Optical-approximation analysis of sidewall-spacing effects on the force between two squares with parallel sidewalls

Using the ray-optics approximation, we analyze the Casimir force in a two-dimensional domain formed by two metallic blocks adjacent to parallel metallic sidewalls, which are separated from the blocks by a finite distance $h$. For $h>0$, the ray-optics approach is not exact because diffraction effects are neglected. Nevertheless, we show that ray optics is able to qualitatively reproduce a surprising effect recently identified in an exact numerical calculation: the force between the blocks varies nonmonotonically with $h$. In this sense, the ray-optics approach captures an essential part of the physics of multibody interactions in this system, unlike simpler pairwise-interaction approximations such as proximity force approximations (PFA). Furthermore, by comparison to the exact numerical results, we are able to quantify the impact of diffraction on Casimir forces in this geometry.

Figure 1

Schematic of a two-dimensional geometry: two metal squares $s\times s$ separated by a distance $a$, and separated from two adjacent metal sidewalls by a distance $h$.

Figure 2

(Color online) Casimir even (red, upper curves) and odd (blue, lower curves) forces vs sidewall separation $h$ normalized by the PFA force $F_{\mathrm{PFA}}=-\hslash c\zeta \left(3\right)s∕8\pi a^3$ (dashed black), computed using the ray-optics (solid) and stress-tensor (dashed) methods. Note that the ray-optics results become exact as $h\to 0$.

Figure 3

(Color online) Casimir force vs sidewall separation $h$ normalized by the PFA force $F_{\mathrm{PFA}}=-\hslash c\zeta \left(3\right)s∕8\pi a^3$, computed using the ray-optics (solid) and stress-tensor (dashed) methods. The Neumann (green), Dirichlet (orange), and total (black) forces are all normalized by the total $\mathrm{Neumann}+\mathrm{Dirichlet}$ PFA force.

Figure 4

(Color online) Casimir force vs separation between squares $a$ with constant sidewall separation $h=0.25$, normalized by the $F_{\mathrm{PFA}}=-\hslash c\zeta \left(3\right)s∕8\pi a^3$ (left) and the corresponding $h=0$ force (right). The force is computed for both even (red, left curves) and odd (blue, right curves) contributions, separately. Inset: Schematic of geometry consisting of two isolated squares with adjacent sidewalls.

Figure 5

(Color online) Schematic of general 2d $\mathrm{squares}+\mathrm{sidewalls}$ lattice. Lines extending from solid circles unto solid circles represent even reflection paths. Lines extending from open solid circles unto open circles represent odd reflection paths. (Here, $h∕a\approx 0.2$ and $s∕a\approx 1$.) A possible even (red) and forbidden odd (blue) path is shown.

Figure 6

(Color online) Schematic of three-reflection paths. Blue (dashed) and/or red represent $(1,2)$ and/or $(2,1)$ paths, and the lengths ${\ell }_i$ shown are used the calculation of the energy.

Figure 7

(Color online) Log-log plot of convergence error $\ln [({\mathcal{E}}_{r+1}-{\mathcal{E}}_r)∕{\mathcal{E}}_r]$ in the even (solid) and odd (dashed) path contributions vs reflection order $\ln r$ for values of $h=0$ (red), $h=0.01$ (blue), and $h=0.1$ (black) (here, ${\mathcal{E}}_r$ means the energy computed up to order $r$). Inset: Absolute error $({\mathcal{E}}_{\mathrm{exact}}-{\mathcal{E}}_r)∕{\mathcal{E}}_{\mathrm{exact}}$ for both even (solid black) and odd (dashed red) contributions for $h=0$.