## 2 Numerical Methods and Theories  
The MSF with circular air holes arranged in regular array can be simulated by the multi-pole method and CUDOS MOF Utilities software, which can calculate the \(n_{\text{eff}}\). The dispersion and loss can be obtained through substituting the real and imaginary parts into different equations, respectively [15]. The multi-pole method can also be used to simulate the all-solid MSF. The basic principle of the multi-pole method is that each component of the electromagnetic field is expressed through Bessel functions in cylindrical coordinates. Then, the Helmholtz equation is solved by using the Maxwell's boundary condition. The waveguide dispersion \(\text{D}_{\text{w}}(\lambda)\) is expressed as follows [16]:  
\[\text{D}_{\text{w}}(\lambda)\ =\ -\frac{\lambda}{c}\frac{\text{d}^{2}\text{Re}[ \text{n}_{\text{eff}}]}{\text{d}\lambda^{2}} \tag{1}\]  
Here, c is light speed and \(\lambda\) is wavelength. \(\text{Re}[n_{\text{eff}}]\) is the real part of the \(n_{\text{eff}}\) of the fundamental mode. The total dispersion \(\text{D}(\lambda)\) of MSF contains \(\text{D}_{\text{w}}(\lambda)\) and the material dispersion \(\text{D}_{\text{m}}(\lambda)\). Their relationship satisfies the following equation [17]:  
\[\text{D}(\lambda)\ =\ \text{D}_{\text{w}}(\lambda)+\text{D}_{\text{m}}(\lambda) \tag{2}\]  
The \(\text{D}_{\text{m}}(\lambda)\) can be expressed as [18]:  
\[\text{D}_{\text{m}}(\lambda)\ =\ -\frac{\lambda}{c}\frac{\text{d}^{2}\text{n}_{ \text{m}}}{\text{d}\lambda^{2}} \tag{3}\]  
Here, \(\text{n}_{\text{m}}\) is the \(\text{Re}[\text{n}_{\text{eff}}]\) of MSF materials, and can be calculated by the Sellmeier equation. The relationship between dispersion slope (DS) and wavelength is as follows [19]:  
\[\text{DS}(\lambda)\ =\ \frac{\text{d}\text{D}(\lambda)}{\text{d}\lambda} \tag{4}\]  
As the wavelength varies, the dispersion, along with DS of MSF, will change. To meet the need of BDC, the dispersion and DS need to be compensated simultaneously. As such, the condition of BDC can be defined as [19]:  
\[\left\{ \begin{array}{l}\text{D}_{\text{SMF}}\text{L}_{\text{SMF}}+\text{D}_ {\text{MSF}}\text{L}_{\text{MSF}}\ =\ 0\\ \text{DS}_{\text{SMF}}\text{L}_{\text{SMF}}+\text{DS}_{\text{MSF}}\text{L}_{ \text{MSF}}\ =\ 0\end{array} \right. \tag{5}\]where L\({}_{\rm SMF}\), DS\({}_{\rm SMF}\), and D\({}_{\rm SMF}\) are the length, DS, and dispersion of SMF, respectively. L\({}_{\rm MSF}\), DS\({}_{\rm MSF}\), and D\({}_{\rm MSF}\) is the length, DS, and dispersion of the compensated MSF, respectively. Therefore, the length of compensated MSF needs to satisfy the relationship of L\({}_{\rm MSF}\) =\(-\)D\({}_{\rm SMF}\)*L\({}_{\rm SMF}\)/D\({}_{\rm MSF}\). In order to realize BDC, the dispersion and DS of the compensated MSF and SMF should have the relationship of D\({}_{\rm MSF}\)DS\({}_{\rm SFM}\) = DS\({}_{\rm MSF}\)D\({}_{\rm SMF}\). Alongside this, the higher the negative dispersion value of the compensated MSF is, the larger the compensation multiple [19]. Meanwhile, the Kappa (\(K\)) value is used to represent the ability of compensating dispersion and DS, expressed as [19]:  
\[K\ =\ \frac{\rm D(\lambda)}{\rm DS(\lambda)} \tag{6}\]  
\(K\)'s unit is n\({}_{\rm m}\). While the \(K\) values of the compensated MSF and SMF are equal, the BDC can be achieved perfectly. The residual dispersion (D\({}_{\rm R}\)) can be expressed as [19]:  
\[\rm D_{R}\ =\ \frac{\rm D_{SMF}L_{\rm SMF}+D_{MSF}L_{\rm MSF}}{L_{\rm SMF}+ L_{\rm MSF}} \tag{7}\]  
The mode area (A\({}_{\rm eff}\)) can be expressed as [20]:  
\[\rm A_{eff}\ =\ \frac{\left(\int\int_{S}[E]^{2}dxdy\right)^{2}}{\left(\int \int_{S}[E]^{4}dxdy\right)^{2}} \tag{8}\]  
Here, E is the propagation amplitude of the transverse electric field in MSF, and S is the mode field area of MSF. The radius of mode field of MSF, \(\omega_{\rm MSF}\), should then be gained by the equation S = \(\pi\omega_{\rm MSF}\)\({}^{2}\).