Stability restoration by asymmetric nonlinear states in non-Hermitian double-well potentials

We introduce a class of one-dimensional complex optical potentials that feature a nonlinearity-induced stability restoration, i.e., the existence of stable nonlinear modes propagating in a waveguide whose linear eigenmodes are unstable. The optical potential is an even function of the transverse coordinate, i.e., the system is parity symmetric but not parity-time symmetric. The stability restoration occurs for asymmetric stationary nonlinear modes that do not respect the parity symmetry. Stable nonlinear states exist either for focusing and defocusing nonlinearities. On the qualitative level the stability restoration can be analyzed using a simple bimodal system. Its solutions enable systematic construction of stable stationary modes and more complex patterns with intensity periodically oscillating along the propagation distance.

Figure 1

(a) Real $\left[w^2\left(x\right)\right]$ and imaginary $\left[w_x\left(x\right)\right]$ parts of an optical potential generated by function (5) with $W_0=2$ and $\ell =2$. (b) Imaginary part of the unstable propagation constant for $W_0=2$ and increasing well separation $\ell$.

Figure 2

Schematic diagram of fixed points of the two-mode system (9) displayed as dependencies $A_0\left(\omega \right)$ in focusing and defocusing media. In terms of the full system, quantities $A_0$ and $\omega$ transform in squared norm and propagation constant of stationary nonlinear modes, respectively; compare this figure with Fig. 3 which shows analogous dependencies produced from full envelope Eq. (1). Arrows show directions which correspond to the increase of the normalized asymmetry parameter $\mathcal{Z}$ from its asymptotic minimal value $\mathcal{Z}=0$ up to the maximal asymptotic value $\mathcal{Z}=1$. Labels $C_{\pm }$ mark the points where the upper and the lower subbranches meet pairwise. Labels $M_{\pm }$ mark the points where the minimal values of $A_0$ are achieved.

Figure 3

Families of stationary nonlinear modes of full Eq. (1) (black curves) on the plane $(U,\beta )$, where $U$ is the energy flow and $\beta$ is the propagation constant, juxtaposed with the analogous dependencies obtained from the fixed points of the two-mode system (gray curves). Solid and dashed segments of curves for the full equation correspond to stable and unstable stationary modes, respectively. Small panels next to the energy flow curves illustrate schematically spatial shapes of nonlinear modes $\left|\psi \right|$ at the corresponding subbranches: weakly and strongly asymmetric states are situated at the upper and lower subbranches, respectively. Points (a)–(c) at the two-mode curves in the focusing medium correspond to solutions used for dynamical simulations in Fig. 6. This figure is obtained for potential given by (5) with $W_0=2$ and $\ell =2$. Nonlinear modes obtained for focusing ($\sigma =1$) and defocusing ($\sigma =-1$) nonlinearities are displayed simultaneously.

Figure 4

Dynamical solutions of full Eq. (1) for initial conditions obtained from slightly perturbed stationary modes $\psi \left(x\right)$ with relative perturbation amplitude $\eta =0.01$, under the focusing nonlinearity with $\sigma =1$: (a) lower subbranch, $\beta =2$ (stable); (b) lower subbranch, $\beta =3$ (unstable); (c) upper subbranch, $\beta =1.7$ (stable); (d) upper subbranch, $\beta =2.3$ (unstable). Four lower panels (a1)–(d1) display the corresponding dependencies $n\left(z\right)$, i.e., the energy-flow fraction stored in both left and right modes as defined by Eq. (32).

Figure 5

Dynamical solutions of full Eq. (1) for initial conditions obtained from slightly perturbed stationary modes $\psi \left(x\right)$ with relative perturbation amplitude $\eta =0.01$, under the defocusing nonlinearity with $\sigma =-1$: (a), (b) lower subbranch, $\beta =0.6$ and 0.2 (both stable); (c), (d) upper subbranch $\beta =0.6$ for perturbation with $\eta =0.01$ (c) and $\eta =0.025$ and $5\%$ (d). Four lower panels (a1)–(d1) display the corresponding dependencies of the energy-flow fraction stored in left and right modes as defined by Eq. (32).

Figure 6

Examples of evolutions computed from full Eq. (1) with initial data obtained as fixed points (${\rho }_0=0$) and periodic orbits (${\rho }_0\ne 0$) of the two-mode reduction. Panels in the first, second, and third rows show the energy flow $U\left(z\right)={\int }_{-\infty }^{\infty }{\left|\mathrm{\Psi }(x,z)\right|}^{2}dx$, amplitudes $|{\tilde{a}}_{R,L}\left(z\right)|$ of projections of the solution $\mathrm{\Psi }(x,z)$ onto the left and right eigenmodes [see Eq. (33)], and the fraction of the energy conserved in the two modes $n\left(z\right)$ defined in Eq. (32). Points (a)–(c) referred to in the upper row correspond to those labeled in Fig. 3. Full dynamical plots are shown in the bottom row. (b) This figure corresponds to the focusing nonlinearity.