Controllable behaviors of Peregrine soliton with two peaks in a birefringent fiber with higher-order effects

Abstract

A coupled variable-coefficient higher-order nonlinear Schrödinger equation in birefringent fiber is discussed, and an analytical Peregrine soliton solution with two peaks is derived by means of the Darboux transformation method. The controllable behaviors of this Peregrine soliton with two peaks are investigated in a periodic dispersion system and a dispersion decreasing fiber with exponential profile. In these systems, the effective propagation distance Z appears a maximum $Z_\mathrm{m}$. Comparing this maximum with values of peak location $Z_\mathrm{p}$, the initial excitation, peak excitation, rear excitation and periodic excitation are constructed.

1 Introduction

Nonlinear wave excitations and various solitons, such as autonomous soliton [1, 2], shock waves [3], spatial soliton [4], dipole soliton [5], Kuznetsov-Ma soliton [6] and light bullet [7], exist in various nonlinear optical systems. As one of research focus, dynamical behaviors of rogue waves (RWs) [8] were intensively studied in different domains of physics, especially in fluid dynamics [9], nonlinear optical systems [10] and plasmas [11].

As is well known, the Peregrine solitons (PS) [12] have been reported as one of the theoretical prototypes to describe RWs. In optics, Solli et al. [13] firstly introduced the concept of RW in a photonic crystal fiber. After then, many groups experimentally observed optical rogue soliton in different contexts [14–16]. Because it is significant and interesting to understand theoretically the control and use of RWs, many authors [17–19] investigated controllable behaviors of RWs. However, these investigation are all related to the standard nonlinear Schrödinger equation (NLSE), and the higher-order effects such as the self-steepening (SS) and self-frequency shift (SFS) are less considered when authors studied RWs.

In this paper, we considered a coupled NLSE with higher-order effects and discussed controllable behaviors of PS with two peaks in a birefringent fiber. Comparing the maximal effective propagation distance $Z_\mathrm{m}$ with values of peak location $Z_\mathrm{p}$, controllable behaviors, such as the initial excitation, peak excitation, rear excitation and periodic excitation, are discussed in a periodic dispersion system and a dispersion decreasing fiber (DDF) with exponential profile.

2 Model and Peregrine soliton solution

Considering the presence of birefringence, single-mode fibers are actually bimodal. This birefringence creates two principal transmission axes within the fiber known as the fast and slow axes. When traveling in a medium, an ultrashort light pulse will induce a varying refractive index of the medium, which will produce a phase shift in the pulse called self-phase modulation (SPM). When two or more optical fields with different frequencies co-propagate in a fiber, the cross-phase modulation (XPM) will be produced through the optical Kerr effect. When short pulse is considered (nearly 50fs), the third-order dispersion (TOD), which will produce asymmetrical broadening in the time domain for the ultrashort soliton pulses [20], cannot be neglected. Moreover, the higher-order nonlinear effects such as the SS [21] and SFS [22] must be considered.

In a real fiber, the variation of the fiber geometry (diameter fluctuations, etc.) brings to the inhomogeneous core medium [23] and thus the governing equation is the following higher-order coupled NLSE with variable coefficients [24]

$$\begin{aligned}&\text {i}q_{jz}-\frac{1}{2}\beta _2(z) q_{jtt}-\gamma (z)\left( \sum ^2_{n=1}a_{nj}|q_n|^2\right) q_j\nonumber \\&\quad +\,\text {i} \beta _3(z) q_{jttt}+\text {i} \chi (z)\left( \sum ^2_{n=1}a_{nj}|q_n|^2\right) q_{jt}\nonumber \\&\quad +\, \text {i} \delta (z)\left( \sum ^2_{n=1}a_{nj}q_{nt}q_j^*\right) q_{j}+\text {i}\Gamma (z)q_j=0, \end{aligned}$$

