# Heterogeneously Integrated Multicore Fibers for Smart Oilfield Applications
## 3 Heterogeneously Integrated Multicore Fibers
### Design of Digital Transmission Cores  
For fiber design, the choice of structure is dependent on the applications. The key consideration for digital signal transmission is DMGD because the computational complexity of DSP increases with it, and the level of DSP complexity determines whether the real-time transmission could be realized. We sweep the parameters, including r\({}_{\text{core}}\), \(\alpha\), and  
Figure 3: Refractive index profiles of various types of fibers. (**a**) step-index fiber, (**b**) graded-index fiber, (**c**) step-index trench-assisted fiber, (**d**) graded-index trench-assisted fiber.  
Figure 2: Schematic of the designed heterogeneous 7-core multicore fiber.  
\(\Delta\)n\({}_{\text{core}}\), and the setting of ranges follows two principles. One is the ranges should be as large as possible and the other is the designed fiber should only support three guide modes. We calculate the DMGD over the C-band based on full vector finite element analysis. The group delay of any propagation mode in FMF is given by [30]  
\[\tau_{g}=\frac{z(n_{eff}-\lambda\frac{dn_{eff}}{d\lambda})}{c} \tag{1}\]  
where \(z\) represents transmission distance, \(c\) is the speed of light in vacuum, \(\lambda\) is the wavelength and \(n_{eff}\) is the effective refraction index. Therefore, the DMGD between \(LP_{mm}\) mode and \(LP_{01}\) mode is given by:  
\[\text{DMGD}=\frac{\left(n_{eff}^{LP_{mm}}-\lambda\frac{dn_{eff}^{LP_{mm}}}{d \lambda}\right)}{c}-\frac{\left(n_{eff}^{LP_{01}}-\lambda\frac{dn_{eff}^{LP_{0 1}}}{d\lambda}\right)}{c} \tag{2}\]  
The DMGD results are shown in Figure 4.  
Within the sweeping ranges, the DMGD of GI fibers and TA-GI fibers have inflection points, while that of SI fibers have a gradual decrease with the increment of r\({}_{\text{core}}\), and \(\Delta\)n\({}_{\text{core}}\). We list the parameters of the three types of fibers at their lowest DMGD levels in Table 1, and the lowest DMGD of the TA-GI fiber is only 10 ps/km, which is 1/18 of the DMGD of SI fiber (183 ps/km) and one percent of that of the GI fiber (1000 ps/km). Although low DMGD could be obtained in the TA-GI fiber, the manufacturing process is more complex than that of SI fibers. In a word, it is not possible to use the GI fiber as the digital transmission core in this scene, and the choice of the SI fiber or the TA-GI fiber is dependent on the DSP complexity tolerance, which is positively correlated with DMGD. In addition, bending loss is considered and calculated by the finite element method. The bending radius is defined as the radius when the loss of the highest guided mode is 0.5 dB/turn, and the bending radius is calculated by the method in [31]. For the SI fiber, GI fiber, and TA-GI fiber, the bending radii are 70 mm, 30 mm and 20 mm. A smaller bending radius means better anti-bending performance. Considering the applied conditions in smart oilfields, the bending radius is m-scale or even larger, the anti-bending performance of the designed fiber is up to standard. A phrase.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **r\({}_{\text{core}}\) (\(\mu\)m)** & \(\Delta\)n\({}_{\text{core}}\) (\%) & \(\Delta\)n\({}_{\text{trench}}\) (\%) & \(\alpha\) & \(\mathbf{w_{1}}\) (\(\mu\)m) & \(\mathbf{w_{2}}\) (\(\mu\)m) \\ \hline SI & 7.1 & 0.27 & \(\backslash\) & \(\backslash\) & \(\backslash\) & \(\backslash\) \\ \hline GI & 10.5 & 0.40 & \(\backslash\) & 2 & \(\backslash\) & \(\backslash\) \\ \hline TA-GI & 10.5 & 0.345 & 0.34 & 2.05 & 2 & 9.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of various types of fibers applied for digital transmission.