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Self-focusing is the ultimate power limit of single mode fiber amplifiers. As fiber technology is approaching this limit, ways to mitigate self-focusing are becoming more and more important. Here we show a theoretical analysis of this limitation in coupled multicore fibers. Significant scaling of the self-focusing limit is possible even for coupled multicore fibers if the out-of-phase mode is chosen. On the other hand the in-phase mode can – depending on the coupling strength – be prone to instabilities.
© 2015 Optical Society of America
1. Introduction
Self-focusing is the ultimate power limit of single mode fiber amplifiers [1, 2]. It can be mitigated in bulk systems, as the propagation length in the crystals can be made shorter than the self-focusing length, but as fibers rely on long propagation length to achieve their advantageous properties, this approach is not an option in fiber amplifiers. So far self-focusing has rarely been an issue because other limitations such as the silica damage threshold, other nonlinearities or thermal issues were the power limiting factors, but with novel fiber designs fiber technology is quickly moving towards the self-focusing limit, which makes a look at possible mitigation strategies valuable. Depending on the assumed nonlinear refractive index, the self-focusing limit is approximately 5 MW (at 1064 nm). Because the Kerr-nonlinearity’s response time is very fast, self-focusing is a peak power limitation.
Multimode [3] and multicore [4–8] fibers present an opportunity to scale the self-focusing limit. At first glance the self-focusing limit seems to scale trivially with just the number of cores. However, this is only the case if the cores are uncoupled. Uncoupled cores behave exactly like multiple fiber amplifiers. For shorter core-to-core distances and low numerical aperture (NA), the cores in multicore fibers are at least slightly coupled. This gives rise to supermodes, which might be prone to the same self-focusing limits as free space intensity distributions.
In this manuscript we will discuss self-focusing of a 6 and a 7 core multicore fiber. We will analyze the self-focusing properties of the in-phase and the out-of-phase mode. The out-of-phase mode allows scaling of the self-focusing limit similar to the scaling possible by the use of vortex beams and higher order TEM modes in bulk media [9]. Furthermore, we show that the scaling of the in-phase mode is limited due to an instability, which occurs lower power levels than critical self-focusing. We finally discuss the impact of the coupling strength on the instability of the in-phase mode.
We are only concerned with self-focusing here. Therefore, we do not take damage thresholds, thermal effects and other nonlinearities into account. The fiber design we are discussing here reflects this. Much larger individual core sizes are necessary in a real situation (which can be provided by modern fiber designs such as large pitch [10], leakage channel [11], chirally coupled core [12] or photonic bandgap [13] fibers), otherwise other limits are reached before self-focusing. Furthermore we do not include effects like self steepening or the different wavelengths of mode locked ultrashort pulses. These effects are certainly also important especially since realistic systems are pulsed and we will look at them in the future.
Self-focusing in a fiber is complicated to analyze due to the waveguide properties of the fiber. The nonlinear waveguide equation can be solved approximately as shown by Dong [14]. One of the main issues arising from the power dependence of the waveguide is that the eigenmodes depend on the power level and the resulting mode field itself. This gives raise to mode size oscillations when propagating a mode which is an eigenmode of the waveguide at low power at a different power level than it was calculated for [15, 16]. Numerically it can be avoided by amplifying the beam during propagation [16] or by considering the nonlinearity and power during the calculation of the eigenmode. In our first simulations we integrated the method by Farrow et al. [16]. However, we later concluded that in practice the exact eigenmode will not be excited in an experimental situation (as this is not even the case at low power levels). Besides that we did not observe any major differences between the adiabatic amplification and the direct launching for the conclusions we are making in this paper.
2. Simulation
We analyzed different modes and by calculating the mode profiles in COMSOL first (6 μm core diameter, 15μm core-to-core distance, NA=0.076, fibers with 6 and 7 cores). Particularly interesting are the in-phase and out-of-phase mode as shown in Fig. 1 and Fig. 2. We then used a split step Fourier propagation algorithm running on a graphics processing unit (GPU) (Nvidia GTX780, which limited us to 32 bit floating point accuracy) to simulate the self-focusing. This approach is similar to what Pelegrina et. al [17] used to simulate asymmetric couplers. The speed of the GPU allowed for a 2048x2048 mesh and a 1 μm step size. The simulated cladding was 125 μm and beyond that region, a supergauss absorbing boundary was used. The power was normalized at each step to ensure that the power does not decrease during propagation. All these operations togeter take about 5.5ms per step. We assumed a signal at 1064 nm and fused silica fiber with a nonlinear refractive index of n2 = 2.2 · 10−20m2/W. As common for fibers, we used the effective mode area
to characterize the beam size. We chose a propagation distance of 8 cm. The propagation distance should be longer than the self-focusing length of course, but still as short as possible to reduce computing time. We found 8 cm is a good compromise. Additionally it could actually be used in fibers for nonlinear compression. As the mode area of non adiabatically amplified fiber does not reach the exact eigenmode, which leads to small periodic oscillations, we plotted the minimum mode area during the propagation as a function of power whenever we want to get an overview. This is experimentally motivated as fiber damage is likely to occur at this point.
