Multicore optical fibers (MCFs) are considered to be the candidate of choice for the next generation of telecommunication networks based on space division multiplexing [1], and in the next few years MCF-based networks will undoubtedly be implemented. They are also proposed for sensing applications [2], in which they enable building, e.g., shape sensors [3] or multiparameter sensors [4]. To allow independent propagation of signals in the cores, they are isolated by different means, such as their distancing, differentiating [5], insulating by trench [6,7] or airholes [8–10] or a combination of these techniques [11]. The parameter that describes the isolation of the cores in a quantitative manner is core-to-core cross talk (XT). Although XT is an intuitive parameter—it defines the relative amount of power that is transferred from the excited core to another core—there are many discrepancies in the literature in terms of XT definition, measurement methodology, and even the unit in which it is expressed [12–14]. As a consequence, there is no reliable way of designing MCFs in terms of the XT parameter, which is confirmed by disagreement between the XT values calculated and measured for fabricated fibers [15].

This article focuses on a model of XT, which is free from simplifications and is suitable for calculating XT in complex MCFs, regardless of the core number and the way of their isolation. The model is based on the basic fact that any light field can be represented as a sum of orthogonal states. In a vacuum, any light field can be represented as a sum of plane waves [16]. The same methodology stays in force when one considers light distribution in a complex medium. The only difference is that there are different solutions of the wave equation, especially in waveguides, in which the role of a plane wave is carried out by modes. As the space is no longer infinite, the number of solutions (modes) that can be supported by the given structure is limited. All modes are orthogonal and their superposition represents an actual field distribution in a fiber. This also applies to MCFs, in which we are talking about supermodes [17] instead of modes. Supermodes have different propagation constants just like modes in standard multimode fibers, and therefore their superposition produces a different power distribution along the fiber. In theory, this is the only source of XT in MCFs. In actual fibers, there is also a second origin of XT, namely, the power coupling between supermodes, which results from imperfections in the fiber, and is thus random in nature. Since the modes’ coupling is strongly dependent on the fiber fabrication process, its influence on XT cannot be quantified at the stage of fiber design. In addition, XT induced by the coupling of supermodes plays an important role mainly when analyzing propagation in MCFs with strongly isolated cores over very large distances.

Since at the stage of fiber design only XT induced by the interference of supermodes can be taken into account, in the further considerations we consider a theoretical case, in which the power coupling between the modes is not allowed. In such a case, the transfer of power between cores is periodic and can be characterized by two parameters, namely, the maximum value of cross talk (${\mathrm{XT}}_{\max }$) and the beat length ($L_b$). ${\mathrm{XT}}_{\max }$ is expressed in decibels for any pair of cores, and this parameter is independent of the fiber length.

The methodology for calculating ${\mathrm{XT}}_{\max }$ in an arbitrary structure is presented below. First, one has to solve the following wave equation directly:

 

Fig. 1. Normalized electric field amplitude distribution of exemplary supermodes (2 of 14), which are present in seven-core fiber with airhole isolation [9].

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The electric field amplitude at any point can be expressed as a superposition of all supermodes propagating in the MCF:

The level of each supermode excitation may be different and is strongly dependent on the initial field distribution. By examining the $P_{\text{core}\text{-}n}(z)$ function

Thus, the final part of determining ${\mathrm{XT}}_{\max }$ is to find the maximum power resulting from the superposition of the supermodes in the core, which is not directly excited. Therefore, ${\mathrm{XT}}_{\max }$ [dB] is defined as the relation between the maximum power level in the investigated core (indexed with $n$) and the maximum power level in the initially excited core (indexed with $m$) according to the equation

Considering the strict model of core-to-core power transfer presented, it is worth noting that analyzing XT at one point of the MCF is not viable. Even if a multicore fiber is treated as being an extremely weakly coupled coupler and ${\mathrm{XT}}_{\max }$ is low, the phenomenon is never a linear function of fiber length. To measure ${\mathrm{XT}}_{\max }$, one ought to find the point of maximum power transfer. For multicore fiber components, which have strictly defined optical length, it is necessary to distinguish between ${\mathrm{XT}}_{\max }$ of the used fiber and the observed XT of the fiber component. Equation (5) and beat length characterizes a two-dimensional MCF structure and is thus independent of fiber length.

Both ${\mathrm{XT}}_{\max }$ and $L_b$ are highly dependent on structure parameters and wavelength, but the result of the supermode superposition also depends strongly on the chosen pair of cores, so for MCFs with more than two cores, it is important to note between which cores the ${\mathrm{XT}}_{\max }$ is being calculated or measured. Furthermore, ${\mathrm{XT}}_{\max }$ is not the only value that depends on the chosen pair of cores—the same situation is observed in the terms of $L_b$, which also varies for different pairs of cores. This results in the different lengths over which the initial phase difference of supermodes is reproduced.

The presented model is suitable for any number of cores, but for a detailed discussion, we will analyze the behavior of supermodes in dual-core fiber. The model being investigated consists of two cores separated by an area of decreased refractive index—$n_a$ and diameter—$d_a$ (Fig. 2). To clarify the influence of splice imperfections to $XT$, we also introduce an initial field offset directed to the center of the fiber (Fig. 2.). Generally, any effects resulting from changing the refractive index $n_a$ are indeed equivalent to distancing the cores or changing the diameter of the area with decreased refractive index ($d_a$). All calculations were performed for the wavelength of 1550 nm by the finite element method with at least $4·{10}^5$ finite elements to ensure convergence. The initial field distribution is considered as a fundamental mode of a single-core fiber with a core radius of 4.1 μm and core doping of 3.5 mol. % ${\mathrm{GeO}}_2$.

