Mode instabilities, i.e. the rapid fluctuations of the output beam of an optical fiber that occur after a certain output power threshold is reached, have quickly become one of the most limiting effects for the further power scaling of fiber laser systems. Even though much work has been done over the last year, the exact origin of the temporal dynamics of this phenomenon is not fully understood yet. In this paper we show that the origin of mode instabilities can be explained by taking into account the interplay between the temporal evolution of the three-dimensional temperature profile inside of the active fiber and the related waveguide changes that it produces via the thermo-optical effect. In particular it is proposed that non-adiabatic waveguide changes play an important role in allowing energy transfer from the fundamental mode into the higher order mode. As it is discussed in the paper, this description of mode instabilities can explain many of the experimental observations reported to date.
© 2012 OSA
Due to the impressive development of fiber laser technology in recent years , optical fibers have earned a solid reputation as a highly power scalable laser concept. This unparalleled progress, that has seen laser systems evolve from low power setups to multi-kW industrial systems in about a decade, has much to do with the extremely high-power handling capability offered by the geometry of the fiber. The very high surface to active volume ratio allows for an efficient heat removal and, therefore, for high-power operation. However, even though the geometry of the fiber relaxes the demands on thermal management, it generates other problems. Thus, the tight confinement of the light in the core of the fiber gives rise to high intensities that interact with the fiber material over long lengths, which increases the impact of non-linear effects. Hence, active fibers for high-power operation (especially in pulsed operation) have to be specifically designed to alleviate the adverse consequences derived from the non-linearity of the material. The most effective way of mitigating non-linear effects in active fibers is to enlarge the core. This results in a twofold advantage: on the one hand it reduces the intensity of the light propagating in the fiber core and, on the other hand, in double-clad fibers if the pump cladding diameter is not changed, it increases the pump absorption, which allows for shorter devices, thus further mitigating the impact of non-linear effects. Unfortunately, realizing fibers with large cores that still support single-mode operation is far from trivial, especially for high-power operation. In fact, even though the most advanced fiber designs have some in-built mechanism of mode discrimination [2,3], fibers with mode-field diameters larger than 50µm typically support the propagation of a few modes. Consequently, in high-power fiber laser systems today the combination of high thermal loads with few-mode operation is to be found for the first time . This can potentially give rise to new phenomena such as the recently observed onset of mode instabilities at high average powers .
The phenomenon of mode instabilities refers to the output beam of a fiber laser system becoming suddenly unstable once that a certain output power threshold has been reached. Thus, it can be observed that with only a small increase of the output power, the once Gaussian-like output beam of the fiber starts to fluctuate. In this regime the intensity profile at the output of the fiber shows a constantly changing beam formed by the coherent superposition of the fundamental mode and one or more higher-order modes . Recent measurements with a high-speed camera have confirmed that there is actually energy transfer between the fundamental mode and the higher-order mode . Furthermore, Fourier analysis of the beam fluctuations has revealed that, near the threshold, these are not random but follow quasi-periodic patterns with well-defined frequencies . However, when increasing the power further, the beam fluctuations seem to become chaotic.
Shortly after the first reports of this effect came out, we published the first hypothesis on the origin of the effect . In this explanation the interference pattern that appears along a fiber due to the beating of two transverse modes gives rise to a long period index grating via the thermo-optic effect or the resonantly enhanced non-linearity of active fibers . Even though this theory could not explain the dynamic behavior of mode instabilities, it provided an explanation for the mechanism responsible for the energy transfer between two orthogonal transverse modes. Shortly afterwards Smith et al.  pointed out that the energy transfer between the modes can only take place if there is a phase shift between the index grating and the interference pattern. In order to obtain this phase shift they introduced the notion of a moving grating, i.e. the index grating remains not static in the fiber but it moves up or downstream. They demonstrated that this moving grating can lead to a strong transfer of energy between the two interfering transverse modes once that a certain power threshold is reached. In their explanation the movement of the grating is originated by the interference of two transverse modes with slightly different frequencies (a few kilohertz typically). Nevertheless, the origin of this second transverse mode with a frequency shift remains somewhat obscure. Spontaneous emission and quantum noise have been proposed as possible sources for this frequency shifted second transverse mode. However, this explanation contradicts the anecdotal evidence that suggests a strong dependence of the mode instability threshold with the coupling conditions in the fiber. Thus, even though it might be eventually possible that mode instabilities evolve from quantum noise or spontaneous emission, it is reasonable to think that these noise sources would provide the highest possible threshold for this effect. This way, it seems actually more likely that the power thresholds seen in the laboratory are mainly dominated by an unavoidable imperfect excitation of the fundamental mode in the active fiber. Another problem with the explanation presented in  is that, as it has been proposed, it leads to a stable full transfer of energy from one mode to the other. Therefore, the mechanism leading to the measured fluctuations in the relative modal power content once that the threshold is reached becomes difficult to explain.
