1. Introduction

Fiber amplifiers have demonstrated an impressive performance scaling potential [1]. One reason is the elongated structure, which enables an efficient heat dissipation. Therefore, fiber laser systems are able to emit high average output powers while maintaining excellent beam quality. This allows, for example, for the development of CW-systems with multi-kWs [2], but also ultrashort pulse amplifiers with kW-class average output power have been demonstrated [3]. However, whenever high peak powers are extracted, hampering nonlinear effects occur that might significantly limit the performance of fiber laser systems [4]. These can be mitigated by using CPA-systems together with large mode field diameters, which helps keeping the peak power low and the impact of nonlinear effects small [5,6]. However, as the average power of fiber laser systems has increased, serious thermal effects, i.e. transverse mode instability, have become the most limiting factor for a further performance scaling of this technology [7].

To overcome these limitations in the scaling of the peak and the average power, coherent beam combination (CBC) has been utilized. With CBC, N independent amplifiers running below their performance limit and emitting separate beams are combined to one final beam. This can be achieved in a filled or tiled aperture scheme, respectively [8]. The scaling potential of this concept has already been demonstrated several times, which includes even kW-class, ultrafast systems [9,10]. These experiments have shown no fundamental detrimental problems when increasing the number of channels in the system. However, there are economic and practical limitations when it comes to increasing the channel count. In particular, the combination of N independent amplifiers demands for N times the space, the components and the costs of a single channel. Consequently, such an approach is not appealing for scaling the channel count by significantly more than an order of magnitude.

Thus, a new concept was introduced to mitigate the problem of space and component multiplicity. The so-called multicore fibers (MCFs) incorporate all active channels in one single fiber sharing the same pump cladding. Additionally, compact splitting and combining elements allow for the coherent combination of all emitted beams [11,12]. Currently these systems are able to reach kW-average powers and allow for the combination of fs-pulses in filled- and tiled-aperture schemes [13,14]. Moreover, phase stabilization systems ensure a stable phase superposition of all emitted modes to minimize the combining losses. However, apart from to the piston phase of the emitted modes, other irregularities, such as differences in spot size, spot displacement, power fraction, etc. will reduce the combined output power [15].

Theoretical considerations have revealed the promising scaling potential of MCFs, but also the challenges when increasing the output power, especially in short rod-type fibers that experience high thermal loads per unit length [16]. In particular, it has been shown that thermal effects arise when the power per core being extracted increases, leading to non-uniform mode shrinking. As an example, Fig. 1 (left) shows the evolution of the effective mode area in the cores of an 80 µm 3 × 3 MCF that is pumped in counter-propagation at 1 kW pump power and 976 nm. The cores are grouped in colors with respect to their position as schematically shown in the inset. It can be clearly seen that the mode area of each core shrinks in the propagation direction. The reason for that is that more power is being extracted towards the fiber end which, in turn, leads to a higher thermal load in this region of the fiber. Moreover, a thermal gradient in the radial direction leads to non-uniform shrinking of the modes. The right-hand side of Fig. 1 shows the intensity profile at the fiber end. As can be seen, it strongly differs from core to core. This is detrimental since combining the beam (modes) emitted by the different cores at a position of high thermal load results in a drastically reduced spatial combining efficiency. This is because the spatial field distribution in each core becomes significantly different from one another, due to the impact of the thermal gradient across the MCF cross-section. Additionally, not only does non-uniform mode shrinking lead to a degradation in the combining efficiency of the whole system but it also influences the amplification efficiency and the nonlinearities. A detailed description on these effects is given in [16], which is also applicable to the results being presented in this work.

 

Fig. 1. Effective mode area (left) in a 3 × 3 1 m long Ytterbium-doped MCF, counter-propagating pumped at 1 kW pump power. The cores are grouped in colors with respect to their position (see inset). The intensity profile at the fiber output (position = 1 m) is shown on the right (with the core positions represented as white circles).

