Casimir effect in twisted photonic gratings with in-plane chirality

TL;DR

The paper investigates rotational and lateral Casimir forces between two twisted one-dimensional photonic gratings made of uniaxially anisotropic material with in-plane chirality, using a reflection-matrix-based Casimir–Lifshitz formalism. It analyzes how the twist angle $\alpha$, anisotropy angle $\theta$, and separation $g$ shape the Casimir energy $E(\alpha,g,\theta)$, from which the force and torque are derived. A key finding is the emergence of a chiral equilibrium at nonzero twist when in-plane chirality is present, corresponding to nearly parallel anisotropy axes with $\alpha_{eq} \approx -2\theta$ in the fitted model. When electrostatic forces from surface charges are included, a joint equilibrium in $(g,\alpha)$ can be achieved, suggesting potential for nanoscale actuation, self-alignment, and reconfigurable chiral photonic systems.

Abstract

We investigate the Casimir effect in a system of two twisted photonic gratings made of uniaxially anisotropic materials. Two distinct configuretions are explored: a stack of symmetric gratings and a stack of in-plane chiral gratings, with the latter realized by choosing specific orientaton of anisotropy axis relative to stripes. We apply the reflection-matrix-based Casimir Lifshitzformalism to explore hoe twiat angle, material anisotropy, and the separation between gratings influence Casimir energy, force and torque. Our calculations reveal that the equilibrium orientation of the gratings is governed by the anisotropy rotation angles, leading to a chiral configuration where the anisotropy axes of the upper and lower gratings are mutually parallel. These findings demonstrate that material anisotropy provids a pwerful mechanism for controlling rotational alignment forces in nanophotonic system.

Figures (6)

• Figure 1: (a) Top and (b) side views of two twisted 1D photonic gratings separated by gap $g$, with anisotropy axes rotated by angles $+\theta$ and $-\theta$.
• Figure 2: (a) Casimir energy versus twist angle $\alpha$ and anisotropy axis angles $\theta$ (raw data). (b) Fitting parameters $a$ and $\phi$ as a function of $\theta$. (c) Casimir energy $\mathcal{E}$ for different $\theta$. Stars denote numerical data, solid lines -- theoretical fits. (d) Casimir torque calculated from fitted curves. (e)--(f) Energy and torque maps over $(\alpha,\theta)$ (analytical fits). $g = 100$ nm.
• Figure 3: Stacks of (a) symmetric and (b) in-plane chiral gratings. (c)-(d): The twist-angle dependencies of the Casimir energy normalized to the absolute value of that for two ideal metallic plates for different gap sizes. (e)-(f): The gap-size dependencies of the fit parameters $a(g)$ and $c(g)$. The Casimir force versus (g)-(i) gap $g$ at $\alpha=90^\circ$ and (j)-(k) $\alpha$ for gaps marked in (g) and (i). (l)-(m) The resultant of the Casimir and electrostatic forces versus $g$ and $\alpha$. Results b,d,f,i,k,m are computed for $\theta = 30^{\circ}$.
• Figure 4: (a)--(b) Dielectric permittivity components $\varepsilon_e$ and $\varepsilon_o$ at real and imaginary frequencies. (c) Spectral Casimir energy (SCE) as a function of imaginary frequency. (d) Spectral modal Casimir energy (SMCE) in the first Brillouin zone for four frequencies. Results for the panels (c) and (d) are calculated for $\alpha = 60^{\circ}$, $\theta = 60^{\circ}$ and $g = 100$ nm.
• Figure 5: (a) Convergence of the expression Eq.11 according to the number of stripes in sum ($n \in [-\text{N},\text{N}]$). (b) The dependence of the electrostatic force Eq.11 on the twist angle $\alpha$ and distance $g$. (c)--(d) The gap-size dependencies of the Casimir, electrostatic and resultant forces for stacks of symmetric lattices and in-plane chiral lattices. Results are computed for $\theta = 30^{\circ}$ and charge density $\sigma = 10^{-5}$ C/m$^2$.