State-of-the-art high power Yb-doped large mode area fibers have been developed to a performance level able to reach the so-called mode instability threshold. In this contribution we will discuss the experimental results regarding the temporal evolution (build up and decay) of this effect to come closer to a comprehensive understanding of its driving mechanisms. Our investigations prove that the relevant time scale for build up and decay of mode instability is in the millisecond range and thus deliver experimental evidence of underlying thermal effects. To the best of our knowledge these are the first systematic, time resolved investigations on that topic.
©2012 Optical Society of America
Until a couple of years ago it was a common notion that active fibers can emit diffraction limited beams independent of their output power . But state-of-the-art high power Yb-doped large mode area fibers have been developed to a performance level able to reach the so-called mode instability threshold. At present, the term 'mode instability' (MI) stands for a threshold-like change in mode quality at a certain power level caused by a dynamic energy transfer from the fundamental transverse mode to higher order modes . Up to now, this phenomenon seems to be mostly independent of the working regime and occurs in narrow-bandwidth continuous wave , as well as in nanosecond broadband pulsed amplifier and laser systems . But eventually this is not only a challenge for Yb-doped fibers. Concerning the progress in scaling high power fiber lasers in the mid-infrared spectral range  it may be only a matter of time until this problem could also be relevant for other rare earth-doped fibers as well. Thus, high power fiber lasers encounter one more fundamentally challenging aspect (beside limitations caused by nonlinearities) with respect to their power scaling capabilities. For a better understanding it is crucial to get profound insight to this effect and the way it typically manifests.
There are a lot of open questions unsettled, for instance, the tangible influence of operation conditions, center wavelength, and bandwidth. Based on experimental observations the fiber design itself is a major factor for the MI threshold level. Consequently, novel fiber designs have been proposed  and in fact Stutzki et al.  could demonstrate an increased threshold level of a factor of three for a specific fiber design. Recently, a hot debate arose about what physical mechanisms can cause the MI after all and the following explanation is actually emerging: mode beating of different transverse modes leads to periodic intensity modulations along the fiber, which results in long-period index gratings enabling energy transfer between different transverse modes under certain conditions. It was demonstrated that the intensity modulation of beating modes can be transformed via the Kerr effect , resonantly induced index changes in doped fibers  and the thermo-optical effect [9,10] to an index modulation having exactly the right period for energy transfer between the beating modes. However, Smith et al.  pointed out that in addition to the right period also a phase shift between the intensity modulation of the beating modes and the induced index grating is required to enable energy transfer. Smith et al.  suggested thermal effects as the most likely origin for index modulations and realized the required phase shift by assuming the beating of two modes at slightly different frequencies in their numerical simulations resulting in a moving index grating. Their retrieved numerical results show dynamical energy transfer similar to the experimentally observed MI. Ward et al.  suggested another explanation for the occurrence of the required phase shift between beating pattern and index grating. Their explanation is based on thermal effects, as well, but they do without the assumption of a second frequency. They indicate the delayed build up of the index grating due to relatively slow thermal effects in combination with changing material properties due to fiber heating as reason for the required phase shift. Although there are different theories that explain the phase shift, the actual mechanism could not be clarified conclusively yet. However, thermal effects are consentaneous considered as the most likely origin for MI. Also experiments showing that the dynamic of MI is correlated to typical thermal time scales in the millisecond range deliver evidence of underlying thermal effects [13,14].
Assuming that MI results from the intensity modulation of beating modes that are transferred to an index gratings due to thermal effects, the occurrence of MI requires two conditions: few-mode guiding resulting in a periodic intensity profile of beating transverse modes and a sufficient heat load per meter. In order to achieve further power scaling of fiber amplifiers the strategy of increasing the mode field diameters and thus to reduce the intensity was commonly pursued. This development, however, favored both conditions for MI: Firstly, a consequence of large mode area fibers is that the separation of the effective indices of different transverse modes decreases, resulting in either inherent few-mode fibers or fibers that become few-mode due to thermally induced core deformations at high power operation [15,16]. Secondly, the heat load per meter is increased because larger mode field diameters result in shorter fibers.
