Microstructure effects for Casimir forces in chiral metamaterials

We examine a recent prediction for the chirality dependence of the Casimir force in chiral metamaterials by numerical computation of the forces between the exact microstructures, rather than homogeneous approximations. Although repulsion in the metamaterial regime is rigorously impossible, it is unknown whether a reduction in the attractive force can be achieved through suitable material engineering. We compute the exact force for a chiral bent-cross pattern, as well as forces for an idealized “omega”-particle medium in the dilute approximation and identify the effects of structural inhomogeneity (i.e., proximity forces and anisotropy). We find that these microstructure effects dominate the force for separations where chirality was predicted to have a strong influence. At separations where the homogeneous approximation is valid, in even the most ideal circumstances the effects of chirality are less than ${10}^{-4}$ of the total force, making them virtually undetectable in experiments.

Figure 1

(Color online) Interactions between two chiral metamaterials (Ref. 20). Shaded box indicates a unit cell; the structures are periodic in the transverse direction. Nearest-neighbor bars on opposing structures are shaded yellow. In the EMA, the forces should obey $F_{\text{SC}}<F_{\text{OC}}$. By contrast, in the pairwise-force approximation the force is determined by the number of overlapping nearest-neighbor bars, as indicated: $F_{\text{OC}}>F_{\text{SC}}$ when the centers are aligned (left column) but $F_{\text{OC}}<F_{\text{SC}}$ when they are displaced by $L/2$ (right column).

Figure 2

(Color online) Relative force difference $\left(F_{\text{OC}}-F_{\text{SC}}\right)/F_{\text{OC}}$ for the left (red) and right (blue) columns of Fig. 1 for PEC (solid) and dispersive gold for $a=1\mu \text{m}$ (dashed). From Fig. 1, the sign difference is due to proximity effects. The inset shows a zoom of the large-$z$ regime, along with the EMA prediction (black).

Figure 3

(Color online) Top: unit cell of isotropic chiral metamaterial: PEC omega particles in vacuum. Shaded cube indicates relative positions in unit cell. Bottom: effective-medium parameters, as a function of imaginary frequency $\omega =i\xi$, deduced directly from imaginary-$\omega$ scattering data. $\mu \left(0\right)\ne 1$ because the particles are PEC.

Figure 4

(Color online) Forces (in dilute approximation) between chiral media from Fig. 3 for the SC (blue) and OC (red) cases. The range of the force across all transverse $\left(x\right)$ displacements is shaded and normalized by the minimum of the force over all $x$. For $z\lesssim 3.1a$, chirality is not well defined (curves overlap). Only when curves are distinct and $x$ independent can the systems be described as chiral metamaterials. Inset: frequency integrand of the relative force difference at $z=3.6a$. The smallness of the $\omega =0$ term is an indicator that this difference is due to chirality and not anisotropy (see text).