(1)

where $q_j(z,t)$ with $j=1,2$ denote two normalized complex mode fields, z and t represent dimensionless propagation and retarded time. In Eq. (1), $\beta _2(z),\gamma (z),\beta _3(z),\chi (z),\delta (z)$ and $\Gamma (z)$ are coefficients of group velocity dispersion (GVD), the nonlinearly coupled terms of the SPM and XPM, TOD, SS and SFS, and loss and gain, respectively. The ($*$) denotes the complex conjugate, and subscripts z and t denote the derivatives with respect to z and t. The constants $a_{nj}$ decide the ratio of the coupling strengths of the XPM to the SPM. For linearly polarized eigenmodes $a_{11}=a_{22}=1,a_{12}=a_{21}= 2/3$, whereas for circularly polarized modes $a_{11}=a_{22}=1,a_{12}=a_{21}= 2$ with elliptically polarized eigenmodes $a_{11}=a_{22}=1,2/3<a_{12}=a_{21}< 2$ [25]. In Ref. [24], dispersion management and cascade compression of femtosecond nonautonomous soliton have been discussed via Darboux transformation method; however, the PS has not been discussed. Note that here we consider the special case with the same wavelength for both polarization components. Moreover, we neglect the degenerate four-wave mixing. If the fiber length $L\gg L_\mathrm{B}$ (beat length), the four-wave-mixing term changes sign often and its contribution averages out to zero. In highly birefringent fibers ($L_B \sim 1$ cm typically), the four-wave-mixing term can often be neglected for this reason [26].

When all coefficients in Eq. (1) are constant, bright soliton has been obtained [27]. Moreover, the coupled NLSE with higher-order effects can also describe short pulse propagation in the dual-core photonic crystal fiber [28, 29]. Different from the numerical study in Refs. [28, 29], here we analytically study the propagation of soliton. The variable-coefficient model here can describe more general situation than the constant-coefficient one in Ref. [27]; however, the solving procedure for the variable-coefficient model (1) here is more complicated than that for the constant-coefficient one in Ref. [27]. Considering bright soliton has been discussed in Ref. [27], we focus on another new soliton structure, that is, the novel PS with two peaks.

Under the condition

$$\begin{aligned}&\beta _3(z):\beta _2(z):\gamma (z):\chi (z):\delta (z)=1:2 \left( \frac{k}{s\gamma _3}-3p\right) :\nonumber \\&\quad -\,\frac{k^2[ps(2a_f+s_a)+k)}{s\gamma _3A_0^2}\exp \left[ 2\int _0^z \Gamma (s) \hbox {d}s\right] :\nonumber \\&\quad -\,\frac{2k^2a_f}{A_0^2\gamma _3}\exp \left[ 2\int _0^z \Gamma (s) \hbox {d}s\right] :\nonumber \\&\quad -\frac{k^2s_a}{A_0^2\gamma _3}\exp \left[ 2\int _0^z \Gamma (s) \hbox {d}s\right] , \end{aligned}$$

(2)

if we use the transformation

$$\begin{aligned} \left\{ \begin{array}{cc}q_1\\ q_2 \end{array}\right\}= & {} A(z)\exp {[\text {i}\phi (z,t)]}Q\left[ Z(z),T(t)\right] \nonumber \\&\times \,\left\{ \begin{array}{cc}|a_{22}-a_{12}|^{\frac{1}{2}}\\ |a_{11}-a_{21}|^{\frac{1}{2}} e^{\text {i}\vartheta }\end{array}\right\} , \end{aligned}$$

(3)

with the amplitude $A(z)=A_0\exp {[-\int _0^z\Gamma (s)\hbox {d}s]}$, phase $\phi (z,t)=p\left[ t+p\left( \frac{k}{s\gamma _3}-4p\right) \int _0^z \beta _3(s)\hbox {d}s\right]$, the effective propagation distance $Z(z){=}-\frac{k^3}{s\gamma _3}\int _0^z\beta _3(s)\hbox {d}s$, intermediate variable $T(t)= k[t+p\left( 9p-\frac{2k}{s\gamma _3}\right) \int _0^z\beta _3(s)\hbox {d}s]$ and constants $k,p,\vartheta$, then Eq. (1) can be transformed into the constant-coefficient higher-order NLSE [30]

$$\begin{aligned}&\text {i} Q_Z+\frac{1}{2} Q_{TT}+|Q|^{2}Q+\text {i}s[a_fQ(|Q|^2)_T\nonumber \\&\quad +\,s_a(|Q|^2Q)_T-\gamma _3Q_{TTT}]=0, \end{aligned}$$

(4)

with three independent parameters $s_a , a_f$ and $\gamma _3$ controlling the relative contribution of SFS, SS and TOD.

