Combined effects of nonparaxiality, optical activity, and walk-off on rogue wave propagation in optical fibers filled with chiral materials

The generalized nonparaxial nonlinear Schrödinger (NLS) equation in optical fibers filled with chiral materials is reduced to the higher-order integrable Hirota equation. Based on the modified Darboux transformation method, the nonparaxial chiral optical rogue waves are constructed from the scalar model with modulated coefficients. We show that the parameters of nonparaxiality, third-order dispersion, and differential gain or loss term are the main keys to control the amplitude, linear, and nonlinear effects in the model. Moreover, the influence of nonparaxiality, optical activity, and walk-off effect are also evidenced under the defocusing and focusing regimes of the vector nonparaxial NLS equations with constant and modulated coefficients. Through an algorithm scheme of wider applicability on nonparaxial beam propagation methods, the most influential effect and the simultaneous controllability of combined effects are underlined, showing their properties and their potential applications in optical fibers and in a variety of complex dynamical systems.

Figure 1

Specific control parameters: the left- and right-hand side of the gain or loss differential term $\eta \left(\xi \right)$, nonparaxial parameter $d\left(\xi \right)$, and TOD $\gamma \left(\xi \right)$, where $\eta \left(\xi \right)=C_{T}sn\left(\xi ,k_7\right),d\left(\xi \right)=dn\left(\xi ,k_5\right),\gamma \left(\xi \right)=cn\left(\xi ,k_6\right)$, and $C_T=1\pm K{T}_c$, with $k_5=0.2,k_6=0.4,k_7=0.5$, and $K{T}_c=0.8$.

Figure 2

The space- and time-modulated group velocity dispersion $P(\xi ,\tau )$ on the left- and right-hand side, respectively, expressed in relation Eq. (24), where $\eta \left(\xi \right)=C_{T}sn(\xi ,k_7),d\left(\xi \right)=dn(\xi ,k_5),\gamma \left(\xi \right)=cn(\xi ,k_6),T_0\left(\xi \right)=sn(\xi ,k_3),T_1\left(\xi \right)=dn(\xi ,k_3),{\rho }_0\left(\xi \right)=dn(\xi ,k_2),{\rho }_1\left(\xi \right)=cn(\xi ,k_1)$, and $C_T=1\pm K{T}_c$, with $k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4,k_7=0.5,\nu =0.2$, and $K{T}_c=0.8$.

Figure 3

The left- and right-hand side amplitude $A\left(\xi \right)$, presented in relation Eq. (29), where $\eta \left(\xi \right)=C_{T}sn\left(\xi ,k_7\right),d\left(\xi \right)=dn\left(\xi ,k_5\right),\gamma \left(\xi \right)=cn\left(\xi ,k_6\right),T_0\left(\xi \right)=sn\left(\xi ,k_3\right),T_1\left(\xi \right)=dn\left(\xi ,k_3\right),{\rho }_0\left(\xi \right)=dn\left(\xi ,k_2\right),{\rho }_1\left(\xi \right)=cn\left(\xi ,k_1\right)$, and $C_T=1\pm K{T}_c$, with $k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4,k_7=0.5$, and $K{T}_c=0.8$.

Figure 4

First-order nonparaxial chiral optical rogue waves on the left- and right-hand side of the rational solution given by Eq. (35), where $\eta \left(\xi \right)=C_{T}sn\left(\xi ,k_7\right),d\left(\xi \right)=dn\left(\xi ,k_5\right),\gamma \left(\xi \right)=cn\left(\xi ,k_6\right),T_0\left(\xi \right)=sn\left(\xi ,k_3\right),T_1\left(\xi \right)=dn\left(\xi ,k_3\right),{\rho }_0\left(\xi \right)=dn\left(\xi ,k_2\right),{\rho }_1\left(\xi \right)=cn\left(\xi ,k_1\right)$, and $C_T=1\pm K{T}_c$, with $k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4,k_7=0.5,\nu =0.2$, and $K{T}_c=0.8$.

