Giant Casimir Torque between Rotated Gratings and the $\theta =0$ Anomaly

We study the Casimir torque between two metallic one-dimensional gratings rotated by an angle $\theta$ with respect to each other. We find that, for infinitely extended gratings, the Casimir energy is anomalously discontinuous at $\theta =0$, due to a critical zero-order geometric transition between a 2D- and a 1D-periodic system. This transition is a peculiarity of the grating geometry and does not exist for intrinsically anisotropic materials. As a remarkable practical consequence, for finite-size gratings, the torque per area can reach extremely large values, increasing without bounds with the size of the system. We show that for finite gratings with only ten period repetitions, the maximum torque is already 60 times larger than the one predicted in the case of infinite gratings. These findings pave the way to the design of a contactless quantum vacuum torsional spring, with possible relevance to micro- and nanomechanical devices.

Figure 1

Geometry of the system: two one-dimensional periodic lamellar gratings having period $D$, filling fraction $f$ and height $h$, placed at distance $d$. The figure shows two samples with $n=5$ periods. (a) Side view of two aligned gratings. (b) Top view of two gratings rotated by an angle $\theta$.

Figure 2

Angle-dependent Casimir energy per unit surface between two infinite gold gratings (see text for grating parameters) at distance $d=100\mathrm{nm}$. The black dot (not in scale on the vertical axis) shows the energy between two perfectly aligned gratings ($\theta =0$). The inset shows the torque as a function of $\theta$ for $\theta \ne 0$.

Figure 3

Casimir energy per unit surface between two finite gold gratings having $n=5$ (red dashed line for two circular gratings having $R=1\mu \mathrm{m}$, blue dashed line for two square gratings having $L=2\mu \mathrm{m}$) or $n=10$ (red solid line for two circular gratings having $R=2\mu \mathrm{m}$, blue solid line for two square gratings having $L=4\mu \mathrm{m}$) unit cells placed at a distance $d=100\mathrm{nm}$. The black dot-dashed line corresponds to two infinite gratings. We stress that here, unlike the result shown in Fig. 2, there is no discontinuity in the limit of $\theta \to 0$. Inset: Casimir energy per unit surface (in absolute value) for $\theta =0\mathrm{deg}$ (upper curve) and $\theta =90\mathrm{deg}$ (lower curve) as a function of the radius $R$ of the finite circular grating. The red points obtained numerically are fitted with a function $A+B/R$ (red dotted lines). The asymptotic values for $R\to \infty$ (red dots, not in scale on the horizontal axis) are compared with the ones obtained theoretically for an infinite grating (black dots). All the points are represented with the error bars coming from the respective numerical techniques.

Figure 4

Casimir torque per unit surface between two finite gold gratings having $n=5$ or $n=10$ unit cells placed at a distance $d=100\mathrm{nm}$ (same color scheme of Fig. 3). The black dot-dashed line corresponds to two infinite gratings, multiplied by a factor of 10 in order to make it more visible.