In this work we present an alternative physical explanation for the movement of the thermally induced grating that does not rely on the presence of a second mode with a frequency shift. Thus, in the next pages it will be shown that the movement of the grating, and therefore the origin of mode instabilities, can be explained taking into account the interplay between the three-dimensional temporal evolution of the temperature profile in the fiber, the related waveguide changes that it causes, and the impact that it has on the interference pattern created by the beating of two transverse modes that have been excited at the beginning of the fiber. As it will be discussed, this theory can explain (at least qualitatively) most of the experimental features exhibited by mode instabilities.
At this point it is worth noting that the phenomenon of mode instabilities is not exclusive of optical fibers, but similar or related effects have also been reported in gas-filled hollow-core compression stages , in intra-cavity high-harmonic generation experiments , and in gas and liquid media (here typically referred to as stimulated thermal Rayleigh scattering) . As far as we are aware, in most of these fields this effect still remains an unsolved problem. However, optical fibers offer unique possibilities to control mode instabilities thanks to their waveguide nature. This has been demonstrated, for example, with the introduction of the so-called large-pitch fibers , which provide a factor ~3 increase of the mode instability threshold when compared to other fiber designs with similar mode-field diameters.
The paper is organized as follows: in section 2 a detailed explanation of the origin of mode instabilities will be provided and in section 3 the theoretical explanation will be supported with simulations. Section 4 provides an overview of the main experimental characteristics of mode instabilities together with a discussion of how they can be understood at the light of the theoretical explanation presented in this paper. Finally some conclusions are drawn.
2. The physical origin of mode instabilities
As described in , when two transverse modes are excited at the input of a few-mode fiber amplifier, a periodic interference intensity pattern is created. This leads to a non-steady absorption of the pump along the fiber, which shows some characteristic oscillations related to the period of the interference pattern. In turn, this inhomogeneous absorption of the pump power generates a non-homogeneous (both transversally and longitudinally) heat load and, consequently, temperature profile. This temperature profile is then transformed into an index grating via the thermo-optic effect. As shown in , this grating always has the right period and symmetry to potentially transfer energy between the interfering modes that ultimately gave rise to it. However, this energy transfer cannot take place in steady state because both the index grating and the interference pattern are in phase (i.e. their maxima and minima fall at the same positions along the fiber). In this context, the present work can be considered the natural follow-up to , in which the dynamic behavior of the temperature gratings and the mechanism leading to energy transfer between the transverse modes (i.e. the movement of the grating) will be detailed.
