## Abstract

We present a dynamic model of thermal modal instability in large mode area fiber amplifiers. This model allows the pump and signal optical intensity distributions to apply a time-varying heat load distribution within the fiber. This influences the temperature distribution that modifies the optical distributions through the thermo-optic effect thus creating a feedback loop that gives rise to time-dependent modal instability. We describe different regimes of operation for a representative fiber design. We find qualitative agreement between simulation results and experimental results obtained with a different fiber including the time-dependent behavior of the instability and the effects of different cooling configurations on the threshold. We describe the physical processes responsible for the onset of the instability and suggest possible mitigation approaches.

©2012 Optical Society of America

## 1. Introduction

Currently the onset of transverse modal instability at a limited power threshold is hindering large mode area fiber average power scaling efforts [1–3]. Key characteristics of the observed instability include: temporal fluctuations on the kHz frequency scale with a degree of chaotic behavior that increases with increasing power, no decrease in overall output power as the fluctuations increase, a dependence on average rather than peak power level, and that increasing the seed power increases the threshold [4]. Attempts to explain this phenomenon have centered on the formation of index gratings that are phase matched to the interference pattern created by two co-propagating transverse modes that resonantly couple power between two or more such modes [4–7]. Symmetry considerations dictate that a phase offset between the optical interference pattern and the grating must exist for this coupling to occur. A frequency difference between the signals travelling in different transverse modes has been proposed as the cause of this offset [6]. Furthermore, the observed dependence of the threshold on average power suggests that thermally-induced index changes may be the dominant mechanism forming the index gratings. In the case of thermally-induced gratings, a temporary phase offset naturally occurs whenever the optical path length of the fiber changes due to environmental fluctuations as the optical interference pattern can shift faster than the temperature distribution. The former is governed by optical propagation equations while the latter is governed by the time dependent heat equation. It is then reasonable to conjecture that the optical intensity field influences the temperature distribution through quantum defect heating and the temperature distribution influences the optical intensity field through the thermo-optic effect leading to a feedback loop causing a dynamic modal instability.

To investigate this possibility we introduce a model wherein the optical intensity field is governed by the propagation and interference of two transverse modes under the perturbing influence of the thermal index variation and gain distribution throughout the fiber. The thermal distribution evolves according to the time dependent heat equation while the gain distribution is determined using two-level rate equations. We assessed this to be the simplest reasonable scheme for investigating the plausibility of this mechanism. A description of this model follows after which some example calculations are presented. We then qualitatively compare the predictions of the model with experimental observations. Finally we comment on possible mitigation techniques revealed by our results.

## 2. Model

Our model begins with the scalar paraxial optical wave equation valid for weakly-guiding large mode area fibers given by Eq. (1):

Substituting ${\beta}_{0}^{2}={\beta}_{1}^{2}+({\beta}_{0}^{2}-{\beta}_{1}^{2})$and invoking Eq. (2) modified to describe ${\varphi}_{1}\left(x,y\right)$ and${\beta}_{1}$ reveals that ${\beta}_{1}^{2}={\beta}_{0}^{2}-2{\beta}_{0}\Delta \beta $which constitutes a somewhat non-standard yet deliberate definition of $\Delta \beta .$ The field evolution along the fiber is then given by Eq. (5)

where we have introduced the complex field amplitude ${c}_{1}.$ Allowing slow variation of the field amplitudes with $z$we arrive at a general expression for the scalar field representing the propagation of two transverse modes given by Eq. (6):The coefficients ${c}_{i}$are normalized such that the signal optical intensity field is given by ${I}_{s}={\left|\psi \right|}^{2}$in which case the power in the modes is given by ${P}_{i}={\left|{c}_{i}\right|}^{2}.$ To study the effect of thermally induced index variations on the evolution of the coefficients we introduce them perturbatively such that $n(x,y,z)={n}_{0}(x,y)+T(x,y,z)\left(dn/dT\right).$ Furthermore, laser gain is introduced locally so as to increase the optical intensity according to the usual two-level rate equations. We neglect resonantly-induced refractive index changes [9] since we are focusing on operating regimes for which these changes are much smaller than the thermally-induced ones. Invoking Eq. (1), Eq. (2), Eq. (4) and Eq. (6) leads to the equation for the evolution of the field