Note that the one-to-one correspondence (3) has been constructed between the complicated variable-coefficient model (1) and the terse constant-coefficient model (4). Therefore, various solutions of Eq.(1) can be obtained by means of the relation (3) and solutions of Eq.(4). Obviously, the advantage of this method is that the correspondence (3) simplifies the solving procedure of complicated variable-coefficient model (1) by firstly solving easier constant-coefficient model (4).

According to the modified Darboux transformation technique in Ref. [30], we can obtain the novel PS solution with two peaks as follows

$$\begin{aligned}&\left\{ \begin{array}{cc}q_1\\ q_2 \end{array}\right\} =A(z)\exp {\{\text {i}\left[ \phi (z,t)+Z\right] \}}\nonumber \\&\quad \times \left\{ \frac{4}{D}(1+2\text {i}Z')-1+\frac{\text {i}s}{D^2}\left[ M(Z',T')-\text {i}N(Z',T')\right] \right\} \nonumber \\&\quad \times \left\{ \begin{array}{cc}|a_{22}-a_{12}|^{\frac{1}{2}} \\ |a_{11}-a_{21}|^{\frac{1}{2}} e^{\text {i}\vartheta }\end{array}\right\} , \end{aligned}$$

(5)

where $D=1+4Z'^2+4T'^2,M(Z',T')=8T'\{4(a_f+6\gamma _3+2s_a)T'^2+12[a_f+2(\gamma _3+s_a)]Z'^2-3a_f-6\gamma _3-4s_a\}, N(Z',T')=32(3a_f+6\gamma _3+5s_a)T'Z'$ with $Z'=Z-sZ_{0},T'=T-sT_{0}$. Here these expressions of Z, T and $\phi$ are shown below Eq. (3), and $Z_0$ and $T_0$ determine the center of solution in $Z-T$ coordinates.

3 Controllable behaviors of PS with two peaks

The higher-order NLSE (4) has PS solution as [30]

$$\begin{aligned} Q= & {} \left\{ \frac{4}{D}(1+2\text {i}Z')-1+\frac{\text {i}s}{D^2}[M(Z',T')\right. \nonumber \\&\quad \left. -\text {i}N(Z',T')]\right\} \exp {(\text {i}Z)}. \end{aligned}$$

(6)

As shown in Fig. 1, this solution becomes asymmetric relative to mirror images along axes $Z=0$ and $T=0$ and possesses two-peak structure. It is also one kind of wave that appeared from nowhere and disappeared without a trace (WANDT) like the original PS [31].

Fig. 1

(Color online) a The PS with two peaks for Eq.(4) in the $Z-T$ coordinates and b the corresponding sectional plot at $Z=2$ and $Z=3$. Parameters are chosen as $\gamma _3=0.2$, $a_f=1$, $s_a=1$, $s=0.25$, $T_0=5$, $Z_0=10$

From solution (5), we know its core is solution (6). However, solution (5) not only possesses the property of WANDT, but also can be controlled to realize different excitations, which does not appear for solution (6) in the frame of higher-order NLSE (4). This new property originates from the restraint for value of Z by the expression $Z(z)=-\frac{k^3}{s\gamma _3}\int _0^z\beta _3(s)\hbox {d}s$, that is, Z is not an arbitrary value from 0 to infinity. Here, we choose $a_{11}=a_{22}=1,a_{12}=a_{21}= 2/3$.

We analyze the controllable behaviors of PS solution (6) in the following periodic distributed amplification system with the periodic varying TOD parameter [32]

$$\begin{aligned} \beta _{3}(z)=\beta _{30}\cos (\eta z)\exp (-\sigma z), \end{aligned}$$

(7)

where the parameters $\eta$ and $\sigma$ control the rate of TOD change inside the fiber. In particular, the constant TOD can be obtained by $\eta =\sigma =0$. When $\sigma =0$, this system (7) is the periodic dispersion system [10]. When $\eta =0$, this system (7) with $\sigma >0$ is the DDF with exponential profile [33].

