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Abstract
In this work we study in detail core-to-core coupling effects in multicore fibers (MCFs) using a simulation tool based on supermodal interference. We pay particular attention to the impact of core area scaling, which plays an important role in prospective amplifier systems. We consider geometrical and optical properties of the MCF structure, including the ability for dense packaging of the cores but also the influence on the core guidance (V-parameter). In general, this study is important to unlock the power and energy scaling potential of the next-generation MCF amplifiers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Over the last years multicore fiber (MCF) systems have attracted an increasing interest in the laser community. Recent experiments have demonstrated the potential of MCF amplifiers with regards to high power and/or high energy extraction [1–3]. At the latest, with the implementation of coherent beam combination (CBC) schemes and the demonstration of ultrafast operation with output powers in the kW-level and excellent beam quality [4], the use of MCFs has become an attractive solution in the field of laser development. Moreover, compared to single-core, coherently combined ultrafast fiber systems [5,6], CBC MCF systems are much more compact and cost efficient.
In spite of these attractive advantages, MCFs face a challenge not present in single-core fibers: the fact that significant core-to-core coupling (optical crosstalk) might occur [7]. This effect occurs due to the overlap of the evanescent fields from neighboring cores. The most recognizable consequence of optical crosstalk is that it leads to an inhomogeneous power distribution among the cores at the fiber output. However, this effect also can result in spatial, temporal and spectral interference effects. Even though there are some applications that exploit such coupling effects, e.g. temperature and strain sensors [8], the generation of orbital angular momentum beams [9] or in tiled-aperture beam combining schemes [10], the performance of most systems (e.g. telecommunication [11], imaging [12] and fiber amplifiers) is negatively affected by core-to-core coupling.
An example of the negative impact of core-to-core coupling in fiber amplifiers is sketched in Fig. 1. This example illustrates the influence of coupling in a typical pulsed filled-aperture CBC system, similar to the one demonstrated in [4]. Herein, an initial seed pulse is split up in equal replicas, according to the number of cores in the MCF. This is achieved with a segmented mirror splitter with adapted partial reflection zones (Fig. 1 left) [13]. In this element the pulses are time-shifted with respect to each other, due to the different optical path lengths in the splitter. After propagation (and amplification) through the different cores, the pulses are re-combined in an additional segmented mirror splitter operated in reverse, which compensates for the time delays between the pulses (Fig. 1 right). In an ideal setup, i.e. when all the cores operate independently from each other, all pulses will combine perfectly, leading to a single temporal feature with high peak intensity. However, in an MCF with core-to-core coupling, the optical crosstalk will result in side pulses, as shown in Fig. 1. Combining these pulses will lead to a drastic performance degradation or, in other words, to a reduction in the combining efficiency, peak power and pulse contrast [14,15].
Fig. 1. Impact of coupling effects in a typical CBC MCF system. The seed pulses for each core originate from the same laser source, which is split up in several replicas with the help of segmented mirrors (left). The combination after the fiber takes place in an inverse segmented mirror configuration (right). Core-to-core coupling effects during propagation in the MCF will lead to side pulses (green) that can be just partially combined. This results in a degradation of the combined power and pulse contrast (right).
This work is structured as follows. We will show experimental observations on the core-to-core crosstalk in different rod-type MCFs. In order to describe these effects, a simulation tool based on supermodal interference is presented. In the last part, the requirements for weakly-coupled MCFs being integrated in CBC systems are evaluated. Particular attention will be paid to the core area scaling potential of MCFs, enabling further power and energy improvements of such systems.
2. Experimental observations
In order to show various degrees of core-to-core coupling, different rod-type MCFs have been drawn in-house. By varying the drawing speed, fibers with different outer diameters, and with that, different core sizes, have been realized. All fibers originate from the same preform, meaning that, apart from their size, the different fiber versions have the same relative geometry and material composition with a core NA of approximately 0.04.