Fig. 1 Seven core fiber. From left to right: refractive index structure, in-phase mode field and out-of-phase mode field
Fig. 2 Six core fiber. From left to right: refractive index structure, in-phase mode field and out-of-phase mode field
3. Design and mode dependence
Even when just looking at the in-phase mode of the 7 core design (Fig. 1) it is quite intuitive that this mode will not significantly increase the self-focusing limit, as center core confines a much larger power than the other cores. On the other hand there is no power in the center for the out-of-phase mode. The power is only spread into six cores, so the maximum increase of the threshold if all the cores collapse individually would be a factor of 6.
Therefore, the seven core design would in the best case have no advantage over a 6 core design lacking the center core (Fig. 2). The out-of-phase mode of the 6 core fiber looks very similar, but the in-phase mode is different. Its power is now evenly spread among all the six cores. Due to the waveguide of the fiber, neither mode of the 6 core fiber is likely to collapse to one central point. Therefore, there is at least no obvious reason why the self-focusing cannot be mitigated to approximately 6 times the single core limit.
We simulated the propagation of the four modes shown in Fig. 1 and 2 and a single core fiber with the same core diameter (6 μm) and NA for comparison. Figure 3 shows the minimum mode area anywhere in the 8 cm of propagation. The single core fiber behaves as expected and the beam shrinks and collapses at about 5 MW. The points after that (especially the increase of mode area) are not physical and caused by the finite grid/step size in the numeric simulation. The behavior of the out-of-phase mode is the same for the seven core and the six core fiber. When using the out-of-phase mode the self-focusing limit can be scaled effectively. The slope of the mode shrinking is significantly reduced and power scaling by a factor of 6 should be possible.
Fig. 3 Minimum mode area anywhere along 8 cm of propagation for the different modes and fiber designs.
On the other hand the in-phase modes seem to be unsuitable to scale the self-focusing limit, regardless of the fiber design. For the seven core fiber this is obvious as discussed before, but for the six core fiber it seems odd at first glance. What is even more surprising is that the mode area shrinks even more rapidly than for the single mode fiber. To understand what is going on here, we analyzed the beam profile in the fiber at a power of 5 MW. During the propagation we recorded the beam profile, mode area, centroid position and four sigma beam size (in x direction: ) (Fig. 4 and for comparison the out-of-phase mode in Fig. 5, the video shows the whole evolution). The effect is not the expected behavior for the term self-focusing. Instead the refractive index change induces an instability analogous to modulation instability, but in the spatial domain. Independently of this work, the effect was also discussed by Lushnikov et al. [18] they purpose to use it for coherent beam combining.
Fig. 4 Evolution of the mode area at 5 MW, center position and 4 sigma size for the in-phase mode [ Media 1].
Fig. 5 Evolution of the mode area at 5 MW, center position and 4 sigma size for the out-of-phase mode [ Media 2].
This result is surprising since we did not seed any noise and for a perfectly symmetric mode there should not be any asymmetry which can then in turn be amplified. Therefore the initialization is caused by numerical noise and/or a small excitation of the other modes. While of course there is no numerical noise in experiments noise is unavoidable in experiments. It would of course be ideal to exactly model the noise initialization as well. However, experience in similar fields like for example thermal mode instabilities [19] shows that even if noise origin is very well defined in theory, it is usually still very challenging to compare the simulations with experiments. Therefore we do not think an anchored quantitative model is currently feasible, and focus on qualitative results relying on numerical noise.
However, it is important to analyze if power in other modes is tolerable. Since the modes we launched into the BPM are calculated for low power, a small power fraction was already excited in the other modes. To increase the power level of the in-phase mode while propagating mostly the out-of-phase mode, we launched a combination of 95% out-of-phase mode and 5% in-phase mode into the fiber (Fig. 6). At the start the phase difference is 0° but due to the different propagation constants all possible phase differences eventually occur in the fiber. The mode size oscillates periodically. This oscillation can be attributed to linear mode beating. The linear mode beating slightly reduces the self focusing limit because it leads to an uneven power spread among the cores, but there is no instability induced on the out-of-phase mode by presence of the in-phase mode.
