## 2 Fundamental Theories  
The diagrammatic sketch of a hexagonally arranged homogeneous six-core MCF is shown in Figure 1. Suppose that only a fundamental mode with a propagation constant of \(\beta\) is supported in each core at our interested wavelength. When the beam is launched into the fiber, the field distribution of a supermode can be considered as a superposition of each isolated mode under the 6 \(\times\) 6 coupling matrix [27; 28; 29; 30]. \[C=\left[ \begin{array}{cccccccc}\beta&\kappa_{12}&\kappa_{13}&\kappa_{14}& \kappa_{15}&\kappa_{16}\\ \kappa_{21}&\beta&\kappa_{23}&\kappa_{24}&\kappa_{25}&\kappa_{26}\\ \kappa_{31}&\kappa_{32}&\beta&\kappa_{34}&\kappa_{35}&\kappa_{36}\\ \kappa_{41}&\kappa_{42}&\kappa_{43}&\beta&\kappa_{45}&\kappa_{46}\\ \kappa_{51}&\kappa_{52}&\kappa_{53}&\kappa_{54}&\beta&\kappa_{56}\\ \kappa_{61}&\kappa_{62}&\kappa_{63}&\kappa_{64}&\kappa_{65}&\beta\\ \end{array} \right]=\left[ \begin{array}{cccccccc}\beta&\kappa_{1}&\kappa_{2 }&\kappa_{3}&\kappa_{2}&\kappa_{1}\\ \kappa_{1}&\beta&\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{2}\\ \kappa_{2}&\kappa_{1}&\beta&\kappa_{1}&\kappa_{2}&\kappa_{3}\\ \kappa_{3}&\kappa_{2}&\kappa_{1}&\beta&\kappa_{1}&\kappa_{2}\\ \kappa_{2}&\kappa_{3}&\kappa_{2}&\kappa_{1}&\beta&\kappa_{1}\\ \kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{2}&\kappa_{1}&\beta\\ \end{array} \right], \tag{1}\]  
where \(\kappa_{\rm pq}\) is the mode-coupling coefficient between the core \(p\) and \(q\). For the six-core fiber given in Figure 1, \(\kappa_{\rm pq}\) may take one of three different values, namely \(\kappa_{1}\), \(\kappa_{2}\), and \(\kappa_{3}\), based on the relative position of two cores(shown in Figure 1 and Equation (1)). By solving the eigenvalues of \(C\), the effective mode indices \(n_{\rm effi}\) (\(i\) = 1, 2, 3, 4, 5, 6) of the six supermodes can be expressed as  
\[ \begin{array}{c}n_{\rm effi}=n_{\rm effi}+\frac{2\kappa_{1}+2\kappa_{2}+ \kappa_{3}}{k_{0}}\\ n_{\rm eff2}=n_{\rm effi}=n_{\rm effi}+\frac{\kappa_{1}-\kappa_{2}-\kappa_{3 }}{k_{0}}\\ n_{\rm effi4}=n_{\rm effi5}=n_{\rm effi0}+\frac{\kappa_{3}-\kappa_{2}-\kappa_{ 1}}{k_{0}}\\ n_{\rm effi6}=n_{\rm effi0}+\frac{2\kappa_{2}-2\kappa_{1}-\kappa_{3}}{k_{0}}\\ \end{array}  \tag{2}\]  
where \(n_{\rm effi0}=\beta/k_{0}\) is the mode index of the individual step index fiber (SIF) cell. From Equation (2), we can see that the second and third supermodes, and the fourth and fifth supermodes are degenerated for a hexagonally arranged six-core MSF. The GVD is one of the most important parameters of optical fibers which limits the speed of each information channel. It can be treated as the sum of material dispersion and waveguide dispersion. The material dispersion can be derived from the Sellmeier equation of silica. Once the \(n_{\rm eff1}\) is solved, the waveguide dispersion \(D_{\rm w}\) (\(\lambda\)) can be calculated by  
\[D_{\rm w}(\lambda)=-\frac{\lambda}{c}\frac{\mathrm{d}^{2}Re[n_{\rm eff1}]}{ \mathrm{d}\lambda^{2}}, \tag{3}\]  
where \(c\) is the speed of light in vacuum, \(\lambda\) represents the wavelength, and \(Re[n_{\rm eff1}]\) is the real part of \(n_{\rm eff1}\)  
Figure 1: The diagrammatic sketch of a hexagonally arranged homogeneous six-core multicore fiber (MCF). In addition to GVD, the losses of the fiber are also crucial for an optical communication system. Confinement loss \(\alpha\) can be obtained by  
\[\alpha=\frac{2\times\pi\times 8.686\times Im[n_{\text{eff1}}]}{\lambda}, \tag{4}\]  
where \(Im[n_{\text{eff1}}]\) represents the imaginary part of \(n_{\text{eff1}}\). As for bending loss, its calculation can be summarized by the three steps. Firstly, the equivalent refractive index profile \(n_{eq}(x,y)\) of the bending fiber needs to be represented by the refractive index profile \(n(x,y)\) of the straight fiber using Equation (5) [31]. \[n_{eq}(x,y)=n(x,y)(1+\frac{x}{R}), \tag{5}\]  
where \(x\) means that we assume the fiber is bending in the positive direction of the \(x\)-axis, and \(R\) is the bending radius.