This model thus accounts for both intra and intermodal non-linear phase modulation and gain. The first term on the right hand side of Eq. (9a)-(9b) represents the change in optical length of the fiber as its temperature changes as well as the optical gain of the modes. The second term represents the resonant energy transfer between the two as well as inter-modal laser gain. The choice of ${\beta}_{0}$as the reference wavevector for the slowly varying envelope approximation may lead to a breakdown in the accuracy of the solutions of Eq. (9a)-(9b) in the presence of a large thermal gradient. To mitigate this possibility we apply a variable transformation given by Eq. (11)

The deliberate choice to include only two modes in the description of the optical field limits the ability of the model to treat cases where the index profile changes appreciably over the length of the fiber. An arbitrary number of modes may be added to our model in the future to treat such cases. Including the LP_{02} mode in particular would enable the model to treat fibers with a thermally-induced shrinking mode field diameter [10]. No power loss results from this limitation however because the resonant coupling terms in Eq. (12a)-(12b) conserve total power.

To solve E Eq. (12a)-(12b) both the temperature and gain distributions throughout the fiber must be known. The thermal distribution evolves according to the time dependent heat equation given by Eq. (13)

with appropriate boundary conditions. Quantum defect heating is assumed to be the sole contributor to the heat load $Q.$ The gain within the doped region of the fiber is given bywhere the population densities ${N}_{i}$in Eq. (14) are determined by the equilibrium solution to the two level rate equations and the quantum defect heating is given by Eq. (15)It is important to note that the population densities and therefore the gain, heating, and temperature vary throughout the fiber as well as in time. We have neglected laser dynamics in our treatment on the grounds that we are primarily considering optical pump and signal intensities far above their respective saturation intensities such that the populations are assumed to reach equilibrium on a time scale smaller than that describing the thermal variations within the fiber.

We now introduce the key approximation of our approach. We treat the spatially-varying temperature distribution as static during the time required for the signal to completely traverse the length of the amplifier. Accordingly the rate equations are solved for the equilibrium population distributions and Eq. (12a)-(12b) are integrated for constant thermo-optic and gain profiles. The resulting modal amplitudes along the fiber are used to calculate the heat load which is then used to update the temperature distribution. More precisely we use a Runge-Kutta method to solve Eq. (12a)-(12b) at a given time step to obtain the optical intensity distribution throughout the fiber assuming uniform pump intensity within the cladding cross section at each point along the length of the fiber. This is then used to calculate the heating distribution. We then use a Crank Nicolson scheme to solve Eq. (13) for the temperature distribution at the next time step. All thermal and optical degrees of freedom are represented on a Cartesian grid. In the counter-pumped case we furthermore approximate the total pump absorption to be independent of the relative amplitudes between the two modes. This alleviates the requirement to iterate the optical solution multiple times at each time step to converge on the appropriate residual pump level to ensure the launched pump level is uniform.

## 3. Modeling results

We model a large mode area photonic crystal fiber (PCF) with a pump cladding diameter of 170 µm, an outer cladding diameter of 400 µm, and doping concentration of 3.5 × 10^{25} m^{−1} which results in a device length of approximately 1.6 meters. This type of fiber is comparable in mode area and device length to some studied previously [4,5,7,8]. The large mode area results in a relatively long beat length between the two transverse modes. The beat length is the primary length scale that drives the longitudinal grid resolution requirement. The modeled fiber has a 7-cell core design resulting in a doped area that is somewhat smaller than the overall core area. The fiber has a pitch of 25 µm and an air hole diameter of 5.2 µm resulting in a mode field area of 2773 µm^{2} for the fundamental LP_{01} mode and 2939 µm^{2} for the LP_{11} mode as shown in Fig. 1
. The beat length between these two modes in the absence of thermally induced index change is 22.6 mm.

