Mode-interference along an active fiber in high-power operation gives rise to a longitudinally oscillating temperature profile which, in turn, is converted into a strong index grating via the thermo-optic effect. In the case of mode beating between the fundamental mode and a radially anti-symmetric mode such a grating exhibits two periodic features: a main one which is radially symmetric and has half the period of the modal beating, and a second one that closely follows the mode interference pattern and has its same period. In the case of modal beating between two radially symmetric modes the thermally induced grating only has radially symmetric features and exhibits the same period of the mode interference. The relevance of such gratings in the context of the recently observed mode instabilities of high-power fiber laser systems is discussed.
© 2011 OSA
High power fiber lasers have undergone an unprecedented development rate in the last decade , in which they have been transformed from Watt-level laboratory curiosities into multi-kW-level systems with high market penetration. The reasons for this unparalleled growth and commercial success have to do with the outstanding characteristics that have become the trademark of fiber lasers: excellent beam quality and stability, robustness, low-maintenance costs, high-efficiency, small footprint and low weight, just to mention a few.
Up to now, fiber lasers have always been able to cope with an ever-increasing demand of higher average powers and higher pulse peak powers. The ability to provide high average powers is mainly due to the excellent thermal management capabilities that fibers have due to their very high surface to active volume ratio . However, even though fiber lasers have been largely free of thermal problems up to now, it is only natural that, with average powers growing increasingly fast, even this technology runs into thermal issues sooner or later.
On the other hand, in order to be able to deliver high-peak powers (or multi-kW average powers in CW operation), fibers need to have large mode field diameters (MFD) to reduce the impact of nonlinear effects such as stimulated Raman Scattering  or Self-Phase Modulation . However, it is extremely challenging to increase the fiber core size while retaining the excellent beam quality characterizing fiber lasers. The reason is that fibers with large MFD become multimode, i.e. they support the propagation of several transversal modes, which typically leads to a degradation of the beam quality. Nevertheless, in the last years significant advances have been made resulting in effectively single-mode fibers with MFDs>100μm . Even though effectively single-mode fibers typically operate in the fundamental mode due to some kind of filtering mechanism [6,7] or delocalization of the high order modes , they still support the propagation of at least a few modes.
The combination of thermal effects and multimode operation in high-average power fiber systems (CW or pulsed) can potentially lead to new and unexpected effects that degrade the operation of fiber laser systems. This is quite likely what has happened with the recent observation of mode instabilities in Large-Mode-Area (LMA) active fibers in high-power operation . The onset of these mode instabilities occurs rather suddenly at a certain power threshold. Thus, for output powers lower than this threshold the output beam remains stable, whereas for output powers beyond this threshold the output beam profile fluctuates in a seemingly chaotic way between the fundamental mode and one (or more) higher order modes (HOM).
In a first effort to understand this effect, we showed that the modal interference between the fundamental mode and a HOM can write a long period index grating in the active fiber  via either the resonantly induced index change of doped fibers  or the thermo-optic effect . This grating has exactly the right period to transfer energy between the interfering modes. A.V. Smith et al. expanded this idea in  by pointing out that in order to get energy transfer between the modes a phase-shift between the mode interference pattern and the induced grating is required. A.V. Smith et al. solved this problem by introducing the notion of a moving grating caused by the interference of two modes with slightly different frequencies. The idea of the moving grating is a very interesting proposal that leads to the appearance of a power threshold for this effect. However, as for today, the physical origin of the frequency difference of the interfering modes remains unclear. Additionally, they also identified thermal effects as the most likely cause of mode instabilities, but no detailed exploration of such thermal effects was carried out.
In this work we show that mode interference between the fundamental mode and a higher order mode generates an oscillating temperature profile along the fiber that, via the thermo-optic effect, is converted into a strong long period index grating. In the case of modal interference with a radially anti-symmetric higher order mode this grating shows two periodic modulations: a main one which is radially symmetric and has half the period of the interference pattern, and a second one that closely follows the interference pattern (and therefore is not radially symmetric) featuring its same period and no phase shift with it. In the case of modal interference between two radially symmetric modes the thermally induced grating only has radially symmetric features and exhibits the same period of the modal beating.