Figure 1 will be used to explain the physical mechanism leading to creation of the grating. Since the mechanism that is going to be described in the following is time dependent, Fig. 1 is divided in four sections (from a to d), each loosely representing a different instant in time. Additionally, the explanation is based on the simultaneous interplay of several elements, and thus each of the four sections of Fig. 1 comprises the schematic depiction of all these different elements. Since this figure can seem confusing at first, a description of the different illustrations displayed in each section of them will be provided beforehand, so that later on the physical explanation of the process can be easily followed. Thus, in Fig. 1(a) and 1(b), three main blocks of illustrations can be found. The upper one is a schematic plot of the axial temperature profile along the fiber. The middle block can be divided in three parts, on the left hand side there is a depiction of the two transverse fiber modes (the fundamental mode (FM) above and a higher order mode (HOM) below) that are excited at the input of the fiber. The different sizes of the symbols representing these modes indicate the modal power distribution, i.e. it has been assumed that at the beginning of the fiber more power goes to the FM (and therefore its bigger symbol) than to the HOM, which is the usual and typically desirable situation. Beside the input modes, there is a representation of the fiber (as a blue rectangle) together with (from Fig. 1(b) onwards) a schematic illustration of the interference pattern (in orange, superposed to the fiber) that results from the beating of the two transverse modes. Additionally, from Fig. 1(b) onwards, on the right hand side of the fiber, a representation of the transverse modes at the output of the fiber is shown. Here the relative sizes of the two modes also give an indication of the modal power content. Finally, at the bottom of each of the sections of Fig. 1 there is another block of illustrations. This depicts the transverse index profile of the core (as a black solid line) at the input facet (on the left) and at the output facet (on the right) of the fiber. Additionally, superimposed to these index profiles there are two dashed horizontal lines representing the effective refractive indexes of the FM (in blue) and of the HOM (in green).
From Fig. 1(c) onwards there is an additional fourth illustration just below the schematic plot of the longitudinal temperature profile. This new illustration also symbolizes the fiber, but in it the thermally-induced index grating has been schematically represented (as an arrangement of solid dark blue rectangles, each standing for a period of the grating) and/or the thermal load (as an arrangement of empty rectangles).
Now that the symbolism of Fig. 1 has been detailed, it is possible to proceed with the explanation of the physical origin of mode instabilities. Figure 1(a) represents an active step-index fiber at rest which is being pumped from the right hand side. Since no signal has been injected yet, the temperature profile is uniform throughout the fiber and it equals the room temperature (this assumption is valid in an inverse three-level system, such as Yb:glass pumped at the zero-phonon line, if the contribution of ASE is neglected and no material absorption is considered). Consequently, the transverse index profile of the fiber remains unchanged along the fiber (and, therefore, it is equal at the beginning and at the end of the fiber as illustrated in Fig. 1(a)). It is worth noting that this extremely simplified initial state is only considered in the explanation of the physical process for the sake of simplicity; the simulations, however, use the actual initial temperature and, therefore, index profile. Additionally it is assumed that an incoming seed beam excites two transverse modes of the fiber in such a way that the FM receives most of the power (as it is usually the case). Then, as shown in Fig. 1(b), these two transverse modes will propagate along the fiber creating an intensity interference pattern along the way. At this point it is important to underline that the period of this interference is anti-proportional to the separation of the effective indexes of the two interfering modes. This interference pattern will result, as the signal starts to be amplified, in a quasi-periodically oscillating heat load (Fig. 1(c)) that mimics the main features of the modal beating pattern . It is worth noting that, even though there is a certain heat load, there has not yet been any increase of temperature. This is because the heat load is quasi-instantaneous but a certain time is required to increase the temperature. This way, finally, after a certain time has elapsed, the temperature increases and it evolves to mimic the heat load and, therefore, the interference pattern that gave rise to it (Fig. 1(d)). Then, this temperature pattern is transformed into an index grating via the thermo-optic effect. Up to this point the process is very similar to the creation of a static thermally induced grating that was described in .
In the following the explanation of the physical process leading to mode instabilities will be divided in three parts: the first one describes the mechanism allowing energy transfer from the FM into the HOM, the second one details the process that provides the required conditions to allow energy transfer from the HOM into the FM, and the third section presents a description of the circumstances that might lead to the reversal of the direction of energy transfer.