Our model describes a wide range of amplifier configurations. In this initial work we consider only counter-pumped configurations with 47.5 Watts of input power in the LP_{01} mode and 2.5 Watts of input power in the LP_{11} mode both inputs being monochromatic. To save the need to time-step throughout the entire warm up period, which can be several seconds, we first solve the time-independent heat equation for the thermal distribution assuming all of the power is in the LP_{01} mode before adding in the LP_{11} power and allowing time to evolve. We have established that this method of initiating the simulation gives the most rapid approach to conditions conducive to equilibrium. Figure 2
shows results for three different pump powers with convective cooling at the boundary at a uniform 200 W/m^{2}K. The time step used was 40 µs and the spatial grid spacing was 111 µm in the longitudinal direction and 5.5 µm in the transverse direction. These were varied to ensure the numerical convergence of representative simulations. These values resulted in a spatial numerical grid with 7.9 × 10^{7} points.

Figure 2(a)-2(b) describe the behavior of the amplifier when pumped with 380 Watts. As shown in the embedded movie (Media 1), during the initial transient behavior power transfers from the LP_{01} to the LP_{11} mode. This coupling requires the phase of the optical intensity to lag the phase of the temperature variations in a particular region of the fiber [6]. This is made possible by the fact that the optical phase can change much faster than the temperature phase. The optical phase shift at a given point in the fiber relative to the phase at the fiber entrance is accumulated throughout the fiber length. Therefore the relative phase change, and hence the coupling behavior, is strongest at the output end of the amplifier. The direction of the power transfer depends on the size of the phase shift. For shifts less than π radians, power couples from the LP_{11} to the LP_{01} mode while for shifts greater than π radians, power couples in the reverse direction. Thus for a monotonically increasing optical phase shift, power transfers back and forth between the modes as the local phase passes multiples of π as the signals propagate down the amplifier. According to this picture, once the temperature distribution reaches equilibrium with the optical phase, no coupling will be possible and the LP_{01} mode will dominate the output as shown in Fig. 2(a)-2(b).

An additional feature is rapid temporary coupling of power between the modes at a spatial frequency corresponding to the inter-modal beat length. This rapid temporary coupling occurs because the effect of the local optical intensity on the temperature is mediated through the heat equation and the laser rate equations rather directly through the intensity. This leads to a coupling constant that reverses sign within one period of the induced grating. Previously published work has shown that no such temporary coupling exists where the index changed is caused purely by the non-linear refractive index [5]. Another important feature of this rapid coupling is that it apparently does not depend on a particular thermo-optic phase shift because it remains present even for amplifiers below the instability threshold that reach equilibrium as shown in Fig. 2(a) and Fig. 4(a) .

Figure 2(c)-2(d) describe the amplifier with 636 Watts of pump. We observe that the beat-length oscillations increase in amplitude and that the LP_{11} mode acquires the most power for a period of about 10 ms after the initial transients. After this the modal output powers are approximately equal out to the end of the run. We do not claim that this represents an equilibrium state, but it is clear that the dynamic behavior persists for a longer time than for the case with lower pumping power. This power level represents the approximate instability onset threshold for this fiber under these conditions. We do not attempt to identify precise threshold values due to the multiple very long-time simulations required to do so.

Figure 2(e)-2(f) describe the amplifier with 952 Watts of pump. Very interestingly, the amplifier appears to reach equilibrium with most of the power in the LP_{11} mode. We chose these power levels to demonstrate three different behavior regimes. To accurately determine the rapidity of the onset of the instability we would need to run the simulation for longer times. For all configurations the total amplifier output is steady in time in agreement with experimental observations [4]. We did not study the convectively-cooled amplifier at pump powers higher than 952 Watts because the maximum temperature of the fiber in this last case approached 500 °C which is probably as high as and perhaps higher than such amplifiers should sustain. We have indeed observed instability in simulations of convectively cooled amplifiers at higher powers that exhibited intolerable fiber core temperatures. Figure 3
shows the temperature profile at the end of the 60 ms simulation for the 952 Watt pump case. We observe that the variation in the profile due to the inter-modal beating is small compared to the overall temperature profile.