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Generally, thermal effects could be reduced by increasing the fiber length while maintaining the same output performance. However, especially high energy ultrafast systems would suffer from high fiber lengths due to rising nonlinear effects. In this contribution we will pay particular attention to short fiber geometries that are prone to high thermal effects and show two ways to counteract the aforementioned problems. First we will examine the behavior of co-propagating pumped coherently-combined MCFs. We will show that the thermal load is distributed more evenly in such fibers, leading to higher achievable output average powers. However, the higher impact of nonlinear effects in this configuration hinder the pulse energy extraction. Therefore, we will present a modification in counter-propagating pumped MCFs that aims at a reduced pump absorption at the fiber end. This leads to lower thermal load, allowing for higher combining efficiencies and, consequently, for higher output powers and pulse energies. Finally, we will compare the results with the predictions of conventional MCFs pumped in counter-propagation.

2. Simulation tool and parameters

The simulation tool described in [16] has been used to calculate the theoretical results that will be presented in this work. The tool calculates the evolution of the LP₀₁ modes propagating along each individual core and the corresponding signal power evolution. The quantum defect heating (assuming a purely radial heat conduction) leads to a stationary temperature profile. The refractive index change that is caused by the thermo-optic effect, influences, in turn, the propagating modes and their power evolution. These thermally induced modal changes affect the thermal load deposition in the fiber. The computing steps are repeated iteratively until convergence is achieved, which in our case means a maximum deviation of 1 % of the temperature, signal and power evolution between two subsequent iteration steps. Once this condition is fulfilled, the steady-state of the laser system can be described.

For reasonable comparisons with recent results, shown in [16], we use the same parameters in our simulations. We consider 1 m long, rod-type MCFs with 2 × 2 to up to 10 × 10 cores. The core dimensions are 30, 50 and 80 µm, with an Yb-doping concentration of $5 \cdot {10^{25}}\; ions/{m^3}$. The cores have a distance to each other of 2.5 times their diameter and possess a flat refractive index profile in cold operation, leading to a V-parameter of 3. Different fiber configurations are shown in Fig. 2.

 

Fig. 2. Cross-section of MCFs with different configurations (different core sizes and core numbers). The pump cladding size (turquoise) is chosen to achieve the same small-signal absorption (20 dB/m) in each case. For configurations with cladding sizes smaller than 1.5 mm an additional over-clad (grey) of 1.5 mm is added.

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An air-clad along the fiber allows for the confinement of the pump light with a small-signal pump absorption of $20\; dB/m$ in all cases, e.g. the cladding size for a 2 × 2 MCF with 30 µm cores is 315 µm. The cladding size scales linearly with the core diameter and the square root of the total number of cores. For small pump cladding sizes (i.e. those smaller than 1.5 mm), an additional glass coating is added to achieve a minimum diameter of 1.5 mm for each MCF. This allows for rigid rod-type fibers that prevent unwanted bending. Due to the rod-type design, an additional polymer coating isn’t necessary, allowing for higher temperatures since the glass is less susceptible to thermal damage. Additionally, it leads to a constant heat conductivity over the whole fiber cross section, which is considered to be $1.38\; W/({m \cdot K} )$ in our simulations. A limitation/interruption of the heat flow due to the air-clad can be neglected, as it was shown in [17]. The fibers are surrounded by water, which acts as the coolant medium at 25°C temperature with a thermal conductivity of $0.58\; W/({m \cdot K} )$. The pump wavelength is set to 976 nm, whereas the signal wavelength is 1030 nm. All core sizes are seeded with the same intensity regardless of their size, i.e. $1\; W$ per core in the case of $30\; \mathrm{\mu} m$ cores.

In this work we will examine the power and energy performance of filled aperture, coherently-combined MCFs using two different approaches, i.e. different pumping schemes. We will compare the new results with the ones that have already been demonstrated for MCFs pumped in counter-propagating direction. To calculate the power scalability of each fiber design, the pump power is swept, i.e. many independent simulation steps have to be performed. For a realistic estimation, as shown in [16], several assumptions have to be made to evaluate the power and energy performance of these systems. Since TMI is expected to occur at high average powers, a maximum output power per core of 300 W is set [18,19]. Moreover, since the optical properties of Ytterbium will drastically change at higher temperature [20], a maximum value of 500 °C will be considered as safe working space. Finally, due to practical reasons, the minimum spatial combining efficiency is set to a self-imposed limit of 70 %. The spatial combining efficiency is the power of all superimposed core modes at the fiber end compared to the total power of all single core modes. The superposition can be understood as a spatial shift of all core modes until their core positions overlap perfectly. Any piston phase shift between the cores is neglected since this can be compensated by phase stabilizing components, such as those presented in [21]. We have to point out that we assume the same polarization state of each core mode since this is crucial for the combining step. In fact, such as situation has to be enforced (if necessary with external optical elements) prior to the beam re-combination. In any case, it is worth mentioning that polarization effects, such as nonlinear polarization rotation, have been observed in previous experiments with such MCF structures [22]. Nevertheless, it was also shown that the different polarization states in the cores at the fiber end could be fully compensated with appropriate polarization control elements.

