## 2 Fiber Parameters
### Dispersion Parameters  
The total dispersion, i.e., chromatic dispersion in any waveguide arises from the combined effect of material dispersion and waveguide dispersion. Material dispersion is a result of the dependence of refractive index on the wavelength of light in that particular material. On the other hand, waveguide dispersion arise from the spectral dependence of the propagation constants of the modes supported by the waveguide. Since the modal properties depend on the geometry of the waveguide, the total dispersion of the propagating modes heavily depend on the fiber geometry in the case of optical fibers.  
In this work, we consider a W-type fiber design, that consists of a narrow core of larger refractive index (RI) at the center, followed by an annular trench of low RI and another annular cladding of intermediate RI at the periphery (see Figure 1). The refractive index \(RI\) profile of the W-type fiber is defined as a function of the radial distance \(r\) (transversal to the fiber) as:  
\[RI(r)=\left\{\begin{array}{ll}N_{1}&0\leq r<\frac{a}{2}\\ N_{2}&\frac{a}{2}\leq r<\frac{b}{2}\\ N_{3}&r\geq\frac{b}{2}\end{array}\right. \tag{1}\]  
For our geometry, \(N_{1}>N_{3}>N_{2}\). We consider a weakly guiding case, i.e., \(N_{1}-N_{2}\) and \(N_{2}-N_{3}\) are small. The mode field \(\psi(r,\phi)\), where \(\phi\) is the azimuthal angle in the cylindrical coordinate system is governed by [23]:  
\[\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\psi}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\phi^{2}}+(k_{0}^ {2}\
u^{2}-\beta^{2})\psi=0, \tag{2}\]  
where \(k_{0}\) is the free space wave number and \(\
u\) is the refractive index of silica. We rewrite Equation (2) as an eigenvalue equation as:  
\[[\
abla_{t}^{2}+k_{0}^{2}\
u^{2}]\psi=\beta^{2}\psi, \tag{3}\]  
where \(\
abla_{t}\) is the transverse Laplacian operator. Then using a mode solver, we found the eigenvalues, i.e., \(\beta\) for different core radii, wavelength and W-ratio. The numerical  
Figure 1: An illustration of the sinusoidal longitudinal variation of the core, trench and cladding radius of a W-type fiber. Input configurations of a dual-pump and single pump PSA are shown in left insets. A qualitative refractive index (\(RI\)) profile in arbitrary units along the transverse direction (\(y\)-direction) is shown in solid red line in th right inset. \(a\) and \(b\) are the core and trench diameters respectively (not to scale).  
calculations for evaluation of \(\beta\) were validated with the data provided in Ref. [39]. Also, within a core radius of 2.3 \(\upmu\)m, a single mode operation was ensured.  
In order to find the dispersion coefficients, the propagation constant \(\beta(\omega)\), is found as a function of frequency, \(\omega\) and is expanded around a central angular frequency \(\omega_{0}\) in a Taylor series. As is standard in PSA calculations, truncating the series up to fourth order terms gives us:  
\[\beta(\omega)\approx\beta_{0}+\beta_{1}(\omega-\omega_{0})+\beta_{2}\frac{( \omega-\omega_{0})^{2}}{2}+\beta_{3}\frac{(\omega-\omega_{0})^{3}}{6}+\beta_{ 4}\frac{(\omega-\omega_{0})^{4}}{24}, \tag{4}\]  
where \(\beta_{m}=\left(\frac{d^{m}\beta}{d\omega^{m}}\right)_{\omega=\omega_{0}}\) with \(m=1,\ 2,\ 3,\ 4\). Throughout our discussion, we will adhere to a maximum value of \(m=4\), except for the analytical modeling, where \(m\) is often truncated to 2. In Figure 2, we plot \(\beta_{2}\) (a), \(\beta_{3}\) (b) and \(\beta_{4}\) (c) as a function of the core radius and the W-ratio.  
For a DOF, the longitudinal variation of the \(m\)-th order dispersion \(\beta_{m}\) is defined as:  
\[\beta_{m}(z)=\overline{\beta}_{m}+\bar{\beta}_{m}\sin\left(\frac{2\pi z}{z_{ \text{mod}}}+\varphi_{\text{mod}}\right), \tag{5}\]  
where \(\overline{\beta}_{m}\) is the average \(m\)-th order dispersion, \(\bar{\beta}_{m}\) is the strength of dispersion modulation, \(z_{\text{mod}}\) and \(\varphi_{\text{mod}}\) are the wavelength and phase of the modulation respectively. \(z\) is the light propagation direction.