More effective cooling enables higher power levels. We next discuss the behavior of amplifiers with conductive cooling that exhibit constant temperature on one edge of the fiber as occurs with fibers wrapped on a cooled mandrel for example. We treat the fiber edge lying at the location lying on the negative *x* axis 200 µm from axis in Fig. 1. This arrangement is consistent with wrapping the fiber on a cooled mandrel. The effect of different cooling symmetries on the amplifier beyond the scope of this paper. Figure 4(a)-4(b) describe the behavior of such an amplifier pumped at 500 Watts. One interesting observation is that the modal contents appear to approach constant finite values as shown in Fig. 4(b). An examination of Fig. 4(a) as well as Fig. 2(a) reveals that this steady value depends on the phase of the beat-length coupling at equilibrium. This suggests that fine-tuning of the pump power may enable optimization of the beam quality by ensuring that a trough in this variation occurs for the LP_{11} modal power. In this case it is important to note that the relative phase of the two modes at the amplifier input is constrained by the fact that the LP_{11} content is likely caused by launching the seed slightly off center. This particular effect may not be observable, however, for fibers exhibiting effective loss discrimination between the modes.

Figure 4(c)-4(d) describe the behavior of a conductively-cooled amplifier pumped at 1060 Watts. One of the interesting features of this simulation is the presence of an apparent temporary equilibrium with most of the power in the LP_{11} mode at the amplifier output followed by a rapid transition to an oscillatory behavior with most of the power in the LP_{01} mode. As the pump power is increased, these oscillations increase in amplitude until they cause a transition to dynamic full-depth oscillation between the modes at the output shown in Fig. 4(e)-4(f) which describe the amplifier when pumped with 1750 Watts.

We propose that longitudinal heat flow combined with the different heat loads caused by the different transverse modes explain the onset of these oscillations. Without longitudinal thermal diffusion, time dependent instability theoretically should not exist in the absence of external perturbation. This may be seen by realizing that in the absence of such heat flow, an equilibrium solution to Eq. (12) may be obtained by calculating the equilibrium *x-y* temperature distribution at each point along the propagation direction using only the optical fields, and hence the heat load, at that longitudinal position. Furthermore, it has been shown [7] that there is significant difference between the temperature profiles of such amplifiers in the presence and absence of longitudinal heat flow. These two facts support the view that longitudinal heat flow is necessary for dynamic modal instability. Nevertheless, we observed temporal oscillations in the modal content of the amplifier with a broader frequency spread at the highest pumping power with longitudinal heat flow artificially turned off in the simulation. We believe this point requires further investigation.

The next link in our proposed instability feedback loop is the effect of modal content on the quantum defect heating load. We have observed that the LP_{11} mode creates less heat per unit length than the LP_{01} mode. This can be seen in the movies linked to Fig. 2 and Fig. 4 where the temperature is shown to be lowered slightly in the regions along the fiber where the modal content is dominated by LP_{11} and elevated where the modal content is dominated by LP_{01}. The phase of the optical interference pattern at any point along the fiber is determined by the relative phases of the two modes which are determined by the initial phases and the accumulated thermal phase shift from the input end of the fiber up to that point. These observations suggest that the ability of changes in modal content to affect the optical phase shift and the ability of the thermo-optic phase shift to affect the modal content through resonant coupling together give rise to the observed instability. We further note that this instability mechanism does not require the higher-order mode to possess a different optical frequency than the fundamental as is the case for a previously proposed mechanism [6].

