Spontaneous reorientations of meta-atoms and electromagnetic spatial solitons in a liquid metacrystal

We show that transverse electromagnetic waves propagating along an external static electric field in liquid metacrystal (LMC) can provoke spontaneous rearrangement of elongated meta-atoms that changes the direction of the anisotropy axis of the LMC. This kind of instability may reorient the meta-atoms from the equilibrium state parallel to a static field to the state along a high-frequency field and back at the different threshold intensities of electromagnetic waves in such a way that bistability in the system takes place. Reorientation of meta-atoms causes a change in the effective refraction index of LMC that creates, in turn, the conditions for the formation of bright spatial solitons. Such spatial solitons are the self-consistent domains of redirected meta-atoms with trapped photons. We find that the instability thresholds as well as energy flux captured by the spatial soliton can be easily managed by variation of the static electric field applied to the LMC. We study the effects of soliton excitation and collisions via numerical simulations.

Figure 1

Schematic view of a meta-atom and an equivalent electric circuit (inset).

Figure 2

(a) Dependence of the critical values of the nonlinear parameter ${\sigma }_{C1}^2\approx {\sigma }_{C2}^2$ (left axis, dash-dotted line) and also their normalized difference, $1-{\sigma }_{C2}^2/{\sigma }_{C1}^2$, and $\Delta \varepsilon$ (right axis, dashed and solid lines, respectively) on the normalized frequency $\omega /{\omega }_s$. (b) Cosine of the equilibrium orientation angle as a function of nonlinearity parameter ${\sigma }_{\text{NL}}^2$: When the HF field increases, the switching from ${\theta }_{\text{eq}}=0$ to ${\theta }_{\text{eq}}=\pi /2$ takes place at ${\sigma }_{\text{NL}}={\sigma }_{C1}$; a decreasing field induces the inverse switch at lower ${\sigma }_{\text{NL}}={\sigma }_{C2}$. The dashed line shows the behavior of unstable ${\theta }_{\text{eq}}=\overline{\theta }$ [see Eq. (15)]. The hysteresis behavior becomes clear in the vicinity of resonance (in the calculations, $\omega =0.99{\omega }_s$ was used). If this is not the case, the critical fields for “on” and “off” switching are quite close to each other.

Figure 3

Dependence of the bright soliton parameters $P/P_0$ (a), $k_{0}b$ (b), and $\beta$ (c) as functions of $\delta k/k_0\equiv k/k_0-\sqrt{{\varepsilon }_l}$. Curves (1), (2), and (3) correspond to $\omega /{\omega }_s=0.9$, 0.95, and 0.99.

Figure 4

Spatial structure of normalized field $|E/E_c|$ in the case of subcritical (a) and supercritical (b) amplitude; the solid line shows the boundaries of the reoriented meta-atom domain. The field amplitude at $z=0$ is taken in the form of a Gaussian beam, $E/E_c=A_{0}exp[-{\left(y\right)}^2/2w^2]$, with $k_{0}w=5$, $A_0=0.9$, and $A_0=2$ for the subcritical and supercritical structures, respectively; $\omega /{\omega }_s=0.95$.

Figure 5

Spatial structure of normalized field $|E/E_c|$ in the case of subcritical (a) and supercritical (b) amplitude; the solid line shows the boundaries of the reoriented meta-atom domain. The field amplitude at $z=0$ is taken in the form of two in-phase Gaussian beams $E/E_c=A_0\{exp[-{(y/w-h)}^2/2]+exp[-{(y/w+h)}^2/2]\}$ with $h=2$; $k_{0}w=5$; $A_0=0.9$ and $1.5$ for the subcritical and supercritical structures, respectively; and $\omega /{\omega }_s=0.95$.

Figure 6

Spatial structure of normalized field $|E/E_c|$ in the case of subcritical (a), intermediate (b), and supercritical (c) amplitude; the solid line shows the boundaries of the reoriented meta-atom domain. The field amplitude at $z=0$ is taken in the form of two out-of-phase Gaussian beams $E/E_c=A_0\{exp[-{(y/w-h)}^2/2]-exp[-{(y/w+h)}^2/2]\}$ with $h=2$; $k_{0}w=5$; $A_0=0.9$, $1.5$, and 2 for cases (a), (b), and (c), respectively; and $\omega /{\omega }_s=0.95$.

Figure 7

Spatial structure of normalized field $|E/E_c|$ in the case of subcritical (a) and supercritical (b) amplitude; the solid line shows the boundaries of the reoriented meta-atom domain. The field amplitude at $z=0$ is taken in the form of two Gaussian beams with the $\pi /2$ phase shift $E/E_c=A_0\{exp[-{(y/w-h)}^2/2]+exp[-{(y/w+h)}^2/2+i\pi /2]\}$ with $h=2$; $k_{0}w=5$; $A_0=0.9$ and 2 for the subcritical and supercritical structures, respectively; and $\omega /{\omega }_s=0.95$.