Stability through asymmetry: Modulationally stable nonlinear supermodes of asymmetric non-Hermitian optical couplers

We analyze the stability of a non-Hermitian coupler with respect to modulational inhomogeneous perturbations in the presence of unbalanced gain and loss. At the parity-time ($\mathcal{PT}$) symmetry point the coupler is unstable. Suitable symmetry breakings lead to an asymmetric coupler, which hosts nonlinear supermodes. A subset of these broken symmetry cases finally yields nonlinear supermodes which are stable against modulational perturbations. The lack of symmetry requirements is expected to facilitate experimental implementations and relevant photonics applications.

Figure 1

(a) Fiber coupler. The wave profiles in the spatial dimensions $x$ and $y$ are determined by the waveguided modes of the two fibers. The “transverse” variable $t$ [Eq. (1)] refers to the time dependence of the wave. (b) Slab waveguide coupler. The wave profiles in the spatial dimension $y$ are determined by the waveguided modes of the two slabs, and their time dependence is harmonic (steady state). The “transverse” variable $t$ [Eq. (1)] refers to the wave profile along the infinite spatial $x$ dimension ($t\equiv x$).

Figure 2

(a) and (b) Growth rate $g$ for the case of self-focusing nonlinearity ($\gamma =1$) and no XPM ($\sigma =0$) for a gain-loss imbalance corresponding to $\alpha =0.2$ for the two nonlinear supermodes. Both NSs are modulationally unstable.

Figure 3

Development of the modulation instability of the nonlinear supermodes under a random-noise perturbation of the order of ${10}^{-2}$ superimposed at $z=0$ for the cases corresponding to (a) Fig. 2 with $\beta =0.2$ and (b) Fig. 2 with $\beta =1$.

Figure 4

Growth rate $g$ for the case of self-defocusing nonlinearity ($\gamma =-1$) and no XPM ($\sigma =0$) for a gain-loss imbalance corresponding to $\alpha =0.2$ for the two nonlinear supermodes. (a) The first NS is shown to be modulationally stable ($g<0$ for all $\omega$) for a range of $\beta$ values. (b) The second NS is always modulationally unstable.

Figure 5

Existence domain of modulationally stable nonlinear supermodes in terms of the parameters $\alpha$ and $\beta$ expressing the asymmetry of the structure. The nonlinear supermodes are stable against homogeneous perturbations (with $\omega =0$) in the whole shaded area and modulationally stable (against perturbations with arbitrary $\omega$) in the blue shaded area. Cases of (a) zero ($\sigma =0$) and (b) nonzero ($\sigma =0.5$) XPM coefficients are shown.

Figure 6

Wave evolution of periodic and localized perturbations superimposed on a modulationally stable nonlinear supermode. Parameter values are as in Fig. 4, and $\beta =1.8$. In the second waveguide, (a) a periodic wave $\tilde{u}=0.5cos\left(t\right)$ and (b) a localized wave $\tilde{u}=0.5\text{sech}\left(0.5t\right)$ have been superimposed on the stable NS at $z=0$.