We also observe that the amplifier exhibits two distinct regions along the length of the fiber. Within the first part of the fiber, the amplitudes are stable in time. Beyond a critical length at which the rapid temporary coupling becomes sufficiently strong, the modal contents oscillate in time. Our model predicts that the length over which oscillating behavior occurs within the fiber should increase with increasing pump power. Figure 5(a)
-5(c) show the spectra of the modal content of the amplifier output at three different power levels exhibiting three different types of behavior. At the lower power level the output is relatively stable for long times as shown in Fig. 4(a) with a corresponding peaked spectrum shown in Fig. 5(a). At an intermediate power level the modal content exhibits relatively regular oscillatory behavior for the longest simulated times. The corresponding spectral peaks shown in Fig. 5(b) are relatively small due to the limitations in the simulation run time. The time series shown in Fig. 4(f) representing the highest output power has no dominant spectral components but is rather described by a noisy peak with a tail extending to several kHz as shown in Fig. 5(c) indicating chaotic oscillatory behavior. This behavior remained unchanged when the cooling point on the fiber was shifted to a point on the *y* axis in Fig. 1. The temperature profile in the conductively-cooled case shows all of the expected features including the overall background profile with local variations imposed by the modal interference pattern as shown in Fig. 6
. The maximum temperature of 100 °C relative to the heat sink should be well tolerable.

There are several other dependencies that may be checked with our model that are beyond the scope of this paper. These include fiber dimensions such as length, core diameter, pump cladding diameter, and outer cladding diameter. The dependence on the seeding configuration, both total power level and modal content may also be investigated. The model may also be used to study the differences between co-pumped and counter-pumped configurations as well as the effect of differential passive loss between the modes and tailored dopant profiles within the core. Rather than trying to study the dependencies of the instability on all of these factors at the same time it makes more sense to conduct this type of analysis in the context of overall evaluation of proposed amplifier configurations.

## 4. Experimental results

We experimentally investigated modal instabilities in an amplifier featuring an SBS-suppressing photonic crystal fiber featuring a segmented acoustically tailored (SAT) core. Details of the core design are provided in [11]. The SAT fiber allowed us to investigate the modal instabilities using a single-frequency seed, as well as phase-modulated and broadband seed sources. Initially, the amplifier was seeded with the output of an intermediate amplifier chain seeded with a single-longitudinal mode non-planar ring oscillator (NPRO).

The PCF exhibited a nominal core diameter and mode field diameter of 39.5 and 30 µm respectively. Borosilicate glass stress rods within the cladding enabled single-polarization operation. The inner pump cladding and outer cladding had diameters of 329 µm and 580 µm respectively with a pumping numerical aperture of 0.6 enabled by an air-clad structure. This fiber is substantially different to the one analyzed in the previous section. The experimental fiber has significantly shorter beat length, longer device length and larger outer cladding diameter than the modeled fiber. Hence modeling the experimental fiber requires at least 1 order of magnitude more computational time and memory than the modeled fiber. We are working to improve and optimize the computer programs to study such fibers in the future. Thus in the present work we seek to qualitatively compare the amplifier we were able to model to the available experimental amplifier.

Figure 7 depicts the experimental setup. The NPRO provided 10 mW of continuous wave seed power with a nominal linewidth of a few kHz that was subsequently amplified using a four stage amplifier chain the final stage of which consisted of the counter-pumped PCF amplifier. For the single-frequency studies, the phase modulator shown in the Fig. 7 was not used. Our measurements indicated that ~30 W of the 50 W of third-stage output power was coupled into the PCF core. We used fiber-coupled diode stacks to launch up to about 1 kW of ~976 nm pump into the 10 meter length of PCF. Investigation into the thermal dependence of the phenomena was prompted by a comparison this fiber design to commercially available fibers used in high power fiber amplifiers. The 976 nm pump absorption of this fiber is at least twice that of 25/400 step index or 40/400 photonic crystal fibers. As such, the heat deposition per unit length is much higher in the fiber discussed here relative to other fiber designs.

To investigate the influence of active cooling on modal instability we employed two different cooling arrangements. At first, the fiber was suspended in the air with a coil diameter of 53 cm and allowed to cool through convection. Each end of the fiber was held in a water cooled chuck, which was mounted on a 5-axis translation stage. The beam was analyzed using a commercial M^{2} measurement device. At power level below 300 Watts, the M^{2} value was less than 1.3 and the near field image showed a Gaussian-like profile consistent with the fundamental fiber mode (LP_{01}). As the signal power approached 380 W, there was a sudden jump in the M^{2} value which was measured at 2.2. We have made multiple attempts to improve mode-matching of the seed laser into the photonic crystal fiber core and modify the fiber coiling geometry to increase higher order mode filtering. These exercises had negligible impact on the modal instability threshold.