Although some investigations on the physical origin of MI already exist, some aspects still need enlightenment, and a profound theory needs a lot of experimental data to prove its worth. In this paper we investigate the build up and decay of MI in order to gain a deeper understanding of its driving mechanism. For such investigations the amplifier was driven in pulsed pump operation. Within each pump cycle MI was observed, which means that the system was above the power threshold for MI while pumped. By this method, statistic investigations on MI during the transient phases, namely build up and decay, are enabled. Our investigations show that MI build up and decay occur on a millisecond time scale which strengthens the notion of a thermal origin. To the best of our knowledge these are the first systematic investigations on MI build up and decay.
2. Experimental setup
The experimental setup used for the following experiments is shown in Fig. 1 . The high power fiber amplifier consists of a double-clad photonic crystal fiber (PCF) with an Yb-doped laser core (diameter of 42 μm, mode field diameter (MFD) = 33 μm) and a pump core (diameter of 500 μm) surrounded by an air cladding. A continuous wave (cw) seed signal of around 10 W at a central wavelength of 1070 nm was coupled into the laser core, while the pump light (diode laser at 976 nm) was coupled into the pump core in counter propagating direction. The amplified signal was separated from the pump light with a dichroitic mirror (DM). For cw-pumping the amplifier was nearly single mode (M2 < 1.7) up to an absorbed pump power of Pabsorbed pump = 750 W (resulting in an output power of Poutput = 590 W) but switched to unstable mode operation for higher pump powers, which was identified by an increased M2-value. These observations are consistent with those presented in  using the same fiber. For the fiber under test a slope efficiency of 79% was measured and the following linear correlation between the absorbed pump power (> 200 W) and the according output power was found:
In order to enable systematic investigations on the MI transient phases, we decided for pulsed-pumping and cw-seed operation. A typical duty cycle of the pulsed-pumping scheme is shown in Fig. 2 (a) and 2(b) below and above the mode instability threshold. Note, that the high noise levels of the pump signals are not characteristic for the laser but are due to low detected signal levels. The detected averaged beam profiles of the amplified signal for stable and unstable mode operation are illustrated in the insets of Fig. 2 (c) and 2(d), respectively. The beam profile below the instability threshold was near Gaussian. At the threshold two fluctuating transverse modes were observed very similar to other reports [2,13]. But for further power increasing above the threshold the mode mix did not only contain one or two higher order modes but several and even contained modes that were not guided in the laser core but in the pump core. The composite of the mode mix is a special characteristic of the fiber geometry but the principle effect of MI is based on the same physical mechanisms.
In pulsed operation mode it is important to choose pulse durations that are long enough to ensure the build up of MI during each pulse. This aspect will be investigated in detail in section 3.1. Furthermore, the off time between following pulses has to be chosen long enough to avoid that the build up of MI is influenced by crosstalk of subsequent pulses and thus to ensure that the build-up time is independent of the duty cycle. This point will be discussed in section 3.2. During the rectangle-like pump pulses the same peak power as for cw-pumping operation is reached. The rise time of the pump pulses is ~1 ms (see inset of Fig. 2 (b)) according to the limit of the pump source dynamic. Since the effective life time of the upper state of the laser process is in the microseconds range (0.5 μs to 25 μs for the power levels under investigation) and thus was much shorter than the rise time of the pump source, the amplifier system directly follows the pump source dynamic. Consequently, the output signal showed the same rise time as the pump (see Fig. 2 (b) and 2(d)). The pulsed-pumping scheme under the mentioned conditions (pulse duration, off time, power level, and effective upper state life time) enables systematic investigations on the transient phases. Advantageously, the pulsed-pumping scheme even enables investigations for fibers with multi-kW threshold power due to the reduced average power.
