Mathematical Problems in Engineering

Volume 2017, Article ID 3517407, 6 pages

https://doi.org/10.1155/2017/3517407

## Influence of the Wavelength Dependence of Birefringence in the Generation of Supercontinuum and Dispersive Wave in Fiber Optics

Facultad de Ciencias, Escuela de Fisica, Universidad Nacional de Colombia, Medellin, A.A 3840, Medellín 20036, Colombia

Correspondence should be addressed to Rodrigo Acuna Herrera; oc.ude.lanu@anucar

Received 19 January 2017; Accepted 23 May 2017; Published 15 June 2017

Academic Editor: Ivan D. Rukhlenko

Copyright © 2017 Rodrigo Acuna Herrera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we perform numerical analysis about the influence of the wavelength dependence of birefringence (WDB) in the Supercontinuum (SC) and dispersive wave (DW) generation. We study different birefringence profiles such as constant, linear, and parabolic. We see that, for a linear and parabolic profile, the generation of SC practically does not change, while this does so when the constant value of the birefringence varies. Similar situation happens with the generation of dispersive waves. In addition, we observe that the broadband of the SC increases when the Stimulated Raman Scattering (SRS) is neglected for all WDB profiles.

#### 1. Introduction

The SC generation has attracted a lot of attention since it has found numerous applications in the fields of telecommunication [1], optical metrology [2], ultrafast coherence spectroscopy, and biological processes [3]. The mechanisms for SC generation in nonlinear fiber optics (FO) have been studied both numerically and experimentally [4–6]. Subsequently, SC generation techniques have brought the design of tunable ultrafast laser sources [7]. In addition, SC generation sources have been used as a simple way to generate multiwavelength optical sources [1]. It is useful for dense wavelength-division-multiplexing (WDM) telecommunications. In the generation of SC, various processes are involved such as self- and cross-phase modulation, four-wave mixing, modulation instability, soliton fission, dispersive wave generation [8], and Raman scattering [9]. All these effects can contribute to creating new frequencies within the pulse spectrum. Numerous SC generation methods have been studied to get a better understanding of the mechanisms for which it is efficiently possible to generate and develop the SC laser. Within these methods, the use of Photonic Crystal Fiber has brought a lot of interests due to its highly nonlinear optics characteristics [10, 11].

The SC generation can be studied with the generalized nonlinear Schrodinger equation (NLSE), which models the propagation of optical pulses in nonlinear FO. NLSE has been used to analyze the influence of various parameters on the SC generation [12, 13]. In those studies, the WDB in the Supercontinuum generation has not been enough studied [14, 15]. In this work, we study numerically the influence of WDB in the SC and DW generation. We analyze different birefringence profiles such as constant, linear, and parabolic. Some of those profiles can be seen in references [16, 17].

We show that, for a linear and parabolic profile, the generation of SC and DW do not change, while they do so as the constant value of the birefringence varies. We also found that the broadband of the SC is wider when the Stimulated Raman Scattering (SRS) is neglected for all birefringence profiles.

The paper is structured as follows. Section 2 presents the fundamental theory of nonlinear pulse propagation in birefringent fiber optics. Section 3 describes the numerical calculation of SC and DW generation for different birefringent FO profiles by using the split-step Fourier method [9] and the four-order Runge-Kutta algorithm [18]. Finally, the conclusions are presented in Section 4.

#### 2. Nonlinear Pulse Propagation in Birefringent Fibers

The nonlinear pulse propagation, in birefringent optical fiber, can be written as follows [9]:where () corresponds to the normalized field amplitude in the ( or ( direction, is the relative time, is the absolute time, and is the propagation length. The constants are the dispersion parameters. is the normalized birefringence parameter, where and and are the birefringence and propagation constant in vacuum, respectively. is the dispersion length, with being the second-order dispersion parameter and being the initial pump pulse width. is the Raman coefficient and is the optical shock time scale. is the pump frequency. The remaining terms in (1) arewhere is the convolution operator, defined, for instance, as

The functions for in (3) are given by, , , andwith , , and . We ignore the SRS and for in (1) in order to analyze the DW behavior. The dispersive wave generation is characterized by the parameters and the soliton order , where is the third-order dispersion parameter. is the nonlinear length, where and are the initial peak power of the pump and the nonlinear coefficient, respectively.

#### 3. Numerical Results

For nonlinear pulse simulations, (1) is numerically solved by combining the split-step Fourier method [9] and the four-order Runge-Kutta algorithm [18]. The basic idea is to divide the equation into a dispersive and nonlinear operator; that is,where

The dispersive and nonlinear operators act together along the fiber. During numerical simulations, the fiber is divided into many* N*-sections with size ; at each section each operator acts independently; that is, at section , and acts; at the next section , , and acts and so on. The caseis solved in the Fourier domain according to

The exponential operator is calculated by the mathematical prescription:where represents the Fourier-transform operator, is obtained from (8) by replacing by , and is the frequency in the Fourier domain. The other caseis solved by using the four-order Runge-Kutta Method [18], where the following algorithm is implemented:for and , where . The constants , , , and are given by

For our simulations, we assume the dispersion terms are the same for and polarization and use the typical parameters shown in Table 1, where the length of propagation is 6 cm. We consider the birefringence profiles shown in Figure 1. They are linear increasing (profile 1), linear decreasing (profile 2), parabolic positive (profile 3), and parabolic negative (profile 4). We first compute the SC generation in both and polarization for six constant birefringence values between and . The results are shown in Figure 2. We see that the generation of SC varies as the birefringence does so. For each birefringence value, we see depths at different wavelengths.