2.1. Energy transfer from the FM to the HOM
Our simulation points out that this type of energy transfer is related to the existence of non-adiabatic longitudinal changes of the transverse index profile along the fiber. For simplicity, from now on, these variations of the transverse index profile will be referred to as waveguide changes. The longitudinal temperature profile along the axis of the fiber typically follows a quasi-exponential trend (see dashed lines in Fig. 2 ) that gets steeper and steeper as the overall temperature rises (until a certain final steady state is reached). As it has been shown in , this non-linear longitudinal temperature trend can strongly distort the quasi-sinusoidal temperature oscillations that are superimposed to it due to modal interference. The result is that, as shown in Fig. 2, the temperature oscillations are quasi-sinusoidal for low-temperatures (green solid line in Fig. 2) but they adopt a sawtooth-like appearance at high temperatures (red solid line in Fig. 2), i.e. the maxima are pushed towards their next minimum, and the minima are pulled towards their previous maximum (as indicated by the red arrows in Fig. 2). This creates strong temperature changes that can take place over very short fiber sections. In turn this means that, periodically along the fiber but especially towards its end where the longitudinal temperature gradients are stronger, the beam finds strong and fast waveguide changes (such as that schematically depicted in Fig. 2 in which the waveguide evolves from the transverse index profile 1 (upper right) to the transverse index profile 2 (lower right) in a very short distance). When the temperature gradients are strong enough, these strong waveguide changes can become non-adiabatic, i.e. the beam cannot adapt itself fast enough to these rapidly changing waveguide conditions. This brings the beam out of phase with the index grating, i.e. the beam is delayed with respect to the index profile. In turn, this induces a downstream movement (i.e. towards the output end of the fiber) of the interference pattern (which will be then, after a certain time, followed by the temperature profile), and thus it creates the necessary conditions to transfer energy from the FM to the HOM. Additionally, as suggested in , these steep temperature changes can eventually lead to some significant amount of longitudinal heat flow that might further contribute to the dynamics of the effect. However, this later point is still to be confirmed.
It is worth noting at this point that this effect becomes stronger the more the HOM content grows towards 50%. At this point the amplitude of the temperature oscillations and, therefore, the non-adiabatic waveguide changes will reach their maximum strength. From this point on, however, the further the HOM content grows towards 100%, the weaker the effect becomes, because the oscillations in the temperature profile become weaker and, therefore, the waveguide changes become progressively more adiabatic.
2.2. Energy transfer from the HOM to the FM
There is an important detail worth noting in Fig. 1(d): the temperature gradient has also a transverse component that, through the thermo-optic effect, modifies the transverse index profile of the fiber (especially towards the output facet, where the temperature increase is the highest in this example). This modification of the index profile of the fiber results in an alteration of the separation of the effective indexes of the transverse modes (typically, with a dominating FM, this separation is increased as schematically indicated in Fig. 1(d) and Fig. 3(a) ). In turn this results in a modification (compression) of the period of the interference pattern. Thus, at this point the modified interference pattern and the index grating (given rise by the unmodified interference pattern) are not in phase anymore. This is mainly because the interference pattern can change nearly instantaneously whereas the index grating requires a certain time to evolve (due in part to the finite thermal conductivity of silica), i.e. the index grating is constantly playing catch-up with the interference pattern. Thus, since the interference pattern and the index grating are not in phase anymore, a certain amount of energy transfer between the interfering transverse modes is allowed (as represented by the larger size of the FM at the output of the fiber in Fig. 3(b)).
The modification of the interference period is ultimately dependent on the local transverse temperature gradient, which varies along the fiber. Therefore, the period of the interference pattern will be modified (compressed) in different degrees at different positions along the fiber (Fig. 3(a)). That is the same to say that the period of the interference will be chirped, with a chirp function that roughly follows the longitudinal temperature profile. This modified interference pattern will generate a new heat load which is slightly shifted with respect to the index grating that has been already thermally induced in the fiber (see Fig. 3(a)). Thus, with more time, the temperature profile, and therefore the index grating, will evolve in the direction given by this new heat load. This evolution causes the local movement of the index grating (Fig. 3(b)). In high-power amplifiers this movement will be typically upstream the fiber (i.e. towards the input of the fiber). Thus, this movement will cause the energy to flow from the higher-order mode into the fundamental mode. It is important to note, additionally, that this effect becomes weaker when the local period of the interference pattern approaches the period determined by the local waveguide characteristics.