The paper is divided as follows: in section 2 the physical origin of this temperature-induced index grating will be discussed, in section 3 the temperature-induced grating will be closely examined, in section 4 the relation between these gratings and the mode instabilities will be discussed and finally conclusions are drawn.
2. Temperature profile in a LMA fiber in high-power operation
Since the purpose of this paper is to analyze the temperature profile along an active fiber in high power operation, no secondary effects such as energy transfer between modes or preferential gain  will be considered. This means that the temperature profile, and the resulting induced index grating, will be calculated in steady-state in an active few-mode fiber in the case when there are two interfering modes that do not exchange energy with each other and when both modes undergo the same amplification by the active medium. Even though these conditions do not necessarily match those found in real-life experiments, they are the best ones to understand the interplay between mode interference and the temperature-induced index grating because they neglect other effects that distort the profile of that grating.
In order to carry out the simulations for this paper, the transversally resolved rate equations  taking into account mode interference were solved in the steady state case. This approach, being significantly less computationally intensive than the beam-propagation method used in , allows calculating the induced grating with a much higher resolution. It should be mentioned, however, that this simple model does not take into account any thermally induced change of the numerical aperture of the fiber (which would result in a local change of the interference period).
In order to calculate the temperature profile T(r,ϕ,z) inside of the fiber, the following equation must be solved :Eq. (3) the transversal distribution of the pump power has been considered homogeneous over the pump core.
To solve the heat equation in Eq. (1), the Newton law of heat transfer has been used at the fiber boundaries . This way, using the transversally resolved rate equations together with equations Eq. (1) to Eq. (3), allows calculating the three-dimensional temperature profile in an active fiber in high-power operation. Furthermore, if the beam profile resulting from the interference between two (or more) fiber modes is calculated along the fiber and fed into the transversally-resolved rate-equations, then the temperature profile caused by this mode interference can be calculated.
In the following a 1m long step index fiber is simulated with 80μm core diameter (completely doped with Yb ions; total ion concentration N = 3.51025 ions/m3), 0.017 numerical aperture, 200μm pump core diameter, 1.2mm outer fiber diameter, passively cooled in air, and pumped with 300Watt at 976nm in the counter-propagating direction. The signal injected as seed of this fiber amplifier is centered at 1064nm and contains 5Watts of power in the LP01 mode and 1Watt of power in the LP11. The HOM content has been chosen in purpose to be very high (around 20%) for illustration purposes, so that the temperature profile and the grating become more visible. Under these circumstances the beam profile along the fiber together with the inversion profile in the x-z plane are shown in Fig. 1(a) and (b) , respectively.
As can be seen in Fig. 1(b), the inversion profile follows the same alternating pattern already shown in . However, as pointed out in , this inversion profile and, therefore, the index grating resulting via the resonantly induced index change are in phase with the beam interference pattern shown in Fig. 1.a. In principle, this excludes the possibility of energy transfer due to an inversion grating in ideal steady state operation. The reason why we observed this energy transfer in our previous work has to do with a coarse discretization of the fiber, which introduced a numerical phase shift of the order of 3% of the period. This, in our opinion, only highlights the extreme sensitivity of this effect to external perturbations. Thus, even though in a perfectly steady system no energy transfer would be observed, in a real system small changes in the excitation of the modes at the fiber input can lead to such energy transfer.
