Modulational instability in $\mathcal{PT}$-symmetric Bragg grating structures with four-wave mixing

Abstract

We investigate the dynamics of modulational instability (MI) in $\cal PT$-symmetric fiber Bragg gratings with a phenomenon of intermodulation known as four-wave mixing (FWM). Although the impact of FWM has already been analyzed in the conventional systems, the inclusion of gain and loss, which induces the notion of $\cal PT$- symmetry, gives rise to many noteworthy outcomes. These include the manifestation of an unusual double-loop structure in the dispersion curve, which was unprecedented in the context of conventional periodic structures. When it comes to the study of MI, which is usually obtained in the system by imposing a small amount of perturbations on the continuous wave by executing linear stability analysis, different regimes which range from conventional to broken $\cal PT$- symmetry tend to create quite a few types of MI spectra. Among them, we observe a unique MI pattern that mimics a tilted two-conical structure facing opposite to each other. In addition, we also address the impact of other non-trivial system parameters, such as input power, gain and loss and self-phase modulation in two important broad domains, including normal and anomalous dispersion regimes under the three types of $\cal PT$- symmetric conditions in detail.

Figures (10)

• Figure 1: (Color online) Nonlinear dispersion relation of $\cal PT$-symmetric FBG system with the effect of FWM is depicted between $\delta$ and $q$ for the conventional case (first column) with $\xi=-1$ (a - d), unbroken $\cal PT$-symmetric regime (second column) with $\xi=-0.2$ (e - h), at the $\cal PT$-symmetric threshold (third column) with $\xi= 0.1$ (i - l), and broken $\cal PT$-symmetric regime (fourth column) with $\xi= 1$ (m - p). The values of the system parameters are $P=1.5$, $\kappa=1$, and $\gamma=2$. The incidence of light is left in this case.
• Figure 2: (Color online) MI gain spectrum in the anomalous dispersion regime with FWM $\xi=1$ for left incidence (top panels) and right incidence (bottom panels) for (a) and (e) $f=-0.1$, (b) and (f) $f=-0.5$, (c) and (g) $f=-1$, and (d) and (h) $f=-3$. The other parameters are $P=1.5$, $\kappa=1$, and $\gamma=2$.
• Figure 3: (Color online) Plots show the role of power on the MI gain spectrum in the anomalous dispersion regime for left incidence (left panels) and right incidence (right panels) for different $\cal PT$-symmetric regimes, including (a-b) conventional, (c-d) unbroken $\cal PT$-symmetric, (e-f) exceptional point, and (g-h) broken $\cal PT$-symmetric regime. The values of other parameters are $f=-0.2$, $\kappa=1$, $\gamma=2$, and $\xi=1$.
• Figure 4: (Color online) The maximum gain of the MI spectra versus $P$ in the anomalous dispersion regime in the positive wavenumber region is shown for (a) left incidence and (b) right incidence for four different $\cal PT$-symmetric conditions. The parameters are the same as given in Fig. \ref{['Figure3']}.
• Figure 5: (Color online) Plots depict how SPM influences the MI gain spectrum in the anomalous dispersion regime for left incidence (left panels) and right incidence (right panels), and for (a-b) conventional, (c-d) unbroken $\cal PT$-symmetric, (e-f) exceptional point, and (g-h) broken $\cal PT$-symmetric regime with $f=-0.5$, $\kappa=1$, $P=3$, and $\xi=1$.