2.3. Reversal of the direction of energy transfer
In the preceding paragraphs two competing effects have been described that are able to transfer energy from the FM into the HOM and vice versa. The interplay between these two effects can lead to time varying fluctuations in the relative mode content at the output of the fiber. Unfortunately, providing a detailed explanation of the complex temporal dynamics of a system containing coupled non-linear effects such as the ones described above is typically not possible. However, in the following we propose a plausible scenario that can cause the reversal of the direction of energy transfer. As it has been mentioned before, when the HOM content starts to grow towards 50%, the visibility of the modal interference and, therefore, the amplitude of the temperature oscillations become larger. This, in turn, strengthens the non-adiabatic waveguide changes, which lead to an even stronger energy transfer in the HOM. Additionally, these non-adiabatic waveguide changes cause the local period of the interference pattern to differ from that that would be determined by the local transverse index profile alone (which will be referred to as natural local period from now on). As the HOM content grows beyond the 50% mark, both the visibility of the modal interference and the oscillations of the temperature profile are reduced (until they completely disappear by 100% HOM content). Thus, the waveguide changes become more and more adiabatic, and the rate at which energy flows to the HOM is reduced, i.e. the index grating becomes weaker. At this point several things can happen. On the one hand, more HOM content will typically lead to a locally reduced absorption of the pump due to the smaller overlap of this mode with the doped region. This in turn results in a locally reduced heat load that will further make the waveguide changes more adiabatic by reducing the temperature gradient. Additionally, a weaker grating will eventually result in a reduced transfer in the HOM, thus potentially allowing more content of the FM to reach the fiber end. Finally, when the waveguide changes become more adiabatic, the effect of the period compression described in section 2.2 will become more and more dominant until it causes the reversal of the movement of the index grating, and thus the reversal of the direction of the energy transfer. Thus, as the energy starts to flow from the HOM into the FM, the visibility of the modal interference will start to grow again as well as, after a certain time, the amplitude of the temperature oscillations. This will cause the waveguide changes to become again non-adiabatic. Thus, as the effect of the period compression weakens (as the local period of the interference approaches the natural local period of the waveguide), the effect of the non-adiabatic index changes will again become dominant. This will in turn cause the energy transfer to change direction again, which starts the cycle anew. At this point we want to state that we do not claim that this is the only scenario leading to the reversal of the direction of energy transfer, but in our opinion it is a plausible one. However, as mentioned above, the dynamics of coupled non-linear effects are typically too complex to be fully described in a simple way; numerical simulations being usually required for a full understanding.
Thus, the physical processes described above are what we believe lead to mode instabilities. However, there are some points that are still under investigation on our side and that will require clarification in the future. One of these points is to study if the model as described above is inherently chaotic (as observed in the measurements) or if, instead, it will always relax to a stable state. Provided that the latter were the case, it might be required to include weak external perturbations (which are, on the other hand, always present in any experimental setup) to trigger the chaotic evolution of the system, as already suggested in . However, it is still soon to draw any conclusions on this point since, as mentioned before, it is still subject of ongoing research.
It is important to underline that in this explanation of the physical process leading to the movement of the grating and to the energy transfer between the transverse modes, only the interplay between the evolution of the three-dimensional temperature profile and the resulting modification in the index profile of the fiber have been considered. Thus in this model, as opposed to , both transverse modes at the beginning of the fiber have identical spectra. Nevertheless, the movement of the index grating, via the Doppler Effect, will lead to a small frequency shift between the modes at the output of the fiber. The crucial detail here is that in the physical process described in this paper, the appearance of a frequency shift between the transverse modes is simply a consequence of the movement of the grating but it is not the mechanism originating this movement.
In the following the viability of the physical process described in this section will be demonstrated with the help of simulations.
3. Simulation results
In order to test whether the mechanism described in the previous section does indeed lead to the movement of the grating and to energy transfer between the transverse modes, a simulation model was set up. This model solves the time and transversely resolved rate equations in which the propagation between the different sections of the fiber is done using a Beam Propagation Method (BPM). Additionally, the temporally resolved three-dimensional temperature profile of the fiber is calculated at each step. Finally, this three-dimensional temperature profile is transformed, via the thermo-optic effect, into a modified index profile of the fiber that is fed back in the BPM algorithm.
