Advances in Condensed Matter Physics

Volume 2017, Article ID 1268230, 9 pages

https://doi.org/10.1155/2017/1268230

## Dynamics of Dispersive Wave Generation in Gas-Filled Photonic Crystal Fiber with the Normal Dispersion

^{1}School of Electrical Engineering, University of South China, Hengyang 421001, China^{2}Oriental Science and Technology College, Hunan Agricultural University, Changsha 410082, China

Correspondence should be addressed to Meng Zhang; moc.361@876420gnemgnahz

Received 18 May 2017; Accepted 1 August 2017; Published 30 August 2017

Academic Editor: Xiaofeng Zhou

Copyright © 2017 Zhixiang Deng and Meng Zhang. 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

The absence of Raman and unique pressure-tunable dispersion is the characteristic feature of gas-filled photonic crystal fiber (PCF), and its zero dispersion points can be extended to the near-infrared by increasing gas pressure. The generation of dispersive wave (DW) in the normal group velocity dispersion (GVD) region of PCF is investigated. It is demonstrated that considering the self-steepening (SS) and introducing the chirp of the initial input pulse are two suitable means to control the DW generation. The SS enhances the relative average intensity of blue-shift DW while weakening that of red-shift DW. The required propagation distance of DW emission is markedly varied by introducing the frequency chirp. Manipulating DW generation in gas-filled PCF by the combined effects of either SS or chirp and three-order dispersion (TOD) provides a method for a concentrated transfer of energy into the targeted wavelengths.

#### 1. Introduction

The high design flexibility of photonic crystal fibers has attracted the attention of many researchers in recent decade [1]. Kagomé-lattice hollow-core PCF reported in 2002 represents a milestone in the development of microstructured fibers [2]. It demonstrates numerous additional significance properties by filling its hollow core with gas [3]. The nonlinearity and the GVD can be remarkably controlled by adjusting gas pressure or replacing of gas species [3, 4]. The Raman-related effect disappears when the hollow core is filled with high pressure monatomic gases such as Ar, also the zero GVD point of fiber can be artificially adjusted from the ultraviolet to the near-infrared [4]. These features highly increase the versatility of hollow-core PCF, which make it an ideal platform to investigate different nonlinear optical effects.

As we all know, optical spectrum broadening and broadband frequency conversion are inherent features for nonlinear optical processes [5]. Dispersive wave (DW), also called nonsolitonic radiation or Cherenkov radiation [6], is particularly important for supercontinuum generation, broadband light sources, and broadband frequency conversion in fiber optics, and manipulating DW generation is a technique with a concentrated transfer of energy into a narrow spectral band. The generation of DW by intense optical pulses propagating, in particular, in photonic crystal fibers has been extensively studied in the past 30 years [6–9]. However, it has been originally presented at the background of the propagation of higher-order soliton. In that situation, higher-order soliton was propagated in the anomalous GVD regime which is generally perturbed by the third- and higher-order dispersion [10–13]. In our previous work, the roles of the self-steepening (SS) effect in the generation and controlling of DW in metamaterials are disclosed [14], and in photonic crystal fiber, the manipulation of DW by the frequency chirped is unfolded [15, 16].

In the time-domain, DW emission is described as the resonant amplification of a linear wave propagating with the same phase velocity as that of the soliton [17]. Recently, in the frequency-domain, phase-matched cascaded four-wave mixing (CFWM) is identified as the nonlinear origins of the DW generation process [18–20]. It was demonstrated that the DW emission is no longer thought to be the exclusive of solitons because the dispersive wave can be emitted even when pumping in the normal dispersion regime in the presence of a zero GVD wavelength [21]. In this regime, the physical origin of DW emission, which is perturbed by high-order dispersion, is intimately related to the dispersive shock waves resulting from the nonlinearity overbalancing a weak second-order dispersion [22]. The expression of the detuned frequency of dispersive waves can be accurately determined by the phase-matching selection rules, which involve the velocity of the dispersive shock waves due to emerging from a gradient catastrophe [22–24]. The roles of high-order dispersions in the generation and controlling of DW are also exposed [24–26]. However, to the best of our knowledge, so far the effect of the self-steepening and initial frequency chirp on DW generation with pumping in normal dispersion regime has not been discussed yet.

In the present paper, we demonstrate that the controllable generation of DW with pumping in the normal GVD dispersion can be realized by two means: either considering the SS effect of fiber or introducing the frequency chirp of the initial input pulse. The paper is organized as follows. In the second section, the defocusing nonlinear Schrödinger equation (dNLSE) for ultrashort pulse propagation in gas-filled photonic crystal fibers with TOD and SS effect is introduced. In the third section, we discuss the controllable DW generation and reveal the roles of the SS effect in the red-shifted and blue-shifted DW generation. In the fourth section, the impact on the DW generation on the basis of the different frequency chirp of the initial input pulse is investigated. Finally, we summarize our results.

