# On the Impact of Wavelength Dependency on Supercontinuum Generation in Photonic Crystal Fibers
## 2 Materials and Methods
### Helmholtz Equations and Photonic Crystal Fiber Characterization  
As explained in the Introduction, the study of the propagation of light through an optical fiber begins with the simulation of its characteristic parameters, which are determined by the internal structure and materials of the fiber. The governing equations of the interaction between this structure and the electric and magnetic fields inside the fiber are the Helmholtz equations [14]:  
\[\{\
abla_{\mathbf{t}}^{2}+k_{0}^{2}e+\
abla_{\mathbf{t}}[\
abla_{\mathbf{t}}( ln(e)]\}\mathbf{e_{t}}=-\gamma^{2}\mathbf{e_{t}} \tag{2}\]  
\[[\
abla_{\mathbf{t}}^{2}+k_{0}^{2}e+\
abla_{\mathbf{t}}(ln(e)\times\
abla_{ \mathbf{t}}\times]\mathbf{h_{t}}=-\gamma^{2}\mathbf{h_{t}} \tag{3}\]  
where \(\mathbf{e_{t}}\) and \(\mathbf{h_{t}}\) are the transversal components of the electric and magnetic fields for the guided mode, \(\gamma\) the complex propagation constant, \(k_{0}\) the wavenumber associated with the input pulse, and \(e\) the dielectric distribution inside the fiber, which depends on its internal structure and constituent materials. By solving one of the equations, [PERSON]'s equations allow expressing this solution in terms of the other field distribution.  
As can be seen, each of the Helmholtz equations can be expressed as an eigenvalue problem, where the eigenvalues are related to the effective index of the PCF and the eigenvectors coincide with the field distributions for the guided modes.  
Starting with the eigenvalue, the effective index can be expressed as:  
\[n_{eff}=-\frac{i}{k_{0}}\gamma=-\frac{i}{k_{0}}(\alpha+i\beta)=\frac{\beta}{k_ {0}}-i\frac{\alpha}{k_{0}} \tag{4}\]  
where \(\beta\) is the real propagation parameter and \(\alpha\) the loss parameter.  
From \(\beta\), the dispersion orders can be calculated through successive derivation [20]:  
\[\beta_{n} =\frac{\partial^{n}\beta}{\partial\omega^{n}} \tag{5}\] \[\omega =\frac{2\pi c}{\lambda} \tag{6}\]  
Figure 1: Model of the studied PCF. The dimensions in this device are: a core radius of 1.275 \(\upmu\)m made of Si, in red in the figure. The air holes have a diameter of 1.2 \(\upmu\)m and the lattice constant is \(\Lambda=2.5\)\(\upmu\)m. The cladding, in blue, is made of MgF\({}_{2}\). The arrow accounts for three periods of the lattice constant (see Ref. [16]).  
On the other hand, the integration of the field distributions obtained from the eigenvectors of the system allow calculating the effective area of the modes [20]:  
\[A_{eff}=\frac{\left(\int_{-\infty}^{\infty}|F(x,y)|^{2}\mathrm{d}x\mathrm{d}y \right)^{2}}{\int_{-\infty}^{\infty}|F(x,y)|^{4}\mathrm{d}x\mathrm{d}y} \tag{7}\]  
where \(F(x,y)\) refers to the optical field. This parameter represents the light-carrying section of the fiber and is related to the nonlinear parameter, where \(n_{2}\) is the second-order nonlinear refraction index:  
\[\gamma=\frac{n_{2}\omega_{0}}{cA_{eff}} \tag{8}\]  
The whole set of parameters presented at this point has a strong dependency on the wavelength, which is introduced in the system through the \(k_{0}\) value in Equations (2) and (3). In addition, another important factor is the \(\epsilon\) parameter, the dielectric constant, which models the electrical response of the constitutive materials in the PCF.