In order to analyze the behavior at the MI threshold a small fraction (< 0.01%) of the high-power near field signal was detected with an InGaAs photo diode (rise time 10 ns) connected to an oscilloscope (analog bandwidth of 500 MHz). Since the beam was larger than the detector only a spatial fraction was detected resulting in a constant signal for stable mode operation and a strongly modulated signal for unstable mode operation due to the spatially fluctuating power distribution of the unstable mode mix as illustrated in Fig. 2 (d) and discussed in [13,14]. Thus, in order to distinguish between stable mode operation and unstable mode operation the standard derivation of the time signal was used in the following. The first column of Fig. 2 illustrates an example for stable-mode operation, where the pump power is below the MI threshold (red dashed in Fig. 2 (a)) resulting in a constant output signal Fig. 2 (c). In the second column an example for instable-mode operation with a pump power above the threshold is plotted (Fig. 2 (b)) resulting in a strongly modulated signal (Fig. 2 (d)). The example time trace for unstable mode operation (Fig. 2 (d)) demonstrates that MI did not set in instantaneously with the pump power turned on. For a short time period, the amplifier shows stable mode operation (indicated by the light gray shaded areas in Fig. 2 (d)), but switches to unstable mode operation with a delay of some milliseconds (indicated by the dark gray shaded areas). This delay is defined as build-up time in the following.
3. Experimental results
3.1 Mode instability build-up time
In order to analyze the build-up time in more detail, Fig. 3 shows the build-up time in dependence on the absorbed pump power. A duty cycle with 20 ms pulse duration and 120 ms off time was chosen for this experiment. The relatively long off time was necessary to avoid that the build-up time was influenced by crosstalk of subsequent pulses and thus to ensure that the build-up time was independent of the duty cycle. This effect will be discussed in detail in the following section 3.2. In order to avoid warm up effects of the amplifier system, such as wavelength drift of the pump diode, the system was running for around 20 min before the measurements were started. Each data point in Fig. 3 was averaged over 120 pulses and the depicted error bars were calculated from the standard derivation.
The build-up time decreased exponentially from 18.3 ms at 800 W to 1.7 ms at 1200 W. An exponential fit of the graph in Fig. 3 indicates that the build-up time converged to 1.6 ms for high pump powers (indicated by the horizontal, gray shaded area). Assuming thermal effects as mechanism responsible for MI a minimal build-up time would be expected because a thermal gradient inside the fiber would not be build up instantaneously. However, the measured build-up time of 1.6 ms was influenced by technical limitations in our experiment: As mentioned before, the rise time of the pump diode was about 1 ms. Thus, due to an estimated error of ~1 ms, the actual build-up time is expected to be shorter (< 1 ms) than measured. In order to measure the build-up time more precisely, faster modulation techniques are required.
Note that the exponential fit of the data describes the development of the measured data within the range shown in Fig. 3 very well, but would fail for further reduced absorbed pump powers resulting in longer build-up times: For cw-pump a MI-threshold power of Pcw threshold = 750 W (indicated by the gray shaded area in Fig. 3) was observed. Consequently, no MI is possible for lower absorbed pump powers and a pole point at 750 W is expected. The concrete behavior, however, can only be explored by an extended measurement range considering longer pulse durations. We observed that for lower pump powers resulting in longer build-up times the standard deviation increases and thus the measurements become less precise. For the pulse duration of 20 ms used in the presented experiment the lowest absorbed pump power, where MI is observable, is 800 W and thus this value was defined as power threshold Pth for MI in our experiments.
3.2 Mode instability life time
In the following experiment the build-up time in dependence on the off time is investigated. In Fig. 4 (a) the build-up time for different off times from 20 ms to 120 ms in steady-state operation is plotted. The pulse duration was set to 20 ms for all measurements and the pulse power was kept constant slightly above the power threshold for MI (Pabsorbed pump = 1.1·Pth). Each data point in the graph was averaged over at least 35 pulses and the depicted error bars were calculated from the standard derivation. The build-up time increases for longer off times by about 1.5 ms from 5 ms at 20 ms off time to around 6.5 ms at 90 ms off time and saturates for off times longer than 90 ms.
On the basis of thermally induced index gratings as reason for MI, the measurements could be explained as follows: An off time less than 90 ms is too short to guarantee a complete depletion of the induced thermal gradient and thus of the index grating to the next pulse. The partly conserved index grating results in a decreased MI build-up time for the following pulse. It was only completely deleted within off times longer than 90 ms, which thus can be identified as the life time of MI. Consequently, for pulses with longer off times the index grating has to be built up from zero, which causes longer MI build-up times. For the presented experiments the build-up time becomes independent of the duty cycle for off times longer than 90 ms. To exclude crosstalk an off time of 120 ms or longer was chosen for all experiments presented in this paper.
