Localized plateau beam resulting from strong nonlocal coupling in a cavity filled by metamaterials and liquid-crystal cells

We investigate the formation of a localized plateau beam in the transverse section of a nonlinear optical ring cavity filled with a metamaterial and a nonlocal medium such as a nematic liquid crystal. We show that, far from the modulational instability regime, localized structures with a varying width may be stable in one and two-dimensional settings. The mechanism of stabilization is related with strong nonlocal coupling mediated by a Lorentzian type of kernel. We show that there exists stable bright and dark localized structures. A reduction of Lugiato–Lefever equation in the regime close to the nascent bistability allows us to analytically derive a simple formula for the width of localized structures in one-dimensional systems. Direct numerical simulations of the dynamical model agree with the analytical predictions.

Figure 1

Ring resonator filled with a left-handed material (LHM) and a nonlocal medium such a liquid crystal. $M$ accounts for the mirrors.

Figure 2

Homogeneous steady-state solutions of Eq. (1) for different values of the detuning parameter $\theta$. The continuous and dashed curves indicate stable and unstable states, respectively.

Figure 3

Localized plateau beam in one transverse dimension. Three different structures in one dimension, for three different values of injection $E_i$. The parameters are $\theta =6.0,D=-1.0,\gamma =1.0,\sigma =0.7$, and $n=2.0$.

Figure 4

Localized plateau beam in two dimensions. Three different structures in two dimensions, for three different values of injection $E_i$. The parameters are $\theta =5.93,D=-1.0,\gamma =1.0,\sigma =0.4$, and $n=2.2$.

Figure 5

Dark LS's in one and two dimensions. In panel (a) the parameters are $\theta =6.0,D=-1.0,\gamma =1.0,\sigma =0.7,n=2.0$, and $E_i=3.08$. In panel (b) the parameters are $\theta =5.93,D=-1.0,\gamma =1.0,\sigma =0.4,n=2.2$, and $E_i=3.034$.

Figure 6

(a) Dark and (b) bright localized structures in two dimensions obtained by numerical simulations of Eq. (1). The parameters are $\theta =1.8,D=-1.0,\gamma =0.3,\sigma =0.4$, and $n=2.2$. The injection-field amplitude is (a) $E_i=1.278$ and (b) $E_i=1.275$.

Figure 7

Numerical value for the Maxwell point as function of the nonlocal intensity $\gamma$. The parameters are $\theta =6.0,D=-1.0,\sigma =0.7$, and $n=2.0$.

Figure 8

Theoretical and numerical size of localized plateau beam in one dimension. The solid line is the theoretical prediction from Eq. (7), the dots are the numerical results from Eq. (1), and the dashed line in panel (a) corresponds to the Maxwell point estimated numerically. In both graphics, the parameters are $\theta =6.0,D=-1.0,\sigma =0.7$, and $n=2.0$. In panel (a), the nonlocal intensity $\gamma =1.0$. In panel (b), the distance to the Maxwell point $E_M-E_i=0.005$.

Figure 9

Interaction between three localized plateau beams in one transverse dimension. The parameters are $\theta =6.0,D=-1.0,\gamma =0.5,\sigma =0.7,E_i=2.94$, and $n=2.0$.