In our simulations we directly deduce the pulse energy from the average output power at a given repetition rate. This is possible since the considered repetition rates are much higher than the inverse lifetime of Ytterbium [23]. However, especially for ultrafast systems, we have to consider not only the pulse energy, but also the nonlinear accumulated phase (B-integral). As described in [16], we will assume pulses of 10 ns with a maximum B-integral of 10 rad as the upper nonlinearity limit. Simulation results that exceed any one of these limitations will not be taken into account, i.e. a maximum pump power is derived at which the system still runs below all the limits mentioned above. In the next sections we will show two different approaches, starting with MCFs that are pumped in the co-propagating direction.

3. MCFs pumped in a co-propagating direction

In our first consideration, the fibers are pumped in co-propagation, as it is shown in Fig. 3. From conventional fiber amplifiers it is well known that co-propagating pumping leads to less thermal load at the fiber end. Concerning MCFs, this behavior can also be expected, as it is represented by the colored cores in Fig. 3. The main heat load can be expected to be close to the fiber beginning (indicated by red color), whereas it will drop towards the fiber end (indicated by blue color). This will have a significant influence onto the modal propagation in the fiber.

 

Fig. 3. MCF scheme that is pumped and seeded in co-propagation. The thermal load in each core is represented by different colors (high thermal load = red, low thermal load = blue).

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Figure 4 shows the evolution of the effective mode area along a 3 × 3 MCF with 80 µm cores that is pumped in co-propagation at 1 kW pump power and 976 nm. The cores have been grouped in colors with respect to their distance to the fiber center, following the color code schematically depicted in the inset. Figure 4 can be directly compared to Fig. 1 (conventional pumping in counter-propagation). At first glance it can be seen that the strength of mode-shrinking varies from core to core and along their longitudinal position. Similar to MCFs pumped in counter-propagation, the inner cores show less shrinking than the outer ones. The reason for that is the thermal gradient in the radial direction (and with it, the refractive index modification) that changes with distance from the fiber center. However, the point of smallest mode field area for the different core modes along the longitudinal direction lies within the fiber (at ∼0.2m in this case). This means that the point of highest thermal load is not located at the fiber end anymore, which is the case in counter-propagating pumped fibers. This is because in co-propagating pumped systems much more amplification takes place at the beginning of the fiber. Due to the reduced pump power towards the fiber end the energy extraction flattens there, leading to lower thermal loads. That is why modal re-expansion intrinsically takes place towards the fiber end in co-propagating pumped MCFs, which will lead to higher combinable output powers than those reached in counter-propagating pumped MCFs, as will be shown in the following.

 

Fig. 4. Evolution of the average effective mode area along a 3 × 3 MCF with 80 µm cores that is pumped in co-propagation at 1 kW pump power. The cores are grouped in colors, as indicated in the inset. Due to high pump absorption on the pump side (fiber position = 0 m), the maximum thermal load occurs within the fiber (∼0.2 m) and not at the fiber end. As can be seen, modal re-expansion takes place towards the fiber end which will lead to high combining efficiencies.