These investigations are confirmed by the results presented in Fig. 4 (b), where the build-up time for MI during the first 950 ms after a relatively long pause of one second was recorded for duty cycles with short off times (20 ms) (black squares) and with long off times (120 ms) (gray dots). Each data point in the graph was averaged over 10 pulses and the depicted error bars were calculated from the standard derivation. For long off times (120 ms) a constant build-up time (around 6.5 ms) for all subsequent pulses is measured. In contrast, for the duty cycle with short off time (20 ms) only the build-up time for the first pulse was the same (around 6.5 ms) as for long off times, while for the following pulses the build-up time was considerably reduced (down to around 4.5 ms) and was comparable to the value measured in steady-state operation (Fig. 4 (a)). This indicates again that for short off times the MI build-up time is reduced by partly conserved temperature gradients from the previous pulses enabling an index grating, while for the first pulse, i.e. directly after turning the pump diode on, as well as for pulses with long off times the index grating has to be built up from zero.
3.3 Two-step pump schemes
The following experiments demonstrate the influence of pre- and post-pulses with powers below the mode-instability threshold on the build-up and decay time, respectively. The first two-step pump scheme experiment was organized as follows: In a first pump step the system was heated up for 40 ms with a pre-pulse of varying power P1 below the power threshold for MI. In a second pump step the power was increased above the power threshold to P2 = 830 W (≈1.05·Pth) for 20 ms. The off time was chosen to 140 ms in order to avoid crosstalk of subsequent pulses according to the previous results in section 3.2. The pumping scheme is sketched in Fig. 5 (a) . In Fig. 5 (b) the MI build-up time after switching the power to P2 above the mode-instability threshold is plotted. A remarkable decrease of the build-up time in dependence on the pre-pulse power was found from 13.5 ms for no pre-pulse (P1 = 0 W) down to 6.4 ms for P1 = 270 W and further down to 2 ms for P1 = 720 W.
These experiments demonstrate that power levels below the mode-instability threshold (P1) influence the build up of MI as soon as the power is raised above the threshold level (to P2). Thus pre-pulses below the threshold do have indirect influence on MI.
In a second experiment with the two-step pumping scheme sketched in Fig. 6 (a) the influence of post-pulses with powers below the MI threshold on the decay time was investigated, where the decay time refers to the delay from reducing the power below the threshold to vanishing MI. During the duty cycle, firstly, the power of the main pulse was set to the constant power P1 = 960 W above the MI threshold (P1 ≈1.2·Pth) for 20 ms. Then, the power was set to a varying post-pulse power P2 (between 250 W and 790 W) below the MI threshold for 40 ms followed by an off time of 140 ms. Figure 6 shows that for post-pulse powers P2 < 500 W mode instability stops with reducing the power below the power threshold (to P2). For post-pulse powers P2 > 500 W mode instability does not instantaneously stop, but the decay time increased to a maximum decay time of more than 10 ms for absorbed pump powers (P2) slightly below the MI threshold.
These results prove that MI does not suddenly stop as soon as the power is reduced below the MI threshold, but that the decay of MI is influenced by post-pulses below the threshold power. However, permanent MI was only observed above the threshold.
At first view the measurements presented in Fig. 6 seem contradictory to the results presented in section 3.2 Fig. 4 because of the different time scales. In Fig. 6 a MI-decay time of a few milliseconds was measured. In Fig. 4 a grating life time of ~90 ms was found. In our opinion the difference between both mentioned time scales is due to the dynamic of the induced grating. In order to realize MI a dynamic grating enabling a phase shift is necessary, which requires sufficient energy deposition to keep the dynamic running. The decay of MI is thus supposed to be related to the decay of the grating dynamic. This correlation could explain the relatively short decay times measured in Fig. 6. In contrast to this, static gratings seem to have a considerably longer life time (~90 ms) in absence of energy deposition. They do not directly enable MI but they seem to reduce the build up of MI as was demonstrated in Fig. 4.