What is interesting to observe in Fig. 1, however, is that the beam saturates the active medium differently depending on its transversal shape (i.e. position along the interference pattern of Fig. 1(a). Thus, as can be seen in Fig. 1(b), when the interference pattern pushes the center of gravity of the beam towards the outside of the core, the inversion becomes strongly transversally inhomogeneous. This is because the inversion is more strongly depleted in certain areas of the fiber core (see darker vertical stripes in Fig. 1(b) than in others because of the transversally inhomogeneous intensity profile of the beam. Due to saturation effects, in those positions along the fiber where the center of gravity of the beam is shifted towards the edge of the core, the overall amount of remaining excited ions is higher than in the fiber sections in which the center of gravity of the beam lies near the fiber axis. This can be seen in Fig. 2(a) , where the total number of excited ions (i.e. excited ions added over the fiber cross-section) at each position along the fiber is shown.
As can be observed, the total number of excited atoms oscillates along the active fiber. As mentioned above this oscillation is caused by the mode interference and, therefore, it shows a periodicity that is related to the modal beating period. In fact, the period of change of the excited ions is half the period of the modal interference. The reason is that there is a local maximum of excited ions every time that the center of gravity of the beam is pushed towards the edge of the core, which happens twice per period of the modal interference (see Fig. 1(a)). In turn, this oscillatory behavior of the excited ions implies that the pump power is not steadily absorbed along the active fiber, as demonstrated in Fig. 2(b). This figure has been obtained by differentiating the evolution of the pump power along the length. Thus, as can be seen, the absorbed pump power also oscillates along the amplifier. In particular, the absorbed power exhibits a minimum approximately every time that the center of gravity of the beam is pushed towards the edge of the core. That is of outmost importance because, according to Eq. (2), it implies that the thermal load of the fiber should also oscillate along the fiber with the same period. This, in turn, means that the temperature profile should also present similar oscillations, as shown in Fig. 3 . In that figure it can be observed that the temperature profile follows closely the evolution of the absorbed power and, correspondingly, it also exhibits maxima at the points where the center of gravity of the beam crosses the fiber axis.
As can be seen in the right hand side plot of Fig. 3, the oscillations of the temperature profile along the amplifier can be very strong (~7°C in ~1cm in this example without considering the longitudinal heat flow, i.e. blue line). These strong temperature gradients in the longitudinal direction suggest that the contribution of the longitudinal heat flow can be significant in this case. The longitudinal heat flow results in a reduction of the amplitude of the temperature oscillations (see red line in Fig. 3(b)). Besides, the longitudinal heat flow causes a slight shift of the maxima and minima in the temperature profile (compare blue and red curves in the inset of Fig. 3). Thus, the maxima of the temperature profile with longitudinal heat flow come later and the minima before than their counterparts of the profile without longitudinal heat flow. On the other hand, this longitudinal heat flow, through its time dependence, might as well be the mechanism conferring these gratings, and the mode instabilities, their dynamic behavior.
3. Temperature-induced index grating
In the following the main characteristics of temperature induced index gratings will be analyzed. In order to do this, two cases will be distinguished: a grating caused by the interference of the fundamental mode with a radially anti-symmetric mode, and a grating caused by the interference of the fundamental mode with a radially symmetric mode.
3.1. Interference between the fundamental mode and a radially anti-symmetric mode
In the example presented in the previous section the temperature profile has been created by the interference between the fundamental mode and a radially anti-symmetric mode. This temperature profile, shown in Fig. 3, creates via the thermo-optic effect a refractive index change mimicking it. Thus, as can be seen in Fig. 4 , an index grating appears in the fiber. In calculating this index grating a temperature dependence of the refractive index dn/dT = 1.210−5 has been considered. Additionally, a more realistic HOM content as in previous simulations has also been assumed. Thus, in this case the power in the fundamental mode (LP01) is as before 5 Watts, but the power contained in the LP11 is just 250mW (so just 5% of the power contained in the LP01). Besides, this simulation takes into account the longitudinal heat flow but not the change of the index of refraction caused by the resonantly induced index change of doped fibers . This has been done to be able to study the thermally induced grating on its own, without other effects distorting it. Moreover, neglecting the resonantly induced index change is acceptable since its impact is much smaller than that of the thermo-optic effect. Actually, according to the literature, a realistic value for the index change due to this resonant effect in our fibers is ~2-410−6 . As it will be seen, the index change expected from the thermo-optic effect is about one order of magnitude stronger. Thus, considering the resonantly induced index change would only result in small alterations of the thermo-optic index change, but it would not modify it substantially.