The core-to-core coupling after a certain fiber length has been experimentally determined, as shown in Fig. 2. Here the intensity profile at the output of different 1.8 m long, rod-type MCFs with core sizes ranging from 15 µm to 30 µm is shown. In each experiment presented in Fig. 2, just one of the corner cores has been excited at the fiber input facet, highlighted in yellow. In spite of this, as can be seen, a significant amount of power is coupled to adjacent cores in the case of the smaller core sizes. Crucially, the coupling becomes weaker for larger cores. In this particular case, a core size larger than 25 µm would be necessary to avoid any measurable core-to-core coupling after 1.8 m. In fact, as it will be seen, the mitigation of core-to-core coupling imposes restrictions on the core size, core NA and core-to-core distance (pitch). However, these parameters also influence the performance of the fiber in laser operation. Therefore, it is important to adopt a holistic approach to optimize the design of MCFs that balances the needs of laser performance against the core-to-core coupling.
Fig. 2. Intensity profile at the fiber end facet of 1.8 m long MCFs with core sizes ranging from 15 µm to 30 µm (from left to right). In all cases only one core was excited at the fiber input (yellow circle). Strong power coupling is observed in the MCFs with the smaller cores, whereas the coupling strength decreases in the fibers with larger cores.
For example, when designing an ultrafast MCF amplifier, coupling effects have to mitigated while, at the same time, meeting other conditions. In particular, the rather short fiber lengths used in ultrafast fiber systems demand for a strong pump absorption. In the case of a MCF this is determined by the size of the pump cladding surrounding the cores. Thus, in order to maximize the pump absorption, a dense packaging of the cores (that allows for a smaller pump cladding area) is necessary. A dense core packaging, however, is more prone to core-to-core coupling. Balancing these contradicting requirements is not an easy task. In this work we will analyze the limits imposed on the core arrangement in order to maintain weakly coupling between them. This study is carried out using a simulation tool based on supermode interference [16]. In the following we will investigate several MCF configurations with different core NAs and core-to-core pitches. Additionally, we will investigate core-to-core coupling in MCFs as a function of the core size.
3. Simulation tool and parameters
In our simulations we consider multicore fibers with a square 3 × 3, step-index core arrangement. This is because the 3 × 3 array is the smallest core arrangement that contains all the core adjacencies. In fact, the 3 × 3 structure is the simplest one in which it is possible to analyze core-to-core coupling between the center core and its direct (orthogonal) and indirect (diagonal) neighbors. These coupling interactions are the most dominant ones even in MCFs containing more cores.
It should be noted that the square arrangement of the cores is compatible with typical splitting and combining elements used in our ultrafast CBC systems [17]. In spite of this, it is worth mentioning that the simulation tool is not restricted to this arrangement, and it could be easily adapted to study other geometries. Besides, we have to mention that any amplification process in the fiber is not taken into account in this study, meaning that also thermal effects are neglected. Thus, the results obtained from this work can be considered as a worst case scenario, since thermal effects tend to de-couple the cores, as it was also shown in [18].
The different array parameters considered in the simulations are shown in Fig. 3. In the simulations the core sizes are swept between 15 µm and 50 µm and the spacing between the cores (pitch) varies between 1.5 and 5 times the core diameter. The cores have a flat step-index profile with an index step Δncore. The cores are surrounded by a glass-cladding that is encircled by a low index material (air in this case), which is necessary to guide pump light. As mentioned above, the simulations performed for this publication do not contain any amplification process which, in turn, means that the pump cladding has no essential function. However, since this structure might influence the coupling between cores, it has to be included in the simulations. The cladding size scales linearly with the core size and the core-to-core spacing, e.g. the cladding size is ∼140 µm for 15 µm cores with a pitch-ratio of 1.5, ∼235 µm for 15 µm core with a pitch ratio of 2.5 and ∼280 µm for 30 µm cores with a pitch ratio of 1.5.
Fig. 3. 3 × 3 MCF with surrounding air-clad. In our simulations we vary the core size (center) as well as the core-to-core distance (right). The cladding size scales linearly with these changes.