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Figure 5 shows the maximum combinable output power of the MCFs pumped in co-propagation (solid lines) compared to the otherwise identical counter-propagating pumped MCFs (dashed lines) for different core sizes. As it was mentioned in section 2, any simulation result that exceeds at least one of the self-imposed limitations (TMI, temperature, combining efficiency or B-integral in the case of energy calculations) is not taken into account. Consequently, the results shown here correspond to the maximum pump power that is feasible while staying below the aforementioned limitations. As expected, pumping MCFs in co-propagation leads to higher combining efficiencies in all cases (since mode-reshaping towards the fiber end occurs, as it can be inferred from Fig. 4), which ultimately results in a higher combined output power. Even large core MCFs significantly benefit from this effect, leading to a maximum average power of 20 kW in the case of 80 µm cores (for the 10 × 10 configuration). This represents nearly a factor 4 improvement in the average power with respect to the results predicted for the counter-propagating pump case. Additionally, since the thermal load is distributed more evenly along the fiber, the absolute temperature in the fibers is lower than in the counter-propagating pumped cases (i.e. TMI is the limitation in all cases). Just in the case of the 10 × 10 MCF with 30 µm cores the maximum temperature of 500 °C is reached, hampering further power scaling (indicated by the small kink at the end of the red solid line).

 

Fig. 5. Maximum combinable output power for different fiber configurations and core diameters. The solid lines show the results for the co-propagating pumped MCFs. The dashed lines show the output powers that are achievable with the identical counter-pump MCFs, as it was presented in [16]. The red area represents the limitation given by the maximum TMI power of more than 300 W per core together with 100 % combining efficiency.

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However, as it is known, the co-propagating pump configuration usually leads to a higher impact of nonlinear effects [24]. This will ultimately limit the energy extraction. Figure 6 shows the maximum combinable pulse energy for the co-propagating pumped (solid line) and counter-propagating pumped (dashed line) MCFs with different core sizes (under the assumptions described before). Here, the repetition rate was lowered until the maximum pulse energy was reached, meaning that the pulse energy is limited by a maximum B-integral of 10 rad.

 

Fig. 6. Maximum combinable pulse energy for different fiber designs and core diameters at the optimum repetition rate. A B-integral of 10 rad is the limit for each case. The solid lines show the results for the co-propagating pumped MCFs. The dashed lines show the output energies that are achievable with conventional MCFs, presented in [16].

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The co-propagating pump configuration is limited to lower pulse energies. This can be explained by the longer effective nonlinear length compared to counter-propagating pumped MCFs. Even though the effective mode area remains larger along the longitudinal direction, the B-integral limitation of 10 rad hampers further energy scaling in the case of co-propagating pumped MCFs in ultrafast systems. In fact, the maximum pulse energy that can be achieved, is just between 50-80 % compared to counter-propagating pumped MCFs. In contrast, when working at higher repetition rates (i.e. quasi CW), the effect of higher a combining efficiency dominates and higher average powers can be achieved, as it was shown in Fig. 5. This results in higher pulse energies at high repetition rates for the co-propagating pumped case, even though the absolute maximum is still not reached.

In order to achieve higher pulse energies together with high average powers, we will introduce a revised fiber configuration in the next section and examine its power and energy scaling potential.

4. Counter-propagating pumped MCFs with a thermally-engineered end section

In this modification the MCFs are pumped in counter-propagation as shown in Fig. 7. In order to mitigate the thermally-induced performance limitations in MCFs, we propose a change of the fiber output end, as shown in the right part. In this section the pump confinement is removed (which can be achieved in the case of an air-clad by collapsing it). By doing so, the natural caustic of the pump beam results in a lower pump intensity and, with it, in a lower pump absorption at the fiber end (due to the reduced overlap of the pump radiation with the MCF cores). As schematically illustrated with the core colors (red and blue), this strategy leads to lower thermal loads at the fiber end and, therefore, to a mitigation of the thermal effects that hamper the coherent combining process, as it was discussed in the introduction.

 

Fig. 7. New fiber design for MCFs pumped in counter-propagation. The removed air-clad (turquoise) at the fiber end leads to a reduced thermal load in the active cores (cool and hot areas represented by blue and red color).

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In our simulations the length of the missing air-cladding is chosen so that the pump spot area at the fiber end facet increases to a factor F of 10 times the original pump cladding area. It is important to note that this condition implies that the length of the MCF fiber section without air-clad will change depending on the number of cores and their size. In terms of the pump caustic (required to calculate the length of the air-clad-free region of the fiber) we assume a free-space pump NA of 0.1. Hence, we can describe our pump beam radius $\omega (z )$ outside of the cladding (but still inside the fiber) as

With ${\omega _0}$ as the regular pump cladding radius, ${n_{fiber}}$ as the refractive index of fused silica and z as the longitudinal coordinate (with $z = 0$ as the starting point of the missing cladding). The length L of the missing cladding section can be calculated as follows:

We have to point out that we keep the free-space NA of the pump beam at 0.1 even for large cladding sizes. That means that the beam parameter product of the pump needs to grow proportionally with the cladding size. Consequently, the pump brightness demands will be significantly lower than it can be expected from commercially available pump sources. Even though this seems in contradiction with the pursuit for a high pump brightness, this could be realized e.g. with large highly-multimode fibers [25], but also large area mode-mixing light pipes or other large beam homogenizers at a given NA should be applicable.

