# On the Impact of Wavelength Dependency on Supercontinuum Generation in Photonic Crystal Fibers
## 2 Materials and Methods
### Generalized Nonlinear Schrodinger Equation and Supercontinuum Generation  
The central idea behind supercontinuum generation is to obtain a broad continuous spectrum from a narrower input pulse after its propagation through a nonlinear media, the PCF in this case. Regarding the pulse, its initial delay should be in the femtoseconds regime, which allows the higher-order dispersion effects, among others, to act over it.  
The governing equation in this case is the GNLSE, the expression of which is [8]:  
\[\begin{split}\frac{\partial E}{\partial z}&=\frac{ \alpha}{2}E+\sum_{n\geq 2}i^{n+1}\frac{\beta^{n}}{n!}\frac{\partial^{n}E}{ \partial T^{n}}\\ &+i\gamma\bigg{(}E(z,t)\int_{-\infty}^{+\infty}R(T)\times|E(z,T -T^{\prime})|^{2}\,dT^{\prime}\bigg{)}\end{split} \tag{9}\]  
The three parts of Equation (9) that appear to the left correspond to the following: In the first term, \(\alpha\) accounts for the loss processes; the second part corresponds to the different partial derivatives that made the group velocity dispersion (GVD) effect; the \(\beta^{n}\) terms and its dependence on the wavelength is the main part of this work.  
Finally, the nonlinear effects including the Raman effect constitute the third part; here, \(\gamma\) is the nonlinear coefficient, which depends on the mode effective area \(A_{eff}\), Equation (7), and the nonlinear refractive index. The Raman effects are included through the nonlinear response function \(R(T)\), which accounts for the instantaneous electronic and delayed Raman contributions, defined as:  
\[R(t)=(1-f_{R})\delta(t)+f_{R}h_{R}(t) \tag{10}\]  
where \(f_{R}=0.043\) and \(h_{R}(t)\) is the Raman response function related to molecular vibrations [22].  
In order to resolve the generalized GNLSE, let us separate it into two parts, each one of these accounting for the two different physical effects this equation applies to: the GVD or linearpart and the self-phase modulation (SPM) or nonlinear part [20].  
After the proposed partitioning, the NLSE has the form:  
\[\frac{\partial E(z,t)}{\partial z}=(\hat{L}+\hat{N})E(z,t) \tag{11}\]  
Here, \(E\) represents the slowly varying envelope of the solution, \(z\) is the propagation distance, and \(\hat{L}\) and \(\hat{N}\) are the differential operators for the linear (GVD) and the nonlinear (SPM) parts of the equation, respectively.  
The separation between both terms is the basis of the time-splitting spectral method used in this work to solve the GNLSE.  
We start with the linear operator \(\hat{L}\):  
\[\hat{L}=\sum_{n\geq 2}i^{n+1}\frac{\beta^{n}}{n!}\frac{\partial^{n}}{\partial T ^{n}} \tag{12}\]  
Figure 2: Wavelength dependence obtained for linear parameters, \(\beta(\lambda)\), calculated, which appear in the generalized NLSE. The differences with a constant value approximation in the range of wavelengths studied are noticeable in all of them.  
In a dispersive medium, the GVD term is a result of the differences in the group velocity in a material with respect to the different frequencies that compound the pulse. For a time-splitting spectral approach, this linear part is resolved in the frequency Fourier space.  
The nonlinear operator \(\tilde{N}\):  
\[\tilde{N}=\alpha+i\gamma\bigg{(}E(z,t)\int_{-\infty}^{+\infty}R(T)\times|E(z,T- T^{\prime})|^{2}\,dT^{\prime}\bigg{)} \tag{13}\]  
In this case, we use the symmetric propagation scheme:  
\[E(\mathbf{x},t+\Delta t)\approx(e^{-\Delta t\tilde{L}/2}e^{-\Delta t\tilde{N }}e^{-\Delta t\tilde{L}/2})E(\mathbf{x},t) \tag{14}\]  
Among the multiple numerical algorithms developed to solve the previous equation, the split-step pseudospectral methods are the most-extended due to its accuracy and comparatively lower computational cost [12]. In this paper, the symmetrical split-step Fourier method (S-SSFM) [13] is applied in the simulations.