In order to analyze the core-to-core coupling in the simulations, the center core is seeded with a Gaussian beam (perfectly matched to its fundamental mode), as depicted in Fig. 4 (top). This beam is then decomposed into a superposition of the supported eigensolutions of the waveguide (i.e. the supermodes, as shown in the lower part of Fig. 4) at a wavelength of 1030 nm. These supermodes [16] are derived by a finite-difference approximation solving the scalar Helmholtz equation [19]. In order to achieve accurate results with this modal solver, the transverse resolution needs to be chosen appropriately, which in our case will be 1/20th of the core diameter. As described in [16], the electric field E(x,y,z) distribution along the waveguide can be then written as:
Fig. 4. Decomposition of a Gaussian beam seeding the center core (top) into a set of supermodes (bottom), each with its own propagation constant. The characteristics of the supermodes depend on the parameters of the waveguide, i.e. core-to-core distance, core index and core size. In this example, three supermodes (highlighted with a red frame) are forming the seed beam.
The power evolution Pi(z) along each core can be determined by considering the electric field distribution E(x,y,z) from Eq. (1) in the area of the core under inspection. Hereby, in order to include evanescent field components, each core area Ai is defined as a circular area around the core center covering a radius of half the pitch Λ of the MCF, leading to a core-bound electric field Ei(x,y,z). Thus, the power Pi in each ith core can be calculated as follows:
In order to illustrate the capabilities of the simulation tool, we present an exemplary simulation result in Fig. 5. The bottom part of the figure shows the typical power evolution along different cores of a 3 × 3 MCF structure when only the central core is seeded. To achieve strong coupling over a fiber length of 1 m, we chose the following fiber parameters: the core diameter is 17.5 µm with a pitch/diameter-ratio of 2.5 and a step index of 6e-4. The cores have been classified and color-coded into the center core (red), the direct/orthogonal neighbors (blue) and the indirect/diagonal neighbors (green), as shown in the top right inset of Fig. 5.Fig. 5. (bottom) Power evolution in the different core groups along a 1 m long, 3 × 3 MCF with 17.5 µm cores and step index of 6e-4. The three different core groups (center core with its direct and indirect neighbors) are color coded, as shown in the upper part, together with the beam intensity profiles at three different fiber locations.
It can be seen that at the beginning of the fiber all the seed power is contained in the center core, i.e. the one that has been seeded. However, after some centimeters of propagation, the power starts to be transferred to the direct neighbors and, finally, to the indirect neighboring cores. In fact, as can be observed, at around 0.5 m all the power is contained in the corner cores, i.e. the indirect neighbors (upper part of Fig. 5). The power evolution in the cores, shown in Fig. 5, repeats itself after a certain propagation length, the so called supermodal beat length. This length depends on the superposition of the contributing supermodes, i.e. on the fiber design and the seed conditions.
4. Simulation results
With the help of the simulation tool presented in the previous section, the experimental results from Fig. 2 can be reproduced. Figure 6 shows the power evolution in the seed (corner) core along the length of a 3 × 3 MCF. The fiber parameters are chosen to match the results from Fig. 1, i.e. the core sizes are between 15 µm and 30 µm with a pitch/diameter ratio of 2.5. In all cases a step index of 5.2e-4 has been chosen, which is in good agreement with the experimental measurements. It can be clearly seen that strong core-to-core coupling occurs for smaller core sizes, as indicated by the strong power fluctuations in the seed core.
Fig. 6. Relative power content in the seed (corner) core along a 1.8 m long MCF with core sizes ranging from 15 µm to 30 µm. All the cores have a refractive index step of 5.2e-4. The smaller the cores, the faster the power is coupled to adjacent cores.
According to these simulation results, the most straightforward solution would be to use MCFs with larger cores. However, with increasing core size higher order transverse modes will be supported, which is an unwanted effect, especially in CBC systems. Thus, to avoid that it is necessary to consider the core size scaling potential of MCFs with constant V-parameter.