Figure 5

Second-order nonparaxial chiral optical rogue waves on the left- and right-hand side of the rational solution given by Eq. (39), where $\eta \left(\xi \right)=C_{T}sn\left(\xi ,k_7\right),d\left(\xi \right)=dn\left(\xi ,k_5\right),\gamma \left(\xi \right)=cn\left(\xi ,k_6\right),T_0\left(\xi \right)=sn\left(\xi ,k_3\right),T_1\left(\xi \right)=dn\left(\xi ,k_3\right),{\rho }_0\left(\xi \right)=dn\left(\xi ,k_2\right),{\rho }_1\left(\xi \right)=cn\left(\xi ,k_1\right)$, and $C_T=1\pm K{T}_c$, with $k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4,k_7=0.5,\nu =0.2$, and $K{T}_c=0.8$.

Figure 6

Nonparaxial chiral optical vector rogue waves with constant coefficients on the right- and left-hand side $\left|{\psi }_1{,}_2(\xi ,\tau )\right|$, where the parameters are $a_1=3,a_2=3,d=10,P=-0.5,\gamma =0.4,\mu =0.3,D=\pm 0.6,C=2,\alpha 3=0.2,\eta =0.5,\sigma =\pm 0.1,k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4$, and $k_7=0.5$. Here, the initial conditions take the form of exact solution Eqs. (42), (43), and (44).

Figure 7

Two-dimensional representations of the nonparaxial chiral optical vector rogue waves with constant coefficients in both sides, where the initial conditions take the form of exact solution Eqs. (42), (43), and (44) with the following parameters: $a_1=1,a_2=1,d=100,P=-0.5,\gamma =0.4,\mu =0.3$, $D=\pm 0.6,C=2,\alpha 3=0.2,\eta =0.5,\sigma =\pm 0.1$, $k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2$, $k_6=0.4$, and $k_7=0.5$.

Figure 8

Nonparaxial chiral optical vector rogue waves with constant coefficients in both sides, where the initial conditions are expressed in the form of exact solutions Eqs. (42), (43), and (44) with the parameters $a_1=1,a_2=1,d=10,P=-0.5,\gamma =0.4,\mu =0.3,D=\pm 0.6,C=2,\alpha 3=0.2,\eta =10,\sigma =\pm 10,k_1=0.3,k_2=0.5,k_3=0.6,k_4=0.4,k_5=0.2,k_6=0.4$, and $k_7=0.5$.

Figure 9

The nonparaxial chiral optical rogue waves with management are derived from Eqs. (49), where the parameters of the base equations are given in relation Eqs. (47) and (46) and the initial conditions take the form of exact solutions given in relation Eqs. (42), (43), and (44), with the following arbitrary constants: $a_1=3,a_2=3,k_5=0.2,k_6=0.4,k_7=0.5,P(\xi ,\tau )=dn(\xi ,k){\tau }^2+cn(\xi ,k)\tau +sn(\xi ,k),K{T}_c=0.8$, and $C_T=1\pm K{T}_c$.

Figure 10

The two-dimensional representation of the nonparaxial chiral optical vector rogue waves with variable coefficients are derived from Eqs. (49), where the parameters of the base equations are given in relation Eqs. (47) and (46) and the initial conditions take the form of exact solutions given in relation Eqs. (42), (43) and (44), with the following arbitrary constants: $a_1=1,a_2=1,k_5=0.2,k_6=0.4,k_7=0.5,P(\xi ,\tau )=dn(\xi ,k){\tau }^2+cn(\xi ,k)\tau +sn(\xi ,k),K{T}_c=0.8$, and $C_T=1\pm K{T}_c$.

Figure 11

Nonparaxial chiral optical vector rogue waves with modulated coefficients are derived from Eqs. (49) where the parameters of the base equations are given in relations Eqs. (47) and (46) and where the initial conditions take the form of exact solutions given in relations Eqs. (42), (43), and (44), with the following arbitrary constants: $a_1=1,a_2=1$ and $k_5=k_6=k_7=1,d\left(\xi \right)=10\times dn(\xi ,k),P(\xi ,\tau )=-(dn(\xi ,k){\tau }^2+cn(\xi ,k)\tau +sn(\xi ,k)),K{T}_c=0.8$, and $C_T=1\pm K{T}_c$.