In conclusion we have investigated the time scales for build up and decay of MI in an Yb-doped high power fiber amplifier. We demonstrate the dependence of the build-up time on the pump power, where the build-up time decreased for increasing pump power in the millisecond range converging to a minimum build-up time of 1.6 ms. Additionally, we identified a life time of MI around 90 ms for the specific fiber under investigation. These investigations confirm that build up and decay of MI takes place on a millisecond time scale, which verifies the theory that MI can be traced back to thermal origin.
Furthermore we demonstrated that pre- and post-pulses influence build up and decay although the power during these pulses was below the mode-instability threshold. The build-up time could be decreased by pre-pulses and the decay time could be increased by post-pulses. Permanent MI, however, was only observed above the threshold.
These observations indicate that physical changes inside the fiber, e.g. temperature gradients, already take place below the MI power threshold. Otherwise pulses below the threshold would have no influence on MI. But only above the threshold these changes seem to be strong enough to impact the light and cause considerable and dynamic energy transfer between transverse modes. Assuming a thermally induced index grating responsible for MI, this means that only above the threshold the grating mechanism is strong enough and shows the necessary dynamic to ensure considerable energy transfer between transverse modes. However, weaker temperature gradients seem to be already build up below the MI threshold.
With the insight gained in the transient phases of MI, the results in Fig. 3 can be reinterpreted as is illustrated in Fig. 7 . In Fig. 3 the dependence of the build-up time on the pump-pulse power was discussed and it was demonstrated that the build-up time decreased with increasing pulse power. From this it can be reasoned that for pulses with pulse durations shorter than the build-up time for a certain pump power no MI would be observable. Consequently, the threshold for MI becomes pulse duration dependent. Considering the same data as in Fig. 3 this new context is reinterpreted in Fig. 7, where the threshold-pump power is plotted in dependence on the pulse duration. The graph indicates an increased MI threshold for decreasing pulse durations and the depicted curve distinguishes the parameter range for stable (gray shaded area) from the one for unstable mode operation. For example, for a pump-pulse duration of 1.7 ms the threshold power would be increased by a factor of 1.6 in comparison to cw-operation, namely from 750 W to 1200 W. For further reduced pulse durations an even higher increase of the MI threshold is expected. Note that as mentioned in section 3.2 the off time between the pulses has to be long enough (> 90 ms) to avoid crosstalk between the pulses. Additionally, it would be insufficient to slightly reduce the power below the power threshold between the pulses, as was discussed in section 3.3 and 3.4. Both shorter off times as well as too high power levels during the off time would result in reduced threshold-pump powers. Based on this argumentation, for applications that do not require high average power but high peak power it would be worth thinking about considerable increasing the MI threshold by operating in short time windows (< 1 ms) separated by long off times (> 90 ms).
Our results suggest that the same effect of MI also takes place in common pulsed high power fiber amplifiers . With typical pulse repetition rates (10 kHz to 100 MHz) the off times between the pulses (10 ns to 100 μs) are too short to avoid crosstalk between subsequent pulses in these amplifier systems. Thus, although these systems provide pulse durations that are too short to allow for MI build up during one single pulse the interaction of several pulses finally leads to MI.
We expect that the exact values for build-up and decay time are characteristic for the experimental system under investigation and are strongly dependent on the fiber geometry. For example, for fibers with larger core diameter it could last longer until thermal gradients would be build up and depleted.
This work has been started within the project “MO-FA – Modenfeldstabile Faserlaser” sponsored by the Thuringian Ministry of Education, Science and Culture (TMBWK, Landesprogramm “ProExzellenz” under contract number: PE203-2-1) and is (partially) funded by the Thuringian Ministry for Economy, Labour and Technology (TMWAT, Project No. 2011 FGR 0103) with a European Social Fund (ESF) grant. We would like to thank the group of Prof. Dr. Limpert (Institute of Applied Physics, Jena, Germany) for helpful discussions. Special thanks go to Hans-Jürgen Otto, Cesar Jauregui, and Fabian Stutzki. Andrea Kliner acknowledges financial support by the Abbe School of Photonics, Jena (Germany).
References and links
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