Additionally, even though the core size is kept constant along the MCF, the behavior of the propagating modes is comparable to the one being observed in tapered fibers (i.e. the mode field area changes with the propagation length). Consequently, similar considerations in terms of adiabaticity have to be taken into account, e.g. avoiding significant losses due to mode coupling. In typical tapered fibers an adiabatic taper criterion has been introduced that depends on the change of the core diameter [26,27]. Following the criteria used for conventional tapers, in our case the fiber section with the missing cladding has to fulfill the following modified adiabatic criterion:

Here, $MF{D_{min}}$ is the minimum LP₀₁ mode field diameter along the fiber and ${L_{beat,max}}$ represents the maximum beat length between the LP₀₁ and the LP₁₁ mode (which is usually the longest modal beat length with the LP₀₁ in a fiber). This simplified criterion corresponds to a worst case scenario. The reason is that both the MFD and the beat length change along the modified fiber section. However, the largest beat length usually corresponds to the largest MFD. In other words, the minimum MFD and the largest beat length do not normally happen at the same point in the fiber. Therefore Eq. (3) can be considered as a very stringent condition for adiabaticity which is always fulfilled in our simulations (when choosing a free-space NA of 0.1). Therefore, we expect no further losses or modal distortions at the fiber end. In the following section we will present the simulation results that can be achieved with this new approach.

As it was shown in Fig. 7, the counter-propagating pumped fiber end is modified to decrease the pump absorption and with it the thermal load. Figure 8(a)) shows the typical evolution of the effective area of the LP₀₁-mode in each core of a 3 × 3 MCF. For a better representation all cores have been grouped in colors with respect to their distance to the fiber center, as shown in the inset. The fiber is seeded from the left (z = 0 m) and pumped in the counter-propagating direction from the right side (z = 1.0 m).

 

Fig. 8. a) Evolution of the average effective mode area along a 3 × 3 MCF with 80 µm cores that is pumped in the counter-propagating direction at 1 kW pump power. The air-cladding has been removed in the last part of the fiber. All cores with the same core-to-center distance have been grouped in colors, as indicated in the inset in a). b) shows the smooth change of the effective area in the last fiber section highlighted in a). The corresponding mode profiles, as depicted in b), are shown in c) and d). A re-expansion of the modes at the fiber end facet can be clearly seen, which improves the spatial combining efficiency.

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With the modification proposed herein, the energy extraction is significantly reduced in the final section of the MCF (where the air-clad has been removed) due to the lower pump absorption. As a consequence, there is a progressive reshaping of the core modes towards their original (cold) shape, as depicted in Fig. 8(b)). Similar to the MCF pumped in the co-propagating direction (Fig. 4) a final mode field area of 4500 µm² for all core modes is achieved. The modal field distributions at the point of highest thermal load as well as at the fiber end are shown in Fig. 8(c)) and d). The lower thermal load at the fiber end enhances the similarity of the core modes which increases the spatial combining efficiency.

As it has been shown in Fig. 8, a significant mode reshaping/re-expansion occurs along the cores in the modified section of the fiber. This leads to an improvement of the combining efficiency and, with it, to an increase of the output power, as it can be seen in Fig. 9. Here, the maximum output power expected from different fiber configurations (taking into account the limits discussed in section 2) is shown for the counter-propagating pump scheme using the regular MCF (dashed line) and for the modified MCF (solid line). The red area represents the limitation given by TMI and a combining efficiency of 100%.

 

Fig. 9. Maximum combinable output power for different fiber configurations and core diameters. The solid lines show the results for the modified MCFs. The dashed lines show the output powers that are achievable with a conventional MCFs pumped in the counter-propagating direction, as it was presented in [16]. The red area represents the limitation given by the maximum TMI power of more than 300 W per core together with 100 % combining efficiency.

