Repulsive Casimir forces with finite-thickness slabs

We use the extended Lifshitz theory to study the behaviors of the Casimir forces between finite-thickness effective medium slabs. We first study the interaction between a semi-infinite Drude metal and a finite-thickness magnetic slab with or without substrate. For no substrate, the large distance $d$ dependence of the force is repulsive and goes as $1/d^5$; for the Drude metal substrate, a stable equilibrium point appears at an intermediate distance that can be tuned by the thickness of the slab. We then study the interaction between two identical chiral metamaterial slabs, with and without substrate. For no substrate, the finite thickness of the slabs $D$ does not significantly influence the repulsive character of the force at short distances, while the attractive character at large distances becomes weaker and behaves as $1/d^6$; for the Drude metal substrate, the finite thickness of the slabs $D$ does not influence the repulsive force too much at short distances until $D=0.05{\lambda }_0$.

Figure 1

Casimir interaction energy per unit area $E(d)/A$ (in units of ${\mathit{hck}}_0^3$) versus $k_{0}d$ between a semi-infinite Drude metal plate and a finite magnetic slab (a,b) without substrate and (c,d) with Drude metal substrate. (b,d) Insets: schematics depicting the arrangements of the slabs or substrates. Curves correspond to different values of the thickness $D$ of the slab. (a,c) Linear-log plot; (b,d) log-log plot. (c) Insets: magnifications of the regions around $k_{0}d\simeq 10$ and $k_{0}d\simeq {10}^2$.

Figure 2

Casimir interaction energy per unit area $E(d)/A$ (in units of ${\mathit{hck}}_0^3$) versus $k_{0}d$ between two identical finite-thickness CMM slabs (a,b) without substrate and (c,d) with semi-infinite Drude metal substrate. (b,d) Insets: schematics depicting the arrangements of the slabs or substrates. Curves correspond to different values of the thickness $D$ of the slab. (a,c) Linear-log plot; (b,d) log-log plot.