Looking at Fig. 4 (left graph) it can be seen that the absolute index change due to temperature in a high power fiber can be beyond 10−3. Additionally, the transversal index change across the fiber core can be as high as 510−5. This can have dramatic consequences in today’s LMA fibers that rely on extremely weak guidance for proper effectively single-mode operation. The impact of this absolute index change in the mode-sets of these fibers should be examined carefully. However, this task is outside of the scope of this paper. Here, we want to concentrate ourselves in the oscillations superimposed to this absolute index change (see Fig. 4 right). At first sight these oscillations seem to be dominantly radially symmetric and they can have amplitudes (i.e. deviations from the background index change) as high as 1.5-210−5. As commented before, these index changes are much higher than those expected for the resonantly induced index change.
Looking at the period of these oscillations it can easily be seen that it is half that of the mode interference giving rise to it. The physical reason for this has already been discussed in the previous section. This fact has an important consequence that becomes clear when analyzing the Bragg condition:Eq. (4) implies that for a grating to be resonant at an specific wavelength, its period has to be larger (a multiple, in fact) than the beating period of the interference pattern of the modes (given by ). This is not the case with this thermally induced grating and, therefore, it cannot couple energy between the fundamental mode and the higher order mode that have originally generated it.
Carefully observing the temperature-induced grating in Fig. 4, a second periodicity can be discovered. This is revealed in the inset of Fig. 4 by the alternatively crossing index profiles along the lines x = + 39µm (red line) and x = −39µm (green line). This second period is due to the transversal asymmetry of the temperature profile, i.e. the center of gravity of the temperature profile also shifts in the x-z plane along the fiber. This is illustrated in Fig. 5(a) , where the normalized transversal index changes at two different positions along the fiber (blue line: z = 0.96m, red line: z = 0.98m) are shown. It can be clearly observed that the center of gravity of both profiles is displaced. This second periodic structure can be made visible by subtracting the radially symmetric component from the index change shown in Fig. 4. In order to achieve this, each index profile along a line parallel to the fiber axis is subtracted from its diametrically opposite and this difference is divided by two. Doing this cancels out the radially symmetrical features of the index change, thus revealing the second periodic structure which is shown in Fig. 5(b).
It can be seen that this secondary grating exhibits oscillations as high as 2.510−5, and that it closely resembles both the interference pattern that gave rise to it and the inversion grating shown in Fig. 1(b) (albeit being almost one order of magnitude stronger than it). Physically, this grating represents the index differences that the beam profile sees at different transversal positions as it propagates along the fiber. In this case, the period of this secondary grating is the same as that of the mode interference. Therefore, it fulfills one of the conditions required to allow energy transfer between the interfering modes. However, as revealed in Fig. 6 , and as it is also the case with the inversion grating, this secondary thermally induced grating has a negligible phase shift with respect to the interference pattern. In Fig. 6 both the thermally induced index grating (in blue) and the intensity of the beam (red line) along the line x = + 39µm (corresponding to the upper white dashed line in the left plot of Fig. 4) are shown. Since the center of gravity of the interference pattern shifts in the x-z plane, the red line in Fig. 6 exhibits a maximum every time that the center of gravity is closest to the x = + 39µm line (i.e. the boundary between core and cladding), and it exhibits a minimum when it is farthest away from the x = + 39µm reference line. Both the index change line and the beam intensity line in Fig. 6 have been normalized to allow for their direct comparison. Thus, in Fig. 6 it is possible to study the position of the maxima of the beam intensity line and of the maxima/minima of the thermally-induced grating. As can be seen, the maxima (and minima) of the curves corresponding to the intensity of the mode interference and to the radially anti-symmetric component of the index grating are almost perfectly aligned. Thus, as mentioned before, the phase shift is negligible, which implies that energy transfer between the interfering modes is not allowed in the steady state case.