Two simplifications have been done in the model in order to reduce the computational complexity (time) of the simulations. The first one is that a step-index fiber has been considered. This is a valid assumption since mode instabilities have been observed in all kinds of fibers: from step-index to photonic crystal fibers. The second simplification is that only radially symmetric modes are considered. This allows using a radially symmetric BPM scheme  that is considerably faster than a full 3D approach. However, since in most (if not all) experiments mode instabilities are reported to happen between the fundamental mode and a radially anti-symmetric mode, the choice of a radially symmetric model requires a more detailed justification. The most likely reason why mode instabilities have been observed so far with radially anti-symmetric modes is because of Transverse Spatial Hole Burning (TSHB). It should not be forgotten that mode instabilities are always observed in strongly saturated fiber systems in which TSHB cannot be neglected. Thus the appearance of TSHB will dramatically reduce the gain that radially symmetric modes undergo (because the much stronger FM depletes the inversion primarily in the center of the fiber, where radially symmetric modes have their maximum of intensity). On the contrary, it is known that radially anti-symmetric modes can undergo gains higher than those of the FM in the presence of TSHB . Therefore, radially anti-symmetric modes will experience a double amplification process: one due to the active fiber and the other one due to the energy transfer enabled by the moving grating. On the contrary, radially symmetric modes will mainly be amplified by the energy transfer process resulting from the thermally induced grating. Thus, it can only be expected that the mode instability threshold of radially symmetric modes is much higher than that of radially anti-symmetric modes. This argument, on the one hand, clarifies why mode instabilities are reported with radially anti-symmetric modes (since they have the lowest thresholds) but, on the other hand it shows that there is no strong physical constrain to use radially symmetric modes to study the physics behind this phenomenon. In fact, as it was demonstrated in , radially symmetric modes also generate thermally induced index gratings. Thus, in our opinion, these modes can be used to test the viability of the physical mechanism described in the previous section.
In the following a 0.25m long step-index fiber is simulated with 80μm core diameter (completely doped with Ytterbium ions; total ion concentration N = 1.31026 ions/m3), 0.027 numerical aperture, 200μm pump core diameter, 1.8mm outer fiber diameter, passively cooled in air, and pumped with 300W at 976nm in the counter-propagating direction. These fiber parameters do not correspond to any realistic fiber. However, they have been chose because the high thermal loads generated in the fiber (due to the high power extraction in a short length) and, therefore, the fast temperature evolution, allows us to accelerate the simulation. Thus, the physics behind mode instabilities can be analyzed with a relatively short computational time.
The signal injected as seed for this fiber amplifier is centered at 1064nm and consists of a 1MHz train of 1ns Gaussian pulses which, when coupled into the fiber, excite pulses with 1000W peak power in the LP01 mode (FM) and of 100W peak power in the LP02 (HOM). The active fiber has been pumped until full inversion is reached before the seed signal is injected. The evolution of the fiber was simulated over 1ms.
Media 1 shows a movie with the evolution of both the axial temperature profile (red line) and intensity interference pattern (blue line) along the fiber (above) together with the evolution of the relative mode content along the fiber (below) over 1ms. Each frame represents the status of the fiber after the front end of each pulse has reached the end of the fiber. In Fig. 4 it can clearly be appreciated that the oscillating temperature profile (which in turn gives rise to an index grating mimicking it) as well as the interference pattern have a phase shift with respect to one another, especially towards the end of the fiber. Media 1 confirms that the physical processes described in the previous section do indeed lead to the movement of both the intensity pattern and the index grating (which is directly proportional to the temperature profile). Thus, as the average temperature in the fiber increases, it can be appreciated how both the intensity pattern and the temperature profile move downstream the fiber. This is because the fast and strong temperature gradients that will occur in this short fiber (much stronger than in a real-world fiber) imply that the movement of the grating is dominated by the presence of non-adiabatic waveguide changes (that can also be appreciated in the red line of the upper graph). Additionally since, as explained in the previous section, the temperature profile/index grating is playing catch up with the movement of the interference pattern and since it evolves from a previous state (i.e. there is a memory effect), it can be potentially blurred if the movement of the interference pattern is fast enough. This blurring of the temperature profile/index grating is something that can be seen happening in the video. On the other hand, the blurring might as well be the consequence of the non-adiabatic waveguide changes blurring the interference pattern. This is a point that needs some clarification in the future.
