The numerical super-mode analysis for FM-MCFs using an exact solution expression

We demonstrate that a Bessel series expansion can serve as an exact solution for multi-core fibers (MCFs) using Graf's additional theorem and Green's theorem. Based on this exact solution, the dispersion equation is derived from the boundary conditions, allowing the estimation of super-modes for multi-mode MCFs with heterogeneous cores. In practice, we apply this equation to few-mode 4-core fibers, confirming the dependence of the propagation constant on core pitch. Moreover, we investigate the instability of the super-mode due to the anti-symmetric structure for smaller core pitches. Additionally, we find that some leakage modes transform into waveguide modes for smaller core pitches.