Conditions for repulsive Casimir forces between identical birefringent materials

Repulsive Casimir-Lifshitz forces are known to exist between two dissimilar materials if a third material, whose dielectric response is intermediate, separates them. However, the force between two identical materials is almost always attractive. Here we show that the force between two identical, semi-infinite birefringent slabs can be repulsive for particular orientations and compare the conditions for repulsion in this system to those of isotropic materials. We examine the dependence of the Casimir-Lifshitz force on retardation and relative orientation in this system and discuss situations in which the force can be changed from attractive to repulsive as a function of both distance and rotation angle.

Figure 1

Schematic of the systems under investigation. Two infinite dielectric slabs interact across a third dielectric of width $d$. We compare the Casimir-Lifshitz interaction in (a) the case of two isotropic slabs with dielectric functions ${\varepsilon }_1$ and ${\varepsilon }_2$ to (b) the case with two identical birefringent materials with dielectric function ${\varepsilon }_{\parallel }$ along the principal axis and ${\varepsilon }_{\perp }$ in the other directions. In (b) the optical axis for material one is along the $x$ axis while that of material two is rotated by $\theta =\pi /2$.

Figure 2

The contribution of a single Matsubara term to the total Hamaker coefficient is plotted as a function of the dielectric coefficients. The red regions represent a positive energy (attractive force) and the blue regions represent a negative energy (repulsive force). (a) Shows the contributions when the interacting materials are isotropic and not necessarily identical. (b) Shows the contributions for two anti-aligned identical birefringent materials. The blue regions correspond to a negative contribution to the free energy (repulsion) for both (a) and (b). For anti-aligned birefringent materials, the greatest negative contribution possible from a single Matsubara term is approximately $-0.45$ zJ. The points indicate the contributions from the first 1000 Matsubara terms for the (a) gold-ethanol-vacuum system and (b) gold gratings interacting across ethanol at room temperature (Matsubara terms $n=10,100,1000$ are indicated by $+,\square ,◯$, respectively).

Figure 3

The ratio of the zero-temperature, long-range Casimir force for the two systems of Fig. 1 scaled to the Casimir force between two perfect conductors, $F_{\mathrm{Casimir}}=-\hslash c{\pi }^{2}240/d^4$, with ${\varepsilon }_{\parallel ,0}$ and ${\varepsilon }_{1,0}$ taken to infinity (as for a perfect conductor). In this case, the DC dielectric constants of the other materials determine the sign of the force. For the isotropic case (a), the condition for repulsion is the usual ${\varepsilon }_{2,0}<{\varepsilon }_{3,0}$. For the case with identical, anti-aligned birefringent materials (b), the repulsive condition is numerically found to be ${\varepsilon }_{\perp ,0}\lesssim 0.27{\varepsilon }_{3,0}$.

Figure 4

The room temperature, nonretarded Hamaker constant between infinite half-spaces of 1D conductors (${\varepsilon }_{\perp }=1, {\varepsilon }_{\parallel }={\varepsilon }_{\mathrm{Au}}$) separated by ethanol. The slabs experience an attractive force when the conduction axes are aligned and a repulsive force when the axes are anti-aligned.

Figure 5

The distance dependence of the Casimir-Lifshitz force between two idealized 1D conductors separated by ethanol. When the conductance directions are perpendicular, the two plates are repelled at short distances ($d\lesssim 70$ nm) and attracted at long distances.

Figure 6

The black lines represent the birefringent crystals with solid and dashed lines corresponding to the ordinary ($\perp$) and extraordinary ($\parallel$) axes, respectively. The gray band represents values of ${\varepsilon }_3\left(i\xi \right)$ that satisfy Eq. (8). The blue lines represent a liquid chosen to maximize the number of Matsubara terms that satisfy Eq. (8). Inset shows the values used in the Ninham-Parsegian oscillator model, with values for ${\omega }_{\mathrm{UV}}$ and ${\omega }_{\mathrm{IR}}$ in eV.