In Fig. 5 the movement of the interference pattern and of the temperature profile/index grating can be better seen. Figure 5(a) shows the evolution of the instantaneous position of one of the maxima of both the interference pattern (in blue) and the temperature profile (in red) over the 1ms simulation time. The maxima that have been tracked in Fig. 5(a) are situated towards the end of the fiber (~0.22m). Please note that only 220µs are plotted because after that the blurring of the grating becomes too strong to find the maxima. As can be observed in Fig. 5(a) there is indeed some movement of the maxima downstream the fiber. The complete shift of the maxima over 220µs simulation time is as large as 5mm, which corresponds more or less to half the modal beating period. Additionally it can be seen that the movement becomes slower with time, as the speed of the temperature change is also reduced. Furthermore, Fig. 5(a) highlights what has already been commented in the previous section: that the interference pattern is the first to move and then this movement is followed by the temperature profile/index grating. This delay creates a phase shift between interference pattern and index grating that allows for energy transfer between the interfering transverse modes, as will be shown in the following.
Figure 5(b) shows the instantaneous shift of three maxima of the temperature profile/index grating originally situated at three different positions along the fiber: one towards the end of the fiber at ~0.22m (red line), another towards the middle of the fiber at ~0.15m (green line), and a last one at the beginning of the fiber at ~0.05m (gray line). What this graph reveals is that even though different maxima of the temperature profile/index grating move in the same direction (downstream the fiber in this case), they do so at different speeds. This points out towards a much more complex movement of the grating than that suggested in .
Even though the movement of the grating, as already commented before, is dominated in this example by the existence of non-adiabatic waveguide changes, Fig. 6 demonstrates the coexistence of the second process described in the previous section. Thus in the figure it can be seen how, as the temperature in the fiber increases (i.e. as time elapses), the local period of the modal interference pattern tends to be compressed due to the transversal index gradient. Figure 6 has been calculated by locally applying a mode solver  to each point along the fiber and by repeating this process for each one of the simulated pulses. Thus, what Fig. 6 reveals is the “natural” local modal beating period, i.e. the period that the interference patter would have, should it not have been affected by propagation. As can be seen, and as it was already discussed in section 2, this effect tends to move the intensity pattern, and subsequently the index grating, upstream the fiber. Thus this effect is in direct competition with the downstream movement of the interference pattern/grating caused by the non-adiabatic waveguide changes.
The evolution of the relative mode content at the output of the fiber over the 1ms simulation time is plotted in Fig. 7 . There it can be seen that the HOM content at the beginning of the simulation was about 2.3% (which is lower than the ~10% that has been set at the input of the fiber mainly due to the effect of preferential gain), but it continuously increases during the simulation until it reaches ~9.64% (i.e. more than a factor of four increase) by the end. Therefore, Fig. 7 confirms that the process generating the movement of the temperature profile/index grating does indeed lead to energy transfer between the interfering transverse modes. Furthermore, the increase of energy in the radiation modes (black line) is an indirect indication of the existence of non-adiabatic waveguide changes.
It is worth noting that Fig. 7 does not show the dramatic energy conversions (of ~100%) characteristic of mode instabilities, nor does it show its complex dynamic behavior (see  and ). The probable reasons for this are, on the one hand, that the typical periods of mode instabilities for fibers with comparable mode-field diameters to the one that has been simulated is of several ms . Additionally, as has been previously discussed, the threshold of mode instabilities for radially symmetric modes is expected to be much higher than that corresponding to radially anti-symmetric modes. Finally, very recent measurements have revealed that the build-up of the index grating can take several ms . All these factors together make it unlikely that mode instabilities can be observed under the conditions simulated in this example. Anyways, even though we have not reproduced the instabilities in the simulation, the numerical results have demonstrated that the physical mechanisms described in this paper lead to energy transfer between the interfering modes.
