# On the Impact of Wavelength Dependency on Supercontinuum Generation in Photonic Crystal Fibers
## 1 Introduction  
In applied optics, the introduction of photonic crystals [1] marked an important milestone in the technological ability to control and manipulate light. The photonic crystal functionality is based on its micrometer or nanometer periodic structure scale, so that electromagnetic waves in the visible range will be affected by effects analogous to those suffered by an electron at a periodic potential. This creates a pattern through which photons cannot propagate in certain directions, with different polarization or for certain wavelengths. This is due to the formation of what are known as photonic band gaps. Their application is quite widespread; it has even recently been shown that they can be used for polarization control in metamaterials [2].  
The idea of bringing these microstructures to the fiber optic waveguides field led to the introduction of the photonic crystal fibers (PCFs) [3, 4], which paved the way to a fresh bunch of new research fields for the photonics community. We can study their use as sensors [5], in high-power generation [6], or as fiber lasers [7], to cite some. Among these new applications, supercontinuum generation by means of a PCF emerged as an easy way to cause a significant broadening in the spectral bandwidth of an input pulse propagating through the fiber [8, 9, 10].  
Since the fabrication process of a PCF is expensive, it is mandatory to previously simulate the characteristics and performance of the design to be sure it will offer the expected results [11].  
The simulation process consists of two clearly differentiated parts: the characterization of the fiber parameters solving the Helmholtz equation and the study with the generalized nonlinear Schrodinger equation (GNLSE) of the propagation of the pulse under the effects of the previously determined parameters [12, 13]. In our work, the Helmholtz equation can be treated as an eigenvalue problem [14], and the GNLSE will be solved using spectral techniques.  
From a PCF characterization model, a set of curves that relates dispersion parameters with the light wavelength inside the fiber has been obtained [15]. Once these parameters are known, the GNLSE shows the pulse evolution in the fiber both in the time and spectral domains.  
There is nothing new in what has been exposed so far; both are well-known problems that have been solved a long time ago. But, while the characterization of the PCF determines the relation between all its parameter and the corresponding wavelength, it is common practice to assume in the propagation a constant working wavelength, usually associated with the center of the pulse [16]. This is an approximation that could be taken in systems where the spectral broadening of the pulse during the simulation is negligible, but the supercontinuum generation is the totally opposite case to that assumption, because our goal is to broaden the spectrum as much as possible [17].  
The aim of this work is to study the influence of the wavelength dependence of non-linear and dispersion parameters on the pulse propagation and its effect on the spectrum. Thus, we analyzed if the impact justifies the implementation of this dependency in the usual methods that solve the GNLSE when supercontinuum generation occurs. As it will be shown in the Conclusions, this is the case.  
The structure of the paper is as follows. Section 2 will introduce the equations mentioned before, starting with the PCF parameters' characterization through the Helmholtz equation and their application to the proposed structure; secondly, the GNLSE is presented, taking into account the different effects that it includes and its associated terms, as well as the spectral method used to solve it. The supercontinuum generation will also be presented there. Section 3 collects all the results from the simulations and the comparison data between the constant and wavelength-dependent cases. The conclusions will be presented at the end of the whole study.