Dark localized structures in a cavity filled with a left-handed material

We consider a nonlinear passive optical cavity filled with left-handed and right-handed materials and driven by a coherent injected beam. We assume that both left-handed and right-handed materials possess a Kerr focusing type of nonlinearity. We show that close to the zero-diffraction regime, high-order diffraction allows us to stabilize dark localized structures in this device. These structures consist of dips in the transverse profile of the intracavity field and do not exist without high-order diffraction. We analyze the snaking bifurcation diagram associated with these structures. Finally, a realistic estimation of the model parameters is provided.

Figure 1

Schematic setup of a ring cavity filled with right-handed and left-handed materials. The cavity is driven by a coherent external injected beam. The lengths of the left- and right-handed layers are $L_l$ and $L_r$, respectively.

Figure 2

Bistable input-output characteristics. The dashed curve indicates unstable solutions and the solid lines indicate stable ones. The associated marginal instability curve $I_s=\left|A_s\right|{}^2$ as a function of wave number $q^2$ is also shown. The hatched area corresponds to an unstable region with respect to modulation instabilities. Parameters are $\Delta =2$, $\theta =0.2$, $\Gamma =1$, and $\delta =0.5$.

Figure 3

(a) Bistable input-output characteristics. Dashed curves indicate periodic solutions and solid lines indicate homogeneous steady states. Stable (unstable) solutions are colored in black (gray). (b) Snaking bifurcation diagram of the 1D DLS. (c) Intensity profiles of 1D DLS obtained for input field amplitudes as indicated in (b). (d) Zoom of the bifurcation behavior around the profile (2) indicated in (b). Parameters: $\Delta =2$, $\theta =0.2$, $\Gamma =1$, $\delta =0.5$, and $L=153.6$ (with 512 discretization points).

Figure 4

Examples of 2D multiple DLS obtained for $\Delta =2$, $\theta =0.2$, $\Gamma =1$, $\delta =0.5$, and $A_i=1.437$. Minima are colored in black and the grid is 256$\times$256. The domain size of the system in both transverse directions is $L_x=L_y=89.6$.