Higher-order dispersion and nonlinear effects of optical fibers under septic self-steepening and self-frequency shift

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schrödinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

Figure 1

The panels show plot of the variation of maximum MI gain as a function of perturbation frequency $\mathrm{\Omega }$ and the SOD term $k_6$ for (a) the normal ($k_2=1$) and (b) the anomalous regimes ($k_2=-1$) with the other parameters being: $P_0=8,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$ and $n_{\mathrm{SFS}6}=0.021$.

Figure 2

The variation of maximum MI gain as a function of perturbation frequency $\mathrm{\Omega }$ and the septic non-Kerr nonlinearity parameter $n_{\mathrm{NL}6}$. Panel (a) shows results for $k_6=0.02$, and panel (b) is plotted for $k_6=-0.02$ with the other parameters being: $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 3

Variation of maximum MI gain versus the perturbation frequency $\mathrm{\Omega }$ and the quintic self-steepening term $n_{\mathrm{SS}4}$. The FOD is considered positive in panels (a) and (b), respectively, obtained for $k_2>0$ and $k_2<0$. Panels (c) and (d) are plotted for a negative FOD coefficient $k_4$ with $k_2>0$ and $k_2<0$, respectively. The other parameters being: $P_0=10,k_6=0.02,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 4

Panels show the variation of maximum MI gain versus the perturbation frequency $\mathrm{\Omega }$ and the septic self-steepening term $n_{\mathrm{SS}6}$ and for (a) $k_6=0.02$, (b) $k_6=-0.02$ with the other parameters being: $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 5

Variation of maximum MI gain versus the perturbation frequency $\mathrm{\Omega }$ and the septic self-frequency shift term $n_{\mathrm{SFS}6}$ and for (a) $k_6=0.02$, (b) $k_6=-0.02$ with the other parameters being: $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{vSS6}=-0.0247,n_{\mathrm{SFS}}=-0.030875$, and $n_{\mathrm{SFS}4}=0.0004$.

Figure 6

The panels show the evolution of the cw under condition (vii) of the soliton stability analysis which predict MI to be violated if $k_2/k_5<0$ with $k_2=-1$ and $k_5=0.00123$. The rest of the parameters are: $k_6=0,P_0=10,k_3=0,k_4=0,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 7

The panels show the evolution of the cw under the effect of the septic nonlinearity term for (${\mathrm{aj})}_{j=1–3}{k}_6=0.0008,{\left(\mathrm{bj}\right)}_{j=1–3}{k}_6=-0.0008$, and $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 8

The panels show the evolution of the cw under the effect of the septic self-steepening term for: ${\left(\mathrm{aj}\right)}_{j=1–3}{k}_6=0.0003,{\left(\mathrm{bj}\right)}_{j=1–3}{k}_6=-0.0003$, and $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{vSS}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SFS}}=-0.030875,n_{\mathrm{SFS}4}=0.0004$, and $n_{\mathrm{SFS}6}=0.021$.

Figure 9

The panels show the evolution of the cw under the effect of the septic self-frequency shift term for: ${\left(\mathrm{aj}\right)}_{j=1–3}{k}_6=0.0008,{\left(\mathrm{bj}\right)}_{j=1–3}{k}_6=-0.0008$, and $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875$, and $n_{\mathrm{SFS}4}=0.0004$.

Figure 10

Effect of the septic self-frequency shift term on the propagation direction of the disintegrated cw for (a) $n_{\mathrm{SFS}6}=0.02,k_6=0.0008,\left(\mathrm{b}\right)n_{\mathrm{SFS}6}=-0.001,k_6=0.0008,\left(\mathrm{c}\right)n_{\mathrm{SFS}6}=-0.02,k_6=-0.0008$, and $P_0=10,k_5=0.00123,k_3=0.009,n_{\mathrm{NL}}=1,n_{\mathrm{NL}4}=1,n_{\mathrm{NL}6}=0.03,n_{\mathrm{SS}}=-0.0247,n_{\mathrm{SS}4}=0.03705,n_{\mathrm{SS}6}=-0.0247,n_{\mathrm{SFS}}=-0.030875$, and $n_{\mathrm{SFS}4}=0.0004$.