Fig. 6 Evolution of the mode area at 5 MW, center position and 4 sigma size for out-of-phase mode (95% of power) and in-phase mode (5% of power). The beam stays stable only regular beating is visible [ Media 3].
4. Coupling strength
An important question for the in-phase mode is how the instability scales with the coupling strength of the fiber. For uncoupled cores the in-phase mode (theoretically this can be defined for any multicore fibers, although it becomes meaningless for low coupling and very large beating length) will stay stable, but at the same time the cores do not necessarily stay phase locked. We investigated the impact of the coupling strength on the in-phase mode by changing the core-to-core distance and propagating the mode for 8 cm again. Figure 7 depicts the minimum mode area along the propagation for different core-to-core distances. For comparison the out-of-phase mode is also shown for 15 μm (the impact of the core-to-core distance is negligible in this case). The complete evolution is shown in Fig. 8.
Fig. 7 Minimum mode area anywhere during 8 cm propagation for different core-to-core distances. For comparison the out-of-phase mode is also plotted.
The mode size oscillations noticeable in Fig. 8 for the 15 μm core-to-core distance are due to the direct launching of the beam in the fiber (without any adiabatic amplification). The relevant part is the behavior of the mode size in the upper right corner (long propagation distance and high power). In this region the fluctuations occur. They seem completely chaotic for the 15 μm core-to-core distance and become more regular with lower coupling. The behavior is quite complex and a simplification would be desirable to understand what is going on.
To simplify the system, we simulated a two-core design (same NA and core diameter). This has the advantage that there is only one possible beat length/coupling strength to be considered because there are only even- and odd-mode. The even-mode corresponds to the in-phase mode in the 6 core fiber and the odd-mode to the out-of-phase mode. Figure 9 depicts the minimum mode area anywhere in 8 cm propagation. Again, there is a sudden mode area shrinking a low power levels, which decreases with increased core-to-core distance. For the 15 μm core-to-core distance, the mode is slightly larger in the beginning, this is caused by the larger overlap of the fields, which causes the intensity do be spread out into a larger area.
Fig. 9 Minimum mode area anywhere along 8 cm propagation of two core fibers for different core-to-core distances
In the field of fiber couplers one regularly has to deal with linear power transfer from one core to the other and back. To describe this process the beat length Lb is a useful parameter
Here βe and βo are the propagation constants of the even and odd modes, ne and no are the effective refractive indices and λ is the wavelength.
Larger Lb corresponds to reduced coupling between the cores. Of course our two core fiber/coupler is nonlinear so we cannot expect the equations for the linear coupler to give exact quantitative results. Furthermore, even the linear coupling is not completely described by Lb (as calculated for low power) because the high power dynamically changes the NA of the cores. However, it is reasonable to expect an influence of the coupling strength and beat length as it approaches the limit of uncoupled cores, where no power transfer between the cores is happening.
Given NA and core diameter the beat length between symmetric and antisymmetric mode can be calculated (Table 1). When comparing the number of periods in table 1 with Fig. 10, one notices that while the exact number of expected periods does not match the number of observed periods in Fig. 10, smaller beat length and larger fiber length leads to a larger number of periods not only for the linear coupler but also for our nonlinear case. Selecting a beat length much larger than the fiber length (here about 1 m) allows a stable power distribution during the propagation even for the even mode.
Fig. 10 Minimum mode size anywhere along 8 cm propagation of two core fibers for different core-to-core distance
Table 1. Effective refractive index difference and beat length between symmetric and antysimmetric mode and expected number of periods in the two core fiber.
5. Conclusion
We have shown an analysis of self-focusing in multi-core fibers. In the 6 core fiber, due to the absence of the waveguide in the center, the collapse of the beam to the center of the fiber is prevented. However, the in-phase mode becomes unstable. This issue can be overcome by selection of the out-of-phase mode. Using the out-of-phase mode it should be possible to significantly increase the self-focusing limit compared with single mode fibers.
Furthermore, we analyzed the origin of the in-phase mode’s instability. There are generally two options to overcome this instability effect: using the out-of-phase mode or reducing the coupling between the modes, so the impact becomes negligible. But in this case there is also no robust phase locking between the cores. On the other hand it could be interesting to investigate if the effect can be controlled well enough to exploit it.
Acknowledgments
Henrik Tünnermann is an International Research Fellow of the Japan Society for the Promotion of Science (JSPS). This research was supported by JSPS KAKENHI Grant Number 25247067 and by the Photon Frontier Network Program of the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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