Computation and Visualization of Casimir Forces in Arbitrary Geometries: Nonmonotonic Lateral-Wall Forces and the Failure of Proximity-Force Approximations

We present a method of computing Casimir forces for arbitrary geometries, with any desired accuracy, that can directly exploit the efficiency of standard numerical-electromagnetism techniques. Using the simplest possible finite-difference implementation of this approach, we obtain both agreement with past results for cylinder-plate geometries, and also present results for new geometries. In particular, we examine a pistonlike problem involving two dielectric and metallic squares sliding between two metallic walls, in two and three dimensions, respectively, and demonstrate nonadditive and nonmonotonic changes in the force due to these lateral walls.

Figure 1

Casimir force between a radius-$R$ cylinder and a plate (inset), relative to the proximity-force approximation $F_{\mathrm{PFA}}$, vs normalized separation $a/R$. The solid lines are the Casimir force computed in Ref. 19 for TE (gray) and TM (blue) polarizations, along with results computed by our method with a simple finite-difference discretization (gray squares). Error bars were estimated for some data points by using computations at multiple spatial resolutions. Inset shows interaction-stress tensor $\Delta ⟨T_{xx}⟩$ at a typical imaginary frequency $w=2\pi c/a$, where red indicates attractive stress.

Figure 2

Casimir force per unit length between metal squares $F/F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), normalized by the total $\mathrm{TE}+\mathrm{TM}$ force per unit length obtained using the PFA, $F_{\mathrm{PFA}}=\hslash cs{\pi }^2/480a^4$. The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions.

Figure 3

Solid lines: Casimir force between 2D gold squares $F/F_{\mathrm{PFA}}$, vs distance from metal plate $h$ (inset), using experimental $\epsilon (\omega )$ [24], normalized by the total force obtained using the PFA. (Here, the PFA force is computed for $x$-infinite gold slabs). The total force is plotted (black squares) along with the TM (red dots) and TE (blue circles) contributions. Dashed lines: force for 2D perfect-metal squares, normalized by the perfect-metal PFA force $F_{\mathrm{PFA}}=\hslash c\zeta (3)s/8\pi a^3$.

Figure 4

(a–f) TM stress map of the 2D-analogous geometry of Fig. 2 for various $h$. The interaction stress tensors $⟨T_{xx}⟩$ (left) and $⟨T_{xy}⟩$ (right) for: (a),(d) $h=0.5a$; (b),(e) $h=a$; and (c),(f) $h=2a$, where (blue, white, red) denotes (repulsive, zero, attractive).