## 3 Structural Design and Numerical Analysis  
The cross-section and equivalent Re[n\({}_{\rm eff}\)] profile of the designed all-solid DCC-MSF is shown in Figure 1. As seen in Figure 1, the MSF is formed by the dielectric cylinder with three different kinds of diameters and Re[n\({}_{\rm eff}\)]. The dielectric cylinder possessing the largest diameter and the smallest diameter in the four cladding forms the inner and outer core of MSF, respectively. At present, according to Handbook of Optical Constants of Solids, by doping fluoride and phosphide into silica properly, the different refractive indices required in this paper can be achieved theoretically. The Re[n\({}_{\text{eff}}\)] of substrate material is n\({}_{0}\) = 1.450. The Re[n\({}_{\text{eff}}\)] of the dielectric cylinder lying in the inner core, outer core, and the cladding is expressed as \(n_{1}\), \(n_{2}\), and \(n_{3}\), and the corresponding diameters are defined as \(d_{1}\), \(d_{2}\), and \(d_{3}\), respectively. Because the inner core region includes the substrate material around the inner core and the inner core material, the equivalent Re[n\({}_{\text{eff}}\)] will be between \(n_{0}\) and \(n_{1}\), as will be the same for the other regions. The space of the adjacent dielectric cylinder is \(\Lambda\). The primitive parameters of the designed MSF are \(n_{1}\) = 1.465, \(n_{2}\) = 1.380, \(n_{3}\) = 1.320, \(d_{1}\) = 2.2 \(\upmu\)m, \(d_{2}\) = 0.51 \(\upmu\)m, \(d_{3}\) = 0.8 \(\upmu\)m, and \(\Lambda\) = 1.8 \(\upmu\)m.  
The change of Re[n\({}_{\text{eff}}\)] of the fundamental and second-order supermode is observed in Figure 2a. The illustrations 1, 3 and 2, 4 are the electric field of the fundamental and second-order supermode at different wavelengths, respectively. While smaller than the phase-matching wavelength (\(\lambda_{\text{p}}\)), the fundamental and second-order supermode transmit in the inner and outer core, separately. There is no energy transferring between the fundamental and second-order supermode. That is to say, the energy of the fundamental and second-order supermode exists in the inner and outer core, respectively. With approaching \(\lambda_{\text{p}}\), the fundamental and second-order supermode are going to couple. Because of mutating at the intersection of two Re[n\({}_{\text{eff}}\)] curves of the fundamental and second-order supermode, the two modes will generate large negative dispersion and positive dispersion respectively, which is seen in Figure 2b. The positive and negative dispersion reach the maximum value at \(\lambda_{\text{p}}\). When longer than \(\lambda_{\text{p}}\), energy transfer will occur. In other words, more and more of the energy of the fundamental supermode transfers from the inner to outer core, and the energy of the second-order supermode goes from the outer to inner core [22; 23]. Because the SMF has some positive dispersion in the common communication band, it can be compensated by utilizing the large negative dispersion generated by the fundamental supermode of compensated MSF. According to the definition of the \(K\) value, BDC is realized perfectly only while the \(K\) value of both the compensated MSF and SMF is equal. Meanwhile, the dispersion and DS are compensated under the circumstances. In terms of Equations (6) and (8), the calculated \(K\) value is shown in Figure 3. The \(K\) value curve of the compensated MSF and SMF exists in the form of a parabolic shape with an upward opening and approximate linear shape, respectively, as seen in Figure 3. The \(K\) value of both the compensated MSF and SMF can't be equal at certain wavelengths under such conditions. However, as the wavelength approaches the point where the \(K\) values are equal, the BDC can be realized approximately due to less variation of the \(K\) value. There are two intersections, at most, of the \(K\) value for both the compensated MSF and SMF, which can be observed in Figure 3. The bandwidth of dispersion compensation realized is wider when there are two intersections in the \(K\) value curve.