Nonperturbative approach to Casimir interactions in periodic geometries

Due to their collective nature Casimir forces can strongly depend on the geometrical shape of the interacting objects. We study the effect of strong periodic shape deformations of two ideal metal plates on their quantum interaction. A nonperturbative approach which is based on a path-integral quantization of the electromagnetic field is presented in detail. Using this approach, we compute the force for the specific case of a flat plate and a plate with a rectangular corrugation. We obtain complementary analytical and numerical results which allow us to identify two different scaling regimes for the force as a function of the mean plate distance, corrugation amplitude, and wavelength. Qualitative distinctions between transversal electric and magnetic modes are revealed. Our results demonstrate the importance of a careful consideration of the nonadditivity of Casimir forces, especially in strongly nonplanar geometries. Nonperturbative effects due to surface edges are found. Strong deviations from the commonly used proximity force approximation emerge over a wide range of corrugation wavelengths, even though the surface is composed only of flat segments. We compare our results to that of a perturbative approach and a classical optics approximation.

Figure 1

(Color online) Transformation of the matrix $\tilde{\mathcal{M}}$ to block-diagonal form. The figure shows a finite part of the matrix, corresponding to the blocks ${\mathcal{B}}_{kl}$ with $k$, $l=-1,0,1$, before and after the permutations of rows and columns. Before the transformation (left box) $\tilde{\mathcal{M}}$ has a band structure with diagonal blocks ${\mathcal{B}}_{ij}$ consisting of $2\times 2$ matrices $N_m$ along the diagonal. (The dependence on the lateral momentum ${\mathbf{q}}_{\perp }$ is not shown here.) The first step of the transformation is to permute the rows and columns which are formed by the first entry $N_m$ in every block ${\mathcal{B}}_{kl}$ (indicated as grid). These entries form after the permutations the first block ${\tilde{\mathcal{M}}}_0$ of $\tilde{\mathcal{M}}$ (right box). The latter permutation process is then repeated for the second and the third entry till the $(N+1)\text{th}$ entry of every block ${\mathcal{B}}_{kl}$, leading to the $N+1$ blocks ${\tilde{\mathcal{M}}}_j$. The momenta $q_1$ within each block ${\tilde{\mathcal{M}}}_j$ are constant for every column and they differ only by integer multiples of $2\pi ∕\lambda$ between columns (of the same block), see labels in the right box. The blocks ${\tilde{\mathcal{M}}}_j$ differ in their momentum shift $j\delta$, $\delta =2\pi ∕W$, which is located in the unit cell $[0,2\pi ∕\lambda [$ since $j=0,\dots ,N=W∕\lambda -1$.

Figure 2

Geometry consisting of a rectangular corrugated plate and a flat plate. The surfaces are translationally invariant along the $x_2$ direction.

Figure 3

(Color online) Casimir force for TM modes (a) and TE modes (b) as a function of $H∕a$ for different corrugation lengths $\lambda ∕a$. Displayed is the change of the force compared to the force between two flat plates, $F_{\text{TM},\text{flat}}=F_{\text{TE},\text{flat}}=-({\pi }^2∕480)H^{-4}$, in units of $F_{\text{TM},\text{flat}}$ and $F_{\text{TE},\text{flat}}$, respectively. The two bold curves enclosing the numerical data are the analytical results $F_0$ for $\lambda \to 0$ (upper curve) and $F_{\infty }$ for $\lambda \to \infty$ (lower curve), see text.

Figure 4

(Color online) Total Casimir force as sum of TM and TE mode contributions in the short-distance regime. Shown is the relative change of the force compared to the total Casimir force $F_{\text{flat}}$ between two flat plates. The data enclosing bold curves have the same meaning as in Fig. 3, but for the total force they are now given by $2F_0$ and $2F_{\infty }$ due to the same contribution of TM and TE modes in these two limits.

Figure 5

(Color online) (a) Ratio of Casimir force from TM and TE modes as a function of the plate distance $H$ for different corrugation lengths $\lambda$. (b) Logarithmic plot of the deviation of the ratio from one at large $H$.

Figure 6

(Color online) Scaling of the force from TM (a) and TE (b) modes close to the upper bound $F_0$ $\left(\lambda \to 0\right)$ and the lower bound $F_{\infty }$ $\left(\lambda \to \infty \right)$ as a function of $\lambda ∕a$ at fixed mean surface distance $H=10a$. Forces are measured in units of $F_{\text{TM},\text{flat}}$ and $F_{\text{TE},\text{flat}}$, respectively.

Figure 7

(Color online) Same plot as in Fig. 6 but for fixed distance $H=100a$.

Figure 8

(Color online) Typical paths of the proximity force approximation and the geometric optics approach for both sinusoidal (a) and rectangular corrugation (b) with $\lambda ⪢a$. Paths with arrows denote distances which are measured normal to one of the surfaces as used for the proximity force approximation. Paths without arrows denote the shortest surface connecting paths of length $\ell \left(\mathbf{x}\right)$ through a point $\mathbf{x}$ located in the gap between the plates.

Figure 9

Two rectangular corrugated plates with the same wavelength $\lambda$ but different amplitudes $a_1$ and $a_2$ and a lateral shift of $b$. The plates are translationally invariant along the $x_2$ direction.