## 3 System Model
### Inter-Core Crosstalk  
Besides the blocking of lightpath establishment, the impact of inter-core crosstalk must also be considered in multi-core fiber environments. We can simultaneously establish lightpaths using the same FSs with different cores within the same fiber. Therefore, using multi-core fiber environments enhances the transmission capacity of elastic optical path networks. However, at the same time, inter-core crosstalk becomes a severe issue for lightpaths using the same FSs in different cores. Inter-core crosstalk impairs the optical signal quality by accumulating at each link along established lightpaths and introducing noise at the receiver nodes. If the crosstalk level at the receiver nodes exceeds a certain threshold, data transmission could fail, even if the lightpaths are correctly established.  
Here, we consider the case where there are two cores, \(c_{i}\) and \(c_{j}\), in a homogeneous multi-core fiber with length \(L\), and an optical signal is transmitted through core \(c_{i}\). In this case, a portion of this signal leaks into core \(c_{j}\). Let \(P_{i}(L)\) and \(P_{j}(L)\) denote the transmitted power in cores \(c_{i}\) and \(c_{j}\), respectively, at the output of the multi-core fiber. In this case, the crosstalk \(\text{XT}_{i,j}\) from core \(c_{i}\) to \(c_{j}\) at the output of the multi-core fiber with length \(L\) is defined as  
\[\text{XT}_{i,j}=\frac{P_{j}(L)}{P_{i}(L)}=\tanh(h_{i,j}L), \tag{2}\]  
where \(h_{i,j}\) denotes the power coupling coefficient [26]. When the value of \(\text{XT}_{i,j}\) is sufficiently small, it can be approximated as \(h_{i,j}L\). The power coupling coefficient \(h_{i,j}\) is given by  
\[h_{i,j}=\frac{2\kappa^{2}b_{r}}{\alpha\Lambda_{i,j}}, \tag{3}\]  
Figure 2: Spectrum fragmentation.  
where \(\kappa\) denotes the mode coupling coefficient, \(b_{r}\) denotes the bending radius, \(\alpha\) denotes the propagation constant, and \(\Lambda_{i,j}\) denotes the core pitch between cores \(c_{i}\) and \(c_{j}\). According to [27], the mode coupling coefficient \(\kappa\) is calculated as  
\[\kappa=\frac{\sqrt{\Delta}}{\sigma_{r}}\frac{X^{2}}{V^{3}}\frac{K_{0}\left( \frac{\Lambda_{i,j}}{c_{r}}W\right)}{K_{1}^{2}(W)}, \tag{4}\]  
where \(\sigma_{r}\) denotes the core radius and \(\Delta\) denotes the relative refractive index difference. \(V\) denotes the normalized frequency, and \(X\) and \(W\) denote the normalized transverse wave number in the core and cladding, respectively. The functions \(K_{0}(x)\) and \(K_{1}(x)\) can be approximately defined as  
\[K_{0}(x)\approx\sqrt{\frac{\pi}{2x}}\exp(-x), \tag{5}\]  
\[K_{1}(x)\approx\sqrt{\frac{3.3}{x}}\exp(-1.1x), \tag{6}\]  
respectively.  
We can calculate the crosstalk level with these equations. For example, in [16], the relationship between the core pitch \(\Lambda_{i,j}\) and the crosstalk \(\mathcal{X}\)T\({}_{i,j}\) was calculated using the parameters listed in Table 1. The authors showed that the power of inter-core crosstalk decreases exponentially as the core pitch increases. Thus, in this paper, we consider the impact of inter-core crosstalk between adjacent cores, but we do not take inter-core crosstalk between non-adjacent cores into account, as shown in Figure 3.