In this case, 3 × 3 MCFs with cores ranging from 15 µm to 50 µm have been simulated with a pitch (i.e. core-to-core distance) of 2.5 times the core diameter and a fixed V-parameter of 3. Seeding the center core with a gaussian beam perfectly matched to its LP01 mode leads to the energy being spread across 3 non-degenerated supermodes. This happens independently of the core size. The upper part of Fig. 7 shows the effective index difference between these supermodes as a function of the core size. It can be seen that the eigenvalues approach each other for larger cores. This implies that the coupling length will increase, i.e. the core-to-core coupling becomes weaker. The lower part of Fig. 7 shows the necessary length to achieve a coupling of 1% of the seed core power to neighboring cores, as a function of the core size. In other words, even with the same V-parameter the core-to-core coupling decreases as the core size increases. However, we have to point out that this phenomenon is just valid in unperturbed systems, e.g. in rod-type fiber systems with no significant bending or other perturbations. Otherwise, detrimental energy exchange between the supermodes can take place, leading to a change in the core-to-core coupling characteristics.
Fig. 7. A gaussian beam seeding the center core is decomposed into the supermodes of the fiber for a 3 × 3 MCF structure. The core sizes vary from 15 µm to 50 µm and the relative pitch is 2.5 times the core diameter. The cores have the same V-parameter of 3. The upper graph shows the refractive index difference of the three interferring supermodes as a function of the core size. It can be seen that the eigensolutions converge towards the same eigenvalue at larger core diameters, leading to longer coupling lengths. The lower graph shows the corresponding length after which 1% of the power in the central core is coupled to other cores.
At this point it is important to stress that the main motivation for this work is the study of CBC systems. In such systems strong coupling and the associated strong core-to-core power fluctuations are considered unwanted effects leading to a degradation of the system performance [14]. In fact, in the example shown in Fig. 5, significant coupling would already occur after a few centimeters of propagation. In the context of CBC systems with fiber lengths in the range of 1 m, the fiber design presented in Fig. 5 is not useful. For this reason, we will focus on weakly coupled structures in the next section.
Also, it is important to remark that the simulations can lead to numerical uncertainties for structures that are very weakly coupled if the transversal resolution is too coarse. The reason is that the eigenvalues for weakly coupled supermodes might still have differences in the propagation constants, which leads to interference in the permille range. Thus, the best way to detect weak coupling is when it reaches the range of a few percent. In fact, in our simulations we will set a limit of 1% coupling as a threshold to consider the cores as decoupled for CBC applications.
5. Requirements for weak coupling in 1 m long rod-type MCFs
In this section we examine the requirements for weak coupling as a function of different MCF parameters. Note that we will stick to the 3 × 3 square core arrangement. However, the results obtained from our 3 × 3 MCF simulations can be extrapolated to weakly coupled MCF structures with more cores since the major coupling partners for each core (i.e. the direct and indirect neighbors) are already included in our model.
In our simulations, the step index of the cores is swept for a given3 × 3 MCF structure, as shown in Fig. 8. Here, the relative power loss in the original seed (center) core after 1 m propagation is plotted for index steps between 2e-3 and 4e-3. The core size is 15 µm with 30 µm pitch. As expected, the strength of the core-to-core coupling depends on the core NA, meaning that the core crosstalk is reduced for larger index steps. In this particular case a step index higher than 3.5e-3 is needed to avoid any significant coupling.
Fig. 8. Relative power loss in the seed (center) core after 1 m propagation in a 3 × 3 MCF with 15 µm cores and 30 µm pitch for different index steps. Strong coupling occurs in the case of small index steps, whereas negligible coupling to the adjacent cores can be expected at around 4e-3.