 

Fig. 2. Dual-core fiber model with $d_{\text{core}}=8.2\mathrm{\mu m}$, $d_a=6\mathrm{\mu m}$, $2\mathrm{\Lambda }=20\mathrm{\mu m}$, and variable $n_a$.

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A dual-core structure supports two orthogonal supermodes, namely, symmetric and antisymmetric (Fig. 3). An effect of the interference of these supermodes is presented in Fig. 4. By definition, two orthogonal supermodes possess different phase velocities, which results in periodic switching of power between cores along the fiber. Therefore, although there is no power coupling between supermodes, their superposition results in core-to-core cross talk. In the presented case (Figs. 3 and 4), strong power transfer between cores arises from a large spatial overlap of excited supermodes. Symmetric refractive index distribution supports the symmetric and antisymmetric supermodes and will always lead to a complete power transfer. According to the definition of ${\mathrm{XT}}_{\max }$ (5), such a fiber has ${\mathrm{XT}}_{\max }$ equal to 0 dB, which means that the power transfer is complete over $L_b/2$—the entire power that is introduced to one core is transferred to the other one.

 

Fig. 3. Symmetric and antisymmetric supermodes’ profiles with the marked refractive index profile supporting them. The $n_a$ in this case is equal to the refractive index of the fiber cladding, and the cores have exactly the same refractive indices.

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Fig. 4. Power transferring for the structure presented in Fig. 3. (a) Pictorial power distribution in particular cores over fiber length. (b) Normalized power in particular cores over fiber length expressed in beat lengths.

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So far we have discussed the symmetric case, in which symmetric refractive index distribution forces the mode profile to be symmetric or antisymmetric as the following relation implies for all supermodes in the fiber:

Thus, only propagation constants and consequently $L_b$ are changed while increasing isolation of the cores. When we decide to break the symmetry in the fiber refractive index distribution, it will lead us to a nonidentical supermode power distribution. In Fig. 5, the case is presented in which the symmetry is broken by a slight change of one core diameter ($d_{\text{core}1}=8.18\mathrm{\mu m}$) with respect to the other ($d_{\text{core}2}=8.2\mathrm{\mu m}$). This in turn causes that quasi-symmetric and quasi-antisymmetric supermodes begin to be clearly distinguishable, not only in terms of intensity but also power distribution. Those nonsymmetric mode profiles can be further differentiated by decreasing $n_a$ [Figs. 5(a), 5(b), and 5(c)].

 

Fig. 5. Quasi-symmetric and quasi-antisymmetric modes’ profiles and refractive index profiles. The difference in $d_{\text{core}}$ between the cores is at the level of 0.25%. The relative difference between $n_a$ and the refractive index of the cladding is equal to 0%, $-0.07\%$, and $-0.97\%$ for graphs (a), (b), and (c), respectively.

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When refractive index distribution supports quasi-symmetric or quasi-antisymmetric supermodes, core-to-core power switching is not complete (Fig. 6). In such case, according to the definition, ${\mathrm{XT}}_{\max }$ is no longer equal to 0 dB but obtains lower values.

 

Fig. 6. Power transferring for the structure presented in Fig. 5(c). (a) Pictorial power distribution in particular cores over fiber length. (b) Normalized power in particular cores over fiber length expressed in beat lengths.

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As symmetry breaking introduces spatial nonuniformity of the power distribution of quasi-symmetric and quasi-antisymmetric supermodes, their overlap integral with the initial field introduced to the fiber is no longer equal. This leads to different ${\mathrm{XT}}_{\max }$ values for various initial excitation conditions, e.g., splicing with the offset (see Fig. 7) or with different fibers.

 

Fig. 7. Power transferring for the structure presented in Fig. 5(c) for two different excitation conditions (offsets).

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Figure 8 presents ${\mathrm{XT}}_{\max }$ for different isolation (understood as refractive index $\mathrm{\Delta }$ of the isolating part) of cores as a function of excitation offset. One can find that increasing the difference in the spatial power overlap of supermode power profiles (e.g., by decreasing $n_a$) causes ${\mathrm{XT}}_{\max }$ to be more sensitive to the change in the excitation field.

 

Fig. 8. Initial field introduction offset increases the ${\mathrm{XT}}_{\max }$ value.

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Another important aspect, which is worth mentioning when discussing $XT$ in MCFs, is the influence of fiber bending on $XT$. Unless we consider a diabatic transformation of the fiber, the bending changes the intensity distribution of each supermode without power transfer between supermodes. It also changes the difference between the effective refractive indices of supermodes and thus also the beat length of the power transfer between cores. One may consider this as a bend-induced cross talk (due to the change of the beat length, the power distribution at the fiber output is also changed) but, in fact, as long as we couple light into the straight part of the fiber and capture light from the straight part of the fiber, which is almost always the case, there is no change in the maximum level of $XT$, which can be observed at the fiber output.

In conclusion, there are two sources of core-to-core power transfer in MCFs. One source is fiber imperfections, which result in the coupling of supermodes between each other and is sometimes discussed from the statistical point of view [18]. The second origin is a result of the superposition of supermodes having different propagation constants, which is the key consideration while designing MCFs.

In this Letter we have discussed the description of core-to-core power transfer with the use of two parameters ${\mathrm{XT}}_{\max }$ and $L_b$. In addition, we have demonstrated a way of calculating these parameters. The method is based on the analysis of the supermode interference and opens up new possibilities for designing and analyzing different MCF structures with complex refractive index distribution. The model presented does not require high computational power as it relies on two-dimensional calculations. At the same time, it is free from simplifications, which means we can obtain exact results. Moreover, perturbations such as fiber bending or splice offset, which play a crucial role in fiber performance, could also be easily investigated. Since the method relies on fully vectoral solutions it can also take into account polarization effects and is not limited to low-refractive-index contrast structures.