A comparison of M^{2} as a function of signal output power was performed for a fiber amplifier under a second cooling condition. The fiber was conductively cooled with an aluminum ring, as shown in Fig. 8
. Water-cooled plates were affixed to the aluminum ring to improve cooling of the fiber. Approximately 2.5 meters of fiber, near the seed input end, was taken off the ring and suspended in air.

Each end of the fiber was held in a water cooled chuck, which was mounted on a 5-axis translation stage. The location of each stage was fixed on the optical table. The section of fiber left in the air was required to allow the fiber to conform to the input and output chuck positions. Improving thermal management of the fiber amplifier by using the cold ring was shown to increase the modal instability threshold beyond 500 Watts. For conductive cooling, the M^{2} value was ~1.4 at power levels approaching 500 W. Figure 9
compares the two cases by showing the measured M^{2} value versus the signal power. As shown, improving thermal management of the fiber amplifier can increase the modal instability threshold. We note that without the benefit of the SAT fiber, the investigation of the modal instability in the conductively-cooled single-frequency amplifier would not have been possible due to the onset of SBS. In fact, in an experiment that utilized a reference fiber with similar dimensions to the SAT fiber but with an acoustically homogeneous core, the SBS threshold was encountered at ~150 Watts. Further power scaling of the SAT fiber was hampered by rapid fluctuations in the near field output of the amplifier. Near field images of the optical intensity profile below and above these modal instabilities are shown in Fig. 10
. For the SAT fiber amplifier described above, these instabilities occurred at ~530 Watts. To rule out linewidth and acoustic tailoring effects on the modal instabilities we assembled an ASE source with an approximate spectral width of 10 nm and used it to seed both the SAT fiber and the reference fiber (i.e. fiber without acoustic tailoring). Since the reference fiber amplifier was SBS limited at ~150 Watts, the ASE source allowed us to reach power levels >500 Watts with this type of fiber. There was little difference in the threshold of the modal instabilities between the SAT and reference fibers.

Additional measurements confirmed that the linewidth of the seed source had little effect on the threshold of the modal instabilities in the SAT fiber. In one study, the linewidth of the ASE source was reduced to 0.5 nm using a notch filter. In another study, the output linewidth from the single-frequency NPRO was broadened by modulating at 1 GHz using the LiNbO_{3} modulator. The modal instability threshold for the conductively cooled SAT amplifier remained constant at 500 Watts within the few Watt uncertainty of the optical power meter regardless of the linewidth.

Finally, we conducted a study to capture the time-dependent behavior of the output signal. It is known that low-speed cameras such as the one used for the M^{2} studies presented above are not fast enough to capture the time dynamics of the energy transfer between the LP_{01} and LP_{11} modes [12]. To increase the time resolution we focused a sample of the output beam into a single-mode fiber and monitored the output using a fast photodiode detector. The output from the fiber is proportional to the power in the fundamental mode. The trace of the output power in time is shown in Fig. 11
exhibiting oscillations with frequencies on the order of several kHz. This trace was obtained near the threshold of 500 Watts and indicates that the amplifier was operating in the intermediate instability regime.

We observed threshold-like transverse modal instability in both simulation and experimental results. In each case, active conductive cooling of the fiber increased the threshold. For the conductively-cooled amplifier cases the onset of the instability was manifest by oscillatory behavior in both the experimental and simulated cases. In the experimental case, the power output was not pushed very far past the onset of the oscillatory instability. In the simulated case, however, pushing the power higher resulted in chaotic oscillations characterized by a decaying frequency spectrum. Thus in the conductively cooled case we find three basic stability regimes characterized by stable operation, periodic oscillations, and chaotic oscillations in modal content. In the convectively cooled case it appears possible that the amplifier could reach equilibrium with most of the power exiting the fiber in the LP_{11} mode. The different behavior in the convectively and conductively cooled cases is not surprising in light of the drastically different temperature profiles observed in the two cases as shown in Fig. 3 and Fig. 6. One important aspect of our simulations is that they assume no external perturbations to the fiber such as those that would be caused by mechanical vibrations and fluctuations in pump and seed powers. Such perturbations may prevent experimental observation of the simulated LP_{11} equilibrium state.