3.2. Interference between the fundamental mode and a radially symmetric mode
The characteristics of the thermally-induced index grating caused by the interference between the fundamental mode and a radially symmetric mode are different from those discussed above. The main different is that the thermally-induced grating only has a radially symmetric component instead of a radially-symmetric and a radially anti-symmetric as in the previous case. The mode interference and its corresponding inversion profile in the x-z plane can be seen in Fig. 7 . In doing this simulation the same fiber parameters as before have been considered. The main difference is that at the input the LP01 and the LP02 modes have been excited at the input with 5Watt and 250mW, respectively. The left graph in Fig. 7 shows the typical breathing of the beam caused by the interference between two radially symmetric modes. In turn this breathing causes some inhomogenities in the inversion profile (Fig. 7(b)) in the longitudinal direction due to the different overlap of the beam with the doped region.
As explained before, this inhomogeneous inversion profile gives rise to an inhomogeneous temperature profile that, subsequently, via the thermo-optic effect generates the index-grating observed in Fig. 8 . This grating has the same period as the mode interference pattern. Additionally, as already mentioned above, this index grating is perfectly radially symmetric, which implies that for symmetry reasons it can only transfer energy between radially symmetric modes. Apart from that, comparing this index change with that shown in Fig. 4, it can be seen that the overall effective index change (background index change) is very similar to the previous case (~2.7e-3). However, the modulation depth of the index grating is much higher than when considering interference with a radially anti-symmetric mode, being as high as ~610−5 in this case.
We have observed that these gratings can exhibit a small amount of local phase shift with respect to the mode interference pattern. This is because the non-linear increase of the background temperature can distort the quasi-sinusoidal index/temperature profile to an extent where it creates local phase shifts with respect to the mode interference pattern shown in Fig. 7(a). In any case, this phase shift is only local and it changes sign along a grating period. This means that a certain energy transfer between the interfering modes is locally allowed, but it will reverse its direction as the beam travels through one period. This will create ripples in the energy transfer diagram similar to those observed in the optically induced gratings reported in . At the end, however, the net energy transfer after each period will be almost zero (the energy transfer in the first semi-period being almost completely reversed in the second semi-period). Almost because the non-linear strong background temperature rise ensures that there is always a small remaining phase shift after each period, i.e. a small energy transfer. Thus, in principle these grating could account for a weak energy transfer between the interfering modes even in steady state. However, in real-world situations these gratings are not expected to be able to cause any significant energy transfer between the interfering modes in steady state operation.
Even though the discussion of this paper has been centered in the counter-propagating pumping case, similar thermally-induced gratings are also found in co-propagating setups. The main difference is that the absolute maximum of temperature, and therefore of index change, is typically reached at some point in the middle of the fiber and not at the pump end as shown in this paper (see Fig. 3, for example).
4. Thermal gratings and mode instabilities
Up to this point it has been shown that mode interference gives rise to an oscillating temperature profile which is, via the thermo-optic effect, transformed into a long period index grating. As has been seen, the interference of the fundamental mode with a radially symmetric or with a radially anti-symmetric mode gives rise to index gratings with different characteristics.
In the case of the grating being created by the interference between the fundamental mode and radially symmetric modes, it has been shown that these gratings only exhibit radially-symmetric features and, therefore, they could only couple light between radially symmetric modes. However, as discussed above, these gratings do not show any significant net phase shift (rather in some cases a local phase shift) and, therefore, the energy transfer between the interfering modes in steady state operation will be negligible.
In the case of the grating being created by the interference between the fundamental mode and radially anti-symmetric modes, it has been shown that, using superposition, this index grating can be decomposed in two components: one that is radially symmetric and one that is radially anti-symmetric. In the following these two components will be analyzed separately in the context of modal instabilities.
