## 3 Structure and Modal Properties of the Six-Core TA-MCF  
Figure 2 shows the diagrammatic sketch of the proposed six-core TA-MCF. The main conception of this fiber involves a simplified microstructure cladding with only one ring of low-index inclusions, added on the basis of the six-core MCF to combine the merits of both MSFs and MCFs. For ease of fabrication, pure, germanium-doped, and fluorine-doped silica glass, which are commonly used in the fabrication of step index fibers (SIFs), were chosen to form the TA-MCF. The matrix is pure silica glass with refractive index \(n_{0}\). The fiber core comprises six hexagonally arranged germanium up-doped high-index silica inclusions (red-colored circles) with refractive index \(n_{1}\) and diameter \(d_{1}\). Outside the fiber cores, there is a hexagonal low-index trench which is composed of twelve fluorine down-doped silica inclusions with refractive index \(n_{2}\) and diameter \(d_{2}\) (gray-colored circles). Every three neighboring inclusions are arranged in regular triangles with inclusion-to-inclusion space denoted as \(\Lambda\). To investigate the low-index trench's influences on the modal properties of the six-core MCF, structural parameters with \(n_{0}=1.45\), \(n_{1}-n_{0}=0.015\), \(n_{2}-n_{0}=-0.015\), \(\Lambda=2.2\)\(\upmu\)m, \(d_{1}/\Lambda=0.80\), and \(d_{2}/\Lambda=0.60\) were initially chosen. Figure 3 shows the mode indices of the fundamental and second-order supermodes for the six-core MCF with and without trench. It can be seen that the effective indices were reduced if the low-index trench was added for both the fundamental and second-order modes. From Equation (2) and Figure 3, we can conclude that the effective index \(n_{\text{fund}}\) of the fundamental supermode is always larger than \(n_{\text{eff0}}\) of the SIF. This indicates that this supermode is always supported by the six-core MCF regardless of the trench. As for the index of the degenerate second-order mode \(n_{\text{sec}}\), whether its value is larger than \(n_{\text{eff0}}\) or not depends on the sign of \(\kappa_{1}\)-\(\kappa_{2}\)-\(\kappa_{3}\). For the six-core MCF without the trench, its \(n_{\text{sec}}\) values became smaller than \(n_{\text{eff0}}\) after 1205 nm. This crossing wavelength had a 219-nm blue shift to 986 nm after the trench was added. More importantly, the two \(n_{\text{sec}}\) curves intersected with the refractive index curve of pure silica at 1298 and 1198 nm. This means that the second-order mode is cut off after those wavelengths, and the introduction of the low-index trench results in a blue shift of the cut-off wavelength and a broader single-mode regime. The electric-field distributions of the second-order mode of TA-MCF at 1.1 \(\upmu\)m and 1.2 \(\upmu\)m are shown in Figure 3b,c. As can be seen from Figure 3b, the light is mainly confined in the core region before the cut-off wavelength. After that wavelength, it is clearly seen that light is coupled out of the cores (Figure 3c). Figure 3: (**a**) The effective indices of the step index fiber (SIF) unit, fundamental supermode, and second-order supermode of the six-core TA-MCF and the six-core MCF; the electric field distributions of the second-order supermode of the TA-MCF at (**b**) 1.1 \(\upmu\)m, and (**c**) 1.2 \(\upmu\)m.  
Figure 2: Diagrammatic sketch of the six-core trench-assisted MCF (TA-MCF) and its equivalent refractive index profile. Figure 4 shows the total dispersion of the SIF unit, the six-core MCF, and the TA-MCF as a function of wavelength. Upon comparing the dispersion curve of the six-core MCF to that of SIF, it can be seen that mode-coupling imposes huge influences on light transmission. By further adding a trench in the MCF, the dispersion value is obviously altered and the dispersion curve becomes flatter at the communication band. This indicates that the TA-MCF proposed herein is promising for achieving wideband flattened dispersion. The optical intensity distribution of the fundamental supermodes was investigated in Figure 5 for MCFs with and without the trench.