## 5. Mitigation strategies

Our model suggests a few promising approaches for mitigating the onset of this instability. The first approach is to minimize the maximum longitudinal thermal gradient within the core thus reducing longitudinal heat flow. One method of doing this is to modify the fiber design to promote constant gain per unit length throughout the fiber. Gain is a function of doping concentration as well as pump and signal intensities therefore the doping concentration or pump cladding diameter could be modified along the fiber length to achieve this goal. The latter of these two approaches, while perhaps easier to implement, would have brightness conservation implications for the pump. Another method would be to create an opposite gradient on the heat sink. Another approach would be to design the fiber core such that the thermo-optic overlap of the two modes is the same. This would cause the local heating load to be independent of the modal content. These two general approaches are compatible with each other and may be employed together to achieve the best results.

On the other hand, our simulation results suggest that filtering strength requirements for mode filtering fibers [3] are more stringent as a result of the observed rapid intermodal coupling. Even differential mode losses of several dB/m may not be able to completely suppress higher order modes over lengths of millimeters. For example in the results depicted in Fig. 4(c)-4(d), the power in the LP_{11} mode rises from nearly zero to about 200 Watts within a few millimeters. Another known approach for suppressing higher order modes is to maximize the overlap of the gain and the fundamental mode while minimizing the overlap with higher-order modes [2]. This is most directly accomplished by concentrating the doped area in the center of the core. According to our model, this would increase the disparity between the modal thermal self-phase modulation coefficients leading to a more rapid optical phase shift for a given temperature change thus possibly enhancing the efficiency of the proposed instability mechanism.

Modal instability now seems to be the limiting factor for further power scaling of narrow-linewidth large mode area fibers. Just as there have been multiple methods developed for increasing the SBS threshold in these fibers, we anticipate methods will emerge for increasing the modal instability threshold with specially designed transverse and longitudinal thermal profiles assuming a role similar to that of acoustic engineering in SBS-suppressing fibers.

## 6. Conclusion

We have presented a theoretical model and experimental observations of dynamic transverse modal instability and in large mode area high average power fiber amplifiers. The model predicts thermo-optic threshold-like onset of modal instability in both convectively and conductively-cooled amplifiers. In the conductively-cooled case, the threshold was observed to be approximately 1.5 times higher than in the convectively cooled case. Depending on the cooling arrangement and amplifier output power three different instability behaviors occur including steady-state power transfer to a higher-order mode, periodic oscillatory behavior in the modal content, and chaotic time variation of modal content. In all studied cases the model and experimental observations were in qualitative agreement. Quantitative comparison was prevented by a disparity between the fiber designs that can currently be modeled and those on hand for experimentation. Our model has illuminated possible physical conditions and processes responsible for the modal instability including longitudinal temperature gradient, effective higher order thermal non-linearity, and disparate thermo-optic overlap between transverse modes. Based on these causes several approaches to mitigating this instability were suggested. Future work will focus on investigating these approaches as well as expanding the range of experimental configurations that can be investigated with the model.

## Acknowledgments

This work was supported in part by a grant of computer time from the DoD High Performance Computing Modernization Program at Air Force Research Laboratory DoD Supercomputing Resource Center, Wright-Patterson Air Force Base, OH. The authors gratefully acknowledge the High Energy Laser Joint Technology Office, the Air Force Research Laboratory and the Laser and Optics Research Center at the United States Air Force Academy for support. The views expressed in this article are those of the authors and do not reflect the official policy or position of the US government or the Department of Defense.

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