Geometry and Spectrum of Casimir Forces

We present a new approach to the Helmholtz spectrum for arbitrarily shaped boundaries and general boundary conditions. We derive the boundary induced change of the density of states in terms of the free Green’s function from which we obtain nonperturbative results for the Casimir interaction between rigid surfaces. As an example, we compute the lateral electrodynamic force between two corrugated surfaces over a wide parameter range. Universal behavior, fixed only by the largest wavelength component of the surface shape, is identified at large surface separations, complementing known short distance expansions which we also reproduce with high precision.

Figure 1

(a) Geometry consisting of two parallel plates with laterally shifted uniaxial rectangular gratings. (b) Approximate validity ranges of PFA and PWS approximations and the sector of asymptotic universality for the lateral Casimir force as estimated from Figs. 2, 4 .

Figure 2

Lateral force $F_{\mathrm{lat}}$ (in units of normal force $F_0$ between flat surfaces) at $b=\lambda /4$ obtained from Eq. (7) for the geometry of Fig. 1a as a function of gap $\delta$ (solid curves). Plotted also are the proximity force (PFA, dash-dotted curves) and pairwise summation (PWS, dashed curves) approximations, and the perturbative result $F_{\mathrm{pt}}$ for sinusoidal profiles (dotted curves) [11]. Inset: DOS for $H=10a$ together with the analytic result for flat plates (dotted curve); see text.

Figure 3

Shape dependence of $F_{\mathrm{lat}}$ on the lateral surface shift $b$ at fixed distance $H=10a$ for different corrugation lengths. The dashed and the dotted curves represent the PWS and the full perturbative result for sinusoidal profiles with arbitrary $H/\lambda$ [11], respectively.

Figure 4

Dependence of $F_{\mathrm{lat}}$ (solid curves) on the corrugation length $\lambda$ at fixed mean distances $H$. Shown are also the approximations (PFA, PWS) for short distances and the perturbative result $F_{\mathrm{pt}}$ for $a\ll H,\lambda$.