Numerical evaluation of Casimir forces using the discontinuous Galerkin time-domain method

Abstract

We present a time-domain scheme for computing Casimir forces within the Maxwell stress tensor formalism, together with a specific realization using the finite-element-based discontinuous Galerkin time-domain method. The approach enables accurate evaluation of Casimir--Lifshitz interactions for a wide range of geometries and material properties at finite temperature. At the core of the method, the electromagnetic Green's tensor is expressed as the system's response to dipolar excitations, thereby recasting the Maxwell stress tensor into a set of classical scattering problems driven by electric and magnetic dipoles. We validate the approach against reference calculations of the Casimir interaction between parallel half-spaces at both zero and nonzero temperature. We further demonstrate its applicability to finite, cylindrically symmetric geometries for which closed-form solutions are unavailable, obtaining accurate agreement with asymptotic predictions based on physical considerations. These findings illustrate the method's potential for studying Casimir interactions in realistic micro- and nanoscale structures, relevant to nanodevice design and experimental settings.

Figures (8)

• Figure 1: Representative tetrahedral meshes generated with Gmsh Geuzaine2009 for (a) plane--plane and (b) cylinder--plane geometries. $L$ is the minimum surface-to-surface separation, while $a$ and $b$ denote the cylinder radius and height, respectively.
• Figure 2: Left column: Relevant quantities characterizing the time-integration scheme for two silver half-spaces separated by $L = 20\,\mathrm{nm}$, excited by an electric dipole oriented parallel to the planes. All quantities are scaled with a reference field strength $E_{0}=1\,\textrm{V/m}$ and a length scale $\ell_{0}=1\,\textrm{nm}$. Each panel depicts a different component of the scheme: (