and one of the LP₁₁ modes. The CoG trajectory shape can reveal this LP₀₁/LP₁₁ energy transfer in Fig. 10(a).

 

Fig. 10. 4QPD measurement at a power of 206 W (transition region). a) XY trajectory of the CoG over a time window of 10 ms. b) Time and frequency domain analysis of the beam movement in the X- and Y-axis.

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Figure 10(b) shows the temporal evolution of the CoG in the X-axis and Y-axis across the fiber core (upper plot), in addition to its corresponding spectral information (lower plot). It can be seen that the fluctuation of the beam profile is periodic and predominant in the X-axis as confirmed by the different amplitudes of the time traces. As shown in the spectral plot, the TMI dynamics occurs at a frequency of around 300 Hz, which is a typical TMI frequency for this type of fibers [14].

4. Conclusion

We have presented the technique of 4QPD as a novel, cost-effective and high-speed method of characterizing TMI. Such a device (i.e., 4QPD) offers a significant increase in the amount of information that it can gather when compared to the standard 2PDs method. On top of that, this device is insensitive to power fluctuations. In fact, in certain cases, the 4QPD can replace an HSC when analyzing the high-speed dynamics of TMI. This technique can measure the TMI threshold and identify the three stability regions even in complex operating regimes. Moreover, the investigations give details about the high-speed spatial and temporal evolution of the CoG across the fiber core. For that matter, the spectral information of TMI dynamics and its evolution with power can also be measured.