Impact of nonlocal interactions in dissipative systems: Towards minimal-sized localized structures

In order to investigate the size limit of spatial localized structures in a nonlinear system, we explore the impact of linear nonlocality on their domains of existence and stability. Our system of choice is an optical microresonator containing an additional metamaterial layer in the cavity, allowing the nonlocal response of the material to become the dominating spatial process. In that case, our bifurcation analysis shows that this nonlocality imposes another limit on the width of localized structures going beyond the traditional diffraction limit.

Figure 1

Influence of the nonlocality to diffraction ratio $\eta$ on the marginal stability curves as given by Eq. (6). The wave vectors inside the curves destabilize the homogeneous steady-state solution of Eq. (5). $\Delta =1.23$. (a) Positive $\eta$. (b) Negative $\eta$.

Figure 2

Maximal intensity of the cavity soliton vs the background intensity for different values of $\eta$. Solid lines indicate stable structures, whereas dashed lines correspond to unstable solutions. $\Delta =1.23$. (a) Positive $\eta$. (b) Negative $\eta$.

Figure 3

Renormalized full width at half maximum $\xi$ vs diffraction strength $\left({\mid \eta \mid }^{-1∕2}\right)$. The width $\xi$ is shown of the higher-branch CS at the change of stability or of the CS at the saddle-node bifurcation when no stable CSs exist. Stable CSs are indicated by solid lines, whereas the unstable CSs as dotted lines. $\Delta =1.23$. (a) Positive $\eta$. (b) Negative $\eta$.

Figure 4

Minimal CS width ${\xi }_{\mathit{min}}$ (solid lines, left axis) and optimal ${\eta }_{\mathit{opt}}$ (dashed lines, right axis) vs detuning $\Delta$. (a) Positive $\eta$. (b) Negative $\eta$.