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It can be seen that the fiber modification improves the power performance for all considered fiber designs. However, unsurprisingly, the highest improvement is achieved for fibers comprising a high number of large cores. For example, in the case of a 10 × 10 MCF with 80 µm cores, 20 kW instead of 5 kW become achievable. Small core MCFs don’t show such a significant improvement, since their performance is more restricted by the absolute temperature limit than by a bad combining efficiency. Additionally, small cores are more resilient to thermally-induced modifications of the modes due to a stronger guidance.

The maximum achievable pulse energy with these fibers is shown in Fig. 10. Here, once more, the dashed and solid lines represent the regular and the modified MCF designs, respectively. The pulse energy calculations have been performed at different repetition rates, whereby the highest repetition rate that provides the highest achievable pulse energy (before hitting the limitations) has been chosen. As in the case of the regular fibers, large core MCFs perform better at lower repetition rates than small core MCFs. Similar to the maximum combinable output power, a much higher pulse energy can be achieved with the modified MCFs. Especially with large cores, an improvement of up to 50 % is possible. For instance, in the case of a 10 × 10 MCF with 80 µm cores, almost 700 mJ instead of 450 mJ are feasible. The reason for this is two-fold. On the one hand, a much higher combining efficiency can be achieved with the modified MCF, as it was explained in the former paragraph. On the other hand, with a larger integrated mode field area along the fiber (due to the reshaping effect in the last section), slightly lower B-integrals are achieved at the same output power. Consequently, the B-integral limitation of 10 rad is shifted towards a higher pulse energy.

 

Fig. 10. Maximum combinable pulse energy for different fiber designs and core diameters at the optimum repetition rate. The solid lines show the results for the modified MCFs. The dashed lines show the output energies that are achievable with a conventional MCFs pumped in the counter-propagating direction, as presented in [16].

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Using the modified fiber design in the case of small core MCFs shows less of an improvement. The reason for that is the B-integral limitation. This limitation is reached faster than in large cores since the integrated effective mode area is much smaller. Thus, the maximum average power will not be achieved with the highest pulse energy and, consequently, the combining efficiency plays a secondary role.

5. Summary

We have proposed two ways to counteract the hampering impact of thermal effects in coherently-combined MCFs. First, a co-propagating pump configuration helps to reduce the thermal load at the fiber end, which leads to an improvement of the spatial combining efficiency. We have shown that the combinable output average power can be significantly increased, especially for large core MCFs. However, since the impact of nonlinear effects is higher in a co-propagating pump configuration, lower maximum pulse energies than with the counter-propagating pump configuration are predicted.

To overcome this limitation, a revised fiber design was introduced. Hereby we reduce the pump absorption at the fiber end of a MCF pumped in the counter-propagating direction. This way the thermal load can also be decreased. When using MCFs with air-clads this can be achieved by collapsing the cladding with a hydrogen flame in the end section of the fiber. The reduced pump absorption leads to a modal re-expansion at the end facet and, with it, to higher combining efficiencies. We have shown that up to 50 % higher energy (than with the conventional MCF pumped in the counter-propagating direction) can be extracted and combined in large core MCFs, e.g. 700 mJ in a 10 × 10 MCF with 80 µm cores are feasible. Even an improvement in the combinable average power was observed compared to conventionally MCFs that are pumped in the counter-propagating direction.

We have to point out that we have just considered single-mode, step index fibers in our simulations but manufacturing such fibers is quite challenging when scaling the core diameter. Nevertheless, the study presented herein should be valid for most MCFs that operate in an effectively single-mode regime with such large mode field diameters, e.g. with embedded Large Pitch structures. In addition to this, we assumed a polarization-independent behavior of the core modes which is crucial for an efficient combining process. This might not be the case in real non-polarization maintaining MCFs, as it has been shown in former experimental studies [22], which then require polarization- compensation elements (e.g. waveplate arrays). Such problems can be circumvented with polarization maintaining MCFs, which will become one aspect of prospective studies.

In summary, the presented revised design is a promising approach that should help in achieving multi-kW, Joule-class level, coherently-combined MCFs systems.