# The Multiband over Spatial Division Multiplexing Sliceable Transceiver for Future Optical Networks
## 4 Numerical XT Evaluation  
MCF transmission arises as a promising SDM technology that can provide increased capacity, efficient use of space, high scalability, simplified network architecture, redundancy, reliability and energy efficiency, by the enabling of data transmission through different cores within the same optical fibre. However, XT can occur when signals from one core interfere with signals in adjacent cores. This interference can significantly degrade signal quality and network performance, and presents a dependence with frequency/wavelength. Hence, a proper design and signal isolation techniques are necessary to mitigate this issue, to achieve low XT levels between neighbouring cores. One effective technique to reduce the XT levels in MCFs is to employ a refractive index trench in the fibre design. A refractive index trench is a modification in the refractive index profile of the optical fibre, by forcing a trench or a low-index region around each core to effectively isolate the cores from one another. This can be strategically used to minimize XT between adjacent cores, as the refractive index trench acts as a barrier between the cores, preventing the leakage of light from one core to the neighbouring cores and hence enhancing the overall signal quality and reliability. The refractive index trenches can be a powerful tool for reducing XT in MCF, and their design and implementation require careful engineering and optimization. As an example, Figure 2 shows the trench-assisted core design of 19-cores 25.4 km MCF considered for the experimental assessment in Section 6. In this context, the mean XT value of a core can be approximated through evaluating a straightforward analytical formula [44]. In fact, the homogeneous core single mode 19-core MCF is designed to achieve XT values between neighbouring cores of \(-\)25 dB, at 1550 nm, for a length of 25.4 km and a bending diameter of 159 mm. Hence, the impact of XT in different transmission bands can be numerically evaluated, also including the structural parameters and specifications of the 19-cores MCF of L = 25.4 km considered for the experimental assessment in Section 6. Specifically, the XT can be approximated through the following equation:  
\[XT_{\mu}\approx\frac{2k_{pq}^{2}R_{b}}{\beta\Lambda}L, \tag{1}\]where \(\Lambda\) is the core pitch (\(\Lambda=34\)\(\mu\)m), \(k_{pq}\) is the mode linear coupling coefficient between two neighbouring cores (defined in Equation (2)), \(R_{b}=7.95\) cm is the bending radius, and \(\beta\) is the propagation constant. The propagation constant is defined as: \(\beta=(2\pi)/(\lambda n_{eff})\), being \(\lambda\) the wavelength of operation and \(n_{eff}\) the core effective refractive index. The mode linear coupling coefficient in trench-assisted structures can be calculated considering the following equation:  
\[k_{pq}=\frac{\sqrt{\Gamma}\sqrt{\Delta_{1}}}{a_{1}}\frac{U_{1}^{2}}{V_{1}^{3} K_{1}^{2}(W_{1})}\sqrt{\frac{\pi a_{1}}{W_{1}\Gamma}}\exp\left[-\frac{W_{1} \Gamma+2(W_{2}-W_{1})w_{tr}}{a_{1}}\right] \tag{2}\]  
From Equation (2) and considering the approximations and simplifications in [44],  
\[\Gamma=\frac{W_{1}}{W_{1}+(W_{2}-W_{1})w_{tr}/\Lambda} \tag{3}\]  
With a relative refractive index difference of \(m=|\Delta_{2}|/\Delta_{1}=2\), \(W_{2}\) and \((W_{2}-W_{1})\) can be approximated as linear functions of \(V_{1}\), being \(V_{1}=\frac{2\pi}{\lambda}a_{1}n_{1}\sqrt{(2\Delta_{1})}\), according to [44]. Specifically, \(W_{2}=1.750V_{1}-0.388\), \((W_{2}-W_{1})=0.607V_{1}+0.608\), \(n_{1}\) is the core refractive index (\(n_{1}=1.45\)), \(a_{1}\) is the core radius and \(w_{tr}\) is the trench width, as shown in Figure 2. Finally, \(K_{1}(W_{1})\) is the modified Bessel function of the 2 nd kind with 1 st order and \(U_{1}^{2}=a_{1}^{2}(k^{2}n_{1}^{2}-\beta^{2})\), being \(k=2\pi/\lambda\). Following this analysis, the estimated XT per wavelength is depicted in Figure 3. A range from 1260 nm to 1675 nm--covering the whole of the O-, E-, S-, C-, L-, and U-bands [26]--has been taken into account.