# Heterogeneously Integrated Multicore Fibers for Smart Oilfield Applications
## 3 Heterogeneously Integrated Multicore Fibers
### The Analysis of the Crosstalk  
To maintain the signal quality, the crosstalk between adjacent cores, which is calculated by the finite element method to evaluate the performance of the multicore fiber, should be lower than \(-\)55 dB/km [35]. As we mentioned above, the cores for digital transmission and analog transmission support three modes while the core for analog transmission only excites the fundamental mode. For sensing cores, we use the standard G.652 fiber. We know that the distance between the edges of cores influences crosstalk greatly and it is a negative correlation with the crosstalk. To simplify the calculation, we suppose that the crosstalk between two three-mode cores is the crosstalk of the whole multicore fiber because the core radius of G.652 fiber (3 \(\upmu\)m) is much smaller than the other two three-mode fibers, which leads to lower crosstalk when the core pitch is fixed. The parameters of cores for digital transmission and analog transmission are listed in Tables 1 and 2. The bending radius is assumed as one meter according to the diameter of the pipeline. The crosstalk \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **r\({}_{\text{core}}\) (\(\upmu\)m)** & **\(\Delta\)n\({}_{\text{core}}\) (\%)** & **\(\Delta\)n\({}_{\text{tench}}\) (\%)** & **\(\alpha\)** & **w\({}_{\text{1}}\) (\(\upmu\)m)** & **w\({}_{\text{2}}\) (\(\upmu\)m)** \\ \hline SI & 8.0 & 0.27 & \(\backslash\) & \(\backslash\) & \(\backslash\) \\ \hline GI & 11.0 & 0.25 & \(\backslash\) & 2 & \(\backslash\) & \(\backslash\) \\ \hline TA-SI & 8.0 & 0.25 & 0.25 & \(\backslash\) & 4 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of various types of fibers applied for analog transmission. (XT) is obtained by using coupled-power theory [36]. The mode-coupling coefficient could be shown as:  
\[\kappa_{pq}=\frac{\omega\varepsilon_{0}\iint_{-\infty}^{+\infty}\left(N^{2}-N_{q} ^{2}\right)E_{p}^{*}E_{q}dxdy}{\iint_{-\infty}^{+\infty}u_{z}\cdot\left(E_{p}^{* }\times H_{p}+E_{p}\times H_{p}^{*}\right)dxdy} \tag{3}\]  
where \(\omega\) is an angular frequency of the sinusoidally varying electromagnetic fields, \(\varepsilon_{0}\) is the permittivity of the medium, and \(u_{z}\) means the outward-directed unit vector. \(E_{p}\) and \(E_{q}\) represent the electric field distribution of core inside the range of core \(p\), and the electric field distribution of core inside the range of core \(q\), respectively. The crosstalk (\(XT\)) between the neighboring cores with length L is estimated as:  
\[XT=\tanh\left(\overline{l}_{pq}\mathrm{L}\right) \tag{4}\]  
\[h_{pq}(z)=\frac{2\mathrm{K}_{pq}^{2}d}{1+\left(\Delta\beta_{pq}^{{}^{\prime}}d \right)^{2}} \tag{5}\]  
where K\({}_{pq}\) is the average value of \(\kappa_{pq}\) and \(\kappa_{qp}\). \(\Delta\beta_{pq}^{{}^{\prime}}\) is the difference of equivalent propagation constant between two cores. \(d\) means the correlation length [37]. Here, \(d\) is assumed to be 0.05 m.  
Figure 6 shows the crosstalk as a function of the pitch between neighboring cores. It could be seen that the crosstalk reduces as the core pitch increases. To compare all the combinations, we set the core pitch as 50 \(\upmu\)m. For each row, the third combination performs better than the other two, which means that we could obtain a better performance if the SI-TA fiber is applied in the analog-transmission core. For example, in the first row, the core for digital transmission is the SI fiber and that for analog transmission is the SI fiber, the GI fiber, and the SI-TA fiber, respectively. The maximum crosstalk of the third combination is \(-63.4\) dB/km, while the other two combinations' maximum crosstalk values are approximately \(-55.4\) dB/km and \(-34.1\) dB/km. Furthermore, comparing the results by column is another dimension.