Repulsive Casimir Force in Chiral Metamaterials

We demonstrate theoretically that one can obtain repulsive Casimir forces and stable nanolevitations by using chiral metamaterials. By extending the Lifshitz theory to treat chiral metamaterials, we find that a repulsive force and a minimum of the interaction energy possibly exist for strong chirality, under realistic frequency dependencies and correct limiting values (for zero and infinite frequencies) of the permittivity, permeability, and chiral coefficients.

Figure 1

Casimir interaction energy per unit area $E/A$ (in units of $hc{k}_0^3$) versus $k_{0}d$; $k_0={\omega }_R/c$. The triangle curve corresponds to $\alpha =A=0.001$, $\kappa =0$ (no chirality),${\omega }_m={\omega }_R$, ${\omega }_{\mathrm{pl}}=0$, ${\omega }_e={\omega }_R$ for material 1, while $\alpha =A=0$, ${\omega }_{\mathrm{pl}}=10{\omega }_R$, ${\omega }_e=0$ for material 2. The diamond curve is the case with $\alpha =A=0.001$, $\kappa =0$, ${\omega }_m={\omega }_R$, ${\omega }_{\mathrm{pl}}=0$, ${\omega }_e={\omega }_R$. The squares curve is the case with $\alpha =A=0.001$, ${\omega }_{\kappa 1}={\omega }_{\kappa 2}=0.6{\omega }_R$, ${\omega }_m={\omega }_{\kappa R}={\omega }_R$, ${\omega }_{\mathrm{pl}}=0$, ${\omega }_e={\omega }_R$. Finally, the circle curve shows repulsion for $k_{0}d<0.0586$ and a stable equilibrium point at $k_{0}d=0.0586$; the parameters are the same as for the square curve except for ${\omega }_{\kappa 1}={\omega }_{\kappa 2}=0.7{\omega }_R$. All the $\gamma$’s equal $0.05{\omega }_R$ except ${\gamma }_{pl}=0.05{\omega }_{pl}$.

Figure 2

(a) The critical value of chirality ${\omega }_{\kappa }^c$ versus $\alpha (=A)$, for two identical CMM plates. (b) The equilibrium distance $k_0{d}_0$ versus ${\omega }_{\kappa }$ for $\alpha =A={10}^{-3}$. For ${\omega }_{\kappa }>{\omega }_{\kappa }^c=0.612{\omega }_R$, the value of the equilibrium distance $k_0{d}_0$ corresponds to the minimum of the energy as shown by the open circle curve in Fig. 1.

Figure 3

Casimir interaction energy per unit area E/A (in units of $hc{k}_0^3$) versus $k_{0}d$ of the two identical CMM plates configuration for different chiral strengths ${\omega }_{\kappa }$’s. $\alpha =0$, $A=0.2$, ${\omega }_{\mathrm{pl}}=0$, ${\omega }_m={\omega }_{\kappa R}={\omega }_R$, ${\gamma }_e={\gamma }_m={\gamma }_{\kappa }=0.05{\omega }_R$, ${\omega }_e={\omega }_R$.