In our study, such an index sweep has been performed for a wide range of parameters, as shown in Fig. 9. Here, the requirements for weak coupling in a 1 m long, 3 × 3 MCFs with core sizes ranging from 15 µm to 50 µm and relative core-to-core distances ranging from 1.5 to 5 have been calculated. The seed wavelength is set to 1030 nm and the center core is seeded with a gaussian beam perfectly matched to its LP01 mode. The colormap indicates the refractive index step Δncore of the cores that is required to achieve a total coupling of (1 ± 0.1) % from the center core to the adjacent cores over the whole fiber length (as it was shown in Fig. 8). For a better visualization, the colormap is saturated for values exceeding an index step of 10−3.
Fig. 9. Core index step required to achieve a weak coupling of 1% for different core sizes and relative core-to-core distances. The fiber length is 1 m. The colormap is saturated for values exceeding 10−3. The white isolines correspond to the transition from single-mode-operation to multimode operation (V = 2.405) and to the next higher-order-mode-transition (V = 3.832).
It can be seen that higher index steps are needed when the cores become smaller, as well as for lower core-to-core distances. Both effects can be understood by considering a scenario where the core index step is kept constant. On the one hand, the evanescent electric field from each core will have a higher overlap with adjacent cores the smaller the core-to-core distance becomes. Thus, one straightforward way to reduce core-to-core coupling consists on increasing the pitch [20]. This approach, however, might introduce an unwanted penalty in pump absorption. On the other hand, the beating of supermodes from MCFs with small cores exhibit shorter beat lengths, indicating an inherently stronger coupling, as it was shown in Fig. 6. Thus, as already mentioned before, another way to reduce core crosstalk is to increase their diameter.
Figure 9 also shows two important V-parameter isolines in white. The upper isoline represents the transition from single-mode-operation to two-mode operation (V = 2.405), meaning that all points above that line correspond to strictly single-mode cores. On the other hand, the lower isoline corresponds to the onset of the next higher order mode (V = 3.832). Thus, strictly single-mode operation together with low core-to-core coupling can be just achieved for pitch/core-ratios (relative pitch) higher than ∼ 2.5 - 3.2, depending on the core dimension. Crucially, this ratio can be lower for larger cores while maintaining the same V-parameter. The reason for this is that the supermodes tend to be more degenerated for larger cores for any given relative pitch and V-parameter, as explained in Fig. 7. In general, the findings shown in Fig. 9 are valid for any weakly coupled MCF design.
6. Summary
In this work we have investigated core-to-core power coupling in short multicore fibers that are typically used in an amplifier configuration, such as e.g. in ultrafast, coherent beam combining systems. Hereby, the influence of different parameters such as core size, core-to-core distance and the core index has been analyzed, using an approach based on supermodal interference.
Our simulations and experiments indicate that MCFs with small cores show stronger coupling over a given propagation distance than those with larger cores, irrespective of whether the core NA or the V-parameter remain constant. This is because the supported supermodes tend to be more degenerated in larger cores, which leads to longer supermode beat lengths along the fiber. This phenomenon synergizes well with the need for core area scaling in MCFs while maintaining single-mode/few-mode operation. Additionally, we have studied the influence of the core-to-core distance on the coupling effects for different core sizes. As can be expected, larger distances will help to reduce the coupling. However, this parameter has to be treated carefully in active MCFs, since it can detrimentally change the pump absorption, which is crucial for high power systems. However, we have shown that MCFs with larger cores, but the same V-parameter can be operated at smaller relative pitches, while preserving weak coupling between the cores. This enables a denser packaging of the cores, allowing for higher pump absorptions.
By implementing additional optical elements to the fiber structure, like trenches, optical barriers or even heterogeneous structures, coupling effects between the cores can be further mitigated [21–23]. Concerning the improvement for dense packaging, the integration of such techniques will be part of prospective studies. In summary, the results presented in this work show how to derive guidelines for the design of next-level high-power MCF systems.
Funding
European Research Council (670557, 835306); Bundesministerium für Bildung und Forschung (13N15244); Deutsche Forschungsgemeinschaft (416342637); Thüringer Aufbaubank (2018FGR0099); Fraunhofer-Gesellschaft (Cluster of Excellence "Advanced Photon Sources").
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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