The radially symmetric component cannot transfer energy between the interfering modes because it has a period that is half their beating length. However, this index grating could transfer energy (even in steady state operation) between the fundamental mode and another HOM with the appropriate effective index (one that provides an index difference with the fundamental mode that is about twice as large as that of the HOM generating the grating). Even though this possibility looks quite remote at first, this can actually be less unlikely than what it might seem. The reason is that these gratings are typically relatively short (at least in fibers with very high MFDs), which imply that their resonances will be broad. Consequently the condition on the refractive index of the third mode is somewhat relaxed. Nevertheless, due to symmetry considerations this index grating can only transfer energy between the fundamental mode and radially symmetric modes. It should be said, however, that at this moment this is just a speculative point and it will be the subject of future work to study its feasibility.
It has also been shown that the radially anti-symmetric component of the thermally induced grating has the right period to transfer energy between the interfering modes giving rise to it, but it exhibits a negligible phase shift with respect to the interference pattern. This last condition prohibits the energy transfer in the ideal steady state case. This does not mean that this secondary grating (i.e. the radially anti-symmetric component of the index change) is not at the root of the mode instabilities observed, however. What it means is that it is simply not possible to analyze what is in essence a dynamic effect with a steady state model. However, this model is useful to reveal the presence of the grating and to analyze it in detail as done in this paper. Anyway, it is our belief that this grating is actually involved in the appearance of these instabilities.
There are several scenarios that can be imagined in which this grating causes the energy transfer observed in the mode instabilities. For example, as proposed in  any movement of this grating would allow that energy transfer. This movement of the grating can be caused by the interference of modes of different frequencies as proposed in , but it might also be due to some kind of thermal destabilization. Alternatively, if the grating remains static, any small changes (in the order of a few percent) in the relative phase between the interfering modes would lead to energy transfer. This small phase changes might be caused by slight modifications of the in-coupling conditions (mechanical vibrations, air turbulence /convection), or alternatively by thermally-induced changes in the waveguide. In any case, it can be inferred that the presence of this grating makes the fiber very sensitive to perturbations (whether external to the fiber or internal to it). This way, it can be interpreted that the index grating at the end amplifies the impact of these perturbations and, therefore, the mode instabilities occur when the amplification factor is higher than the damping of the manifestation of these perturbations.
Independently of the physical effect making the energy transfer possible symmetry considerations in the case of the radially anti-symmetric component of the index change lead to the conclusion that this secondary grating can only couple energy between the fundamental mode and radially anti-symmetric modes. However, it is important to note that also in this case the energy transfer does not have to occur between the two modes giving rise to the index grating, but this grating might also allow the energy transfer to a third radially anti-symmetric mode. As discussed in the case of the radially symmetric component, this mechanism would also work in the steady state case.
As mentioned in , the fact that the frequency of the modes instabilities lay in the kHz region points towards the thermal origin of this effect. But additionally, the fact that the vast majority of the modal instabilities observed to date show fluctuations between the fundamental mode and a radially anti-symmetric mode  is a strong indication that the radially anti-symmetric grating shown in Fig. 5(b) is at the root of those mode instabilities.
In this paper it is shown that mode interference in a few-mode LMA fiber gives rise to an oscillating temperature profile along its length. This temperature profile, in turn, generates a long period index grating via the thermo-optic effect. The interference of the fundamental mode with both radially symmetric and radially anti-symmetric higher order modes has been studied. In the case of mode interference between the fundamental mode and a radially anti-symmetric mode such a thermally induced index grating exhibits two periodic features: one that is radially symmetric and has half the period of the modal beating, and a second one that closely follows the mode interference pattern and has its same period. In theory none of these two periodic features can transfer energy between the interfering modes in the steady state case. In the first case because of the grating having the wrong period, and in the second case because of the grating having no phase-shift with the interference pattern. In spite of this, it is our belief that these thermally induced gratings are at the root of the modal instabilities recently reported. Accordingly several possible scenarios for this to happen are discussed in the text.
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). F. J. and C. J. acknowledge financial support from the Abbe School of Photonics.
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