In DDF, from expression of Z below Eq.(3), we obtain $Z=-k^3\beta _{30}[1-\exp (-\sigma z)]/(s\gamma _3\sigma )$. When $\sigma >0$, the maximum value is $Z_\mathrm{m}=-k^3\beta _{30}/(s\gamma _3\sigma )$ as $z\rightarrow \infty$. Moreover, peaks of PS appear some fixed locations $Z_\mathrm{p}$ along the Z-axis in the frame of higher-order NLSE in Fig. 1. Therefore, we can adjust the value of the maximum $Z_\mathrm{m}$ compared with values of peak locations $Z_\mathrm{p}$ in order to control the degree of excitation of the PS structure in Fig. 1.

When $Z_\mathrm{m}<Z_\mathrm{p}$, the critical value of exciting PS is not enough. As shown in Figs. 2a, b, the PS is initially excited and maintains this shape self-similarly. If $Z_\mathrm{m}=Z_\mathrm{p}$, the PS is excited to the magnitude of peak, and sustains its two-peak structure along the propagation distance in a self-similar form (See Fig. 2c, d). When $Z_\mathrm{m}>Z_\mathrm{p}$, the excitation of rear of PS with two peaks does not finish. The PS is excited to the rear part, and the rear part looks like a tail and sustains its magnitude a long distance from Fig. 2e, f. Thus, the full excitation of the PS cannot be realized.

Fig. 2

(Color online) Controllable behaviors of PS with $|u|=|q_1|$ in DDF: a initial excitation, c peak excitation and e rear excitation. b, d and f are sectional plots corresponding to a, c and e at different z. Parameters are chosen as $k=0.4,\beta _{30}=-0.05,\Gamma =-0.01,A_0=0.5,p=1, \eta =0$ with a $\sigma =0.035$, b $\sigma =0.0256$ and c $\sigma =0.005$. Other parameters are chosen as those in Fig. 1

In periodic dispersion system, from expression of Z below Eq. (3), we obtain $Z=-k^3\beta _{30}\sin (\eta z)/(s\gamma _3\eta )$. This expression hints that Z is limited to the range of $|Z|<|Z_\mathrm{m}|=|k^3\beta _{30}/(s\gamma _3\eta )|$. When $|Z_\mathrm{m}|<Z_\mathrm{p}$, the critical value of exciting PS is not reached, and only initial shape of PS is excited. This maximum is modulated by sin-function, and thus, this initial part appear periodically in Fig. 3a. If $|Z_\mathrm{m}|>Z_\mathrm{p}$, the PS will be firstly excited at $z=-\arcsin [Z_\mathrm{p}s\gamma _3\eta /(\beta _{30}k^3)]/\eta$ and then recur self-similarly in Fig. 3b. From it, we find that the orientation of PS changes alternately, and PS exhibits asymmetric layout relative to mirror image.

Fig. 3

Controllable behaviors of PS with $|u|=|q_1|$ in periodic system: a periodic initial excitation and b periodic complete excitation. Parameters are chosen as $\sigma =0$ with a $\eta =1.2$ and b $\eta =0.2$. Other parameters are chosen as those in Fig. 2

From these examples, we know that controllable behaviors can be realized by adjusting the parameter $\sigma$ in DDF and the parameter $\eta$ in periodic dispersion system when other parameters are fixed.

4 Conclusions

In summary, we discussed a coupled variable-coefficient higher-order NLSE in birefringent fiber, and obtained an analytical Peregrine soliton solution with two peaks via the Darboux transformation method. Moreover, we studied controllable behaviors of PS with two peaks in a DDF and a periodic dispersion system. In the DDF, the effective propagation distance Z appears a maximum $Z_\mathrm{m}$. Comparing this maximum with values of peak location $Z_\mathrm{p}$, the initial excitation, peak excitation and rear excitation are constructed by adjusting the parameter $\sigma$ in DDF, and periodic initial excitation and complete excitation are built by modulating the parameter $\eta$ in periodic dispersion system when other parameters are fixed. These results will stimulate study RWs for shorter pulse width when higher-order effects are considered.