#### 2. Numerical Model

To technically elucidate the mechanism of resonant dispersive wave emission, our numerical model is based on the following normalized form of the defocusing nonlinear Schrödinger equation with the Raman term removed [27]: Note that the Raman scattering effect is absent in the noble gas-filled PCF such as Ar considered here. We have introduced the normalized variables , , and , where is the duration of the launched pulse, is the power of the input field, is the group velocity at the central frequency , and is the th-order dispersion length and is the nonlinear length. Note that the defocusing feature arises from the assumption of the normal group velocity dispersion and nonlinear coefficient . and represent the normalized nonlinear coefficient and the normalized three-order dispersion coefficient, respectively. The nonlinear term in the right-hand side of (1) consists of the Kerr effect term and the shock derivative term , which gives rise to a frequency-dependent nonlinear coefficient. However, in our numerical simulation, the dispersion expansion can be truncated to the first correction to GVD, that is, third-order dispersion, whereas all the higher-order dispersive terms can be safely neglected.

To quantify ensemble frequency changes of the DW during propagation process, we introduce the intensity-weighted central frequency of the DW as a function of propagation distance [28]:Dispersion effects are described by the first term on the right-hand side of (1), where the range of integration (from to ) is selected to be no more than −30 dB compared to the maximum intensity of the DW and represents a function of the spectral intensity of the DW with propagation distance.

In order to obtain more intuitive information of the intensity distribution of DW, the performance of the continuum spectral distribution is characterized by the decibel scale of the relative intensity [29] and the relative average intensity of DWThe decibel scale used here permits us to clearly show the low-intensity radiation. However, if the intensity spectrum is presented in the form , its value is so small that can be submerged with the background of the strong pulse.

#### 3. Manipulating the Dispersive Wave Generation by Self-Steepening Effect

It was well-known that the SS effect in gas-filled photonic crystal fiber can not be ignored in practice [4]. The pulse propagating in the normal dispersion regime is also affected by the higher-order dispersion and nonlinear terms. Thus, the combined effects of the SS and TOD on the DW generation should be discussed.

To discuss the DW generation in the fiber, we employ the standard split step Fourier method to solve the dNLSE numerically. In the numerical simulation, the normalized input pulse is employed. For convenience, we only consider the DW generation under the condition of . The reason for the selection can be that efficient generation of DWs from a pump in the normal GVD region typically requires the nonlinear length of the pump to be much shorter than the dispersion length of the pump. To gain a physical understanding of the effects, the loss of fiber is neglected. If not otherwise specified, only the normal GVD is considered.

##### 3.1. Dispersive Wave Generation for Positive Dispersion Slope

In the normal GVD of gas-filled PCF, DW emitted by dispersive chock waves owing to the positive dispersion slope (i.e., the positive TOD) will be frequency downshifted with respect to the pump. Therefore, it is called red-shift DW.

Figure 1 shows how the self-steepening effect influences the red-shift DW generation; here the positive TOD () is considered. In Figures 1(a) and 1(b), we plot the spectral evolution of the pulse as a function of normalized propagation distance when the self-steepening effect is switched off (i.e., set ) and included (i.e., set ), respectively. Obviously, in the initial stages of propagation, the spectrum of the pump pulse displays strong and symmetrical broadening due to self-phase-modulation-induced pulse compresses in the presence of weak normal dispersion, but as the spectral tail of the broadened pulse overlaps with the phase-matched frequency, the occurrence of resonant energy transfer process can be seen. As the propagating distance increases, the DW emitted by dispersive shock waves begins to emerge in the red-shifted band. As can be seen by comparing Figure 1(a) with Figure 1(b), the SS effect is important for red-shift DW generation. For the two cases that the SS effect is excluded/included, the evolutions of the output spectra have the following characteristics: to begin with, the pulse spectra become narrow. When the SS effect is included, the spectrum narrowing is obvious. Therefore, this is a disadvantage to the supercontinuum generation. Secondly, the central frequency of the red-shift DW moves away from the spectral body of the residual pump pulse. In other words, the SS effect leads to the increase of the frequency detuning from the pump. The cause of this result is that the frequency of DW is determined by the phase-matched condition associated with the nature group velocity, but SS increases the group velocity at resonant frequency eventually because it affects the group velocity in an intensity dependent fashion [30]. As can be seen from Figure 1(c), the shifting is not remarkable, which indicates that the influences of the SS effect on the frequency shifts of red-shift DW are less important than TOD. Finally, one conspicuous observation is that the relative average intensity of red-shift DW is weakened. The spectral intensity of the DW is related to the strength of the pump spectrum at the phase-matched frequency [31, 32]. Since the SS effect asymmetrically weakens the spectrum toward the red, the efficiency of DW generation dropped. As shown in Figure 1(d), the relative average intensity of red-shift DW is −56 dB when the SS effect set and drops to −75 dB when the SS effect set .