In this section some of the experimental characteristics of mode instabilities will be listed and, then, it will be analyzed how the current theoretical understanding of mode instabilities can (or cannot) explain them. The list of experimental characteristics has been compiled either from published papers (references will be given) or from anecdotal evidence.
- ▪ During mode instabilities there is a continuous back and forth transfer of energy between the interfering transverse modes : As explained in section 2, there are two competing effects that force the index grating to move in opposite directions. Thus, depending on which effect is dominant the energy will flow from the FM into the HOM or the other way around.
- ▪ There is a dependence of the fluctuation speed of the mode instabilities with the mode-field diameter of the modes, i.e. in larger fibers the mode instabilities are slower : This can be explained by taking into account the thermal origin of mode instabilities. Due to the finite thermal conductivity of silica, the thermalisation time becomes larger the larger the mode area.
- ▪ Fibers with larger mode-field diameters exhibit, in general, lower mode instability thresholds : This can be understood given that larger modes are typically more sensitive to waveguide changes. This implies that smaller temperature gradients (both transverse and longitudinal) are required to strongly modify them and, therefore, to set the index grating in motion.
- ▪ Near the threshold the spectrum of the mode instabilities shows several resonances with a certain bandwidth : The appearance of these peaks is due to the frequency shift caused by the movement of the index grating. Additionally, their finite bandwidth can be understood taking into account the complex movement of the grating revealed by our simulations, i.e. different sections of the grating move at different speeds.
- ▪ Mode instabilities have been observed exclusively between the fundamental mode and radially anti-symmetric modes: Most likely this is due to the presence of transverse spatial hole burning favoring the amplification of radially anti-symmetric modes, as discussed in section 3. This would explain why these modes have the lowest mode instability threshold and, therefore, why almost all mode instability experiments published to date have been reported the appearance of this kind of modes. However, this is a point that still requires further clarification.
- ▪ Above the power threshold for mode instabilities the beam fluctuations evolve towards chaos : Possibly the combination of the non-linear coupling between the beam propagation and the temperature (i.e. they influence one another in a non-linear way as can be inferred from the discussion presented in section 2) with the memory effect of the temperature profile/index grating (which has to evolve from a previous state) leads to a chaotic system. However, this is still a hypothesis and it needs to be confirmed with future research. The truth is that our current models allow us to understand the origin of mode instabilities, but the investigations are not so advanced yet as to explain the complex temporal dynamics of this effect.
- ▪ The effect of mode instabilities seems to have a build-up time and lifetime of some ms : This, once again can be explained taking into account the thermal origin of the effect: as the fiber heats up, the index grating gains in strength and it progressively transfers more and more energy to the HOM. Thus, there is a build-up time of the index grating. On the other hand, if the pump or the seed are suddenly switched-off, the index grating will survive for a time until a homogeneous temperature profile is reached in the fiber.
In this paper a new understanding of the physical origin of mode instabilities has been presented and discussed. Energy transfer between the two transverse modes is allowed by the movement of an index grating, which has been ultimately originated by the interference of these two transverse modes. The movement of the grating can be explained by the complex interplay between the temporal evolution of the three-dimensional temperature profile in the fiber, the waveguide changes that it causes, and the effect that these changes have on the interference pattern (which in turn gives rise to the temperature profile). The validity of this model has been supported by simulations showing that the proposed physical process does indeed lead to the movement of the index grating and to the energy transfer between the transverse modes. Finally, some of the most important experimental characteristics of mode instabilities have been analyzed at the light of this new theoretical explanation of the effect. The detailed understanding of the physical origin of mode instabilities should help in the quest of finding efficient methods to mitigate it.
The authors what to thank Andrei N. Starodoumov for useful discussions and for making us aware of the existence of stimulated thermal Rayleigh scattering. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement n°  and the Thuringian Ministry of Education, Science and Culture under contract PE203-2-1 (MOFA) and contract B514-10061 (Green Photonics). F.J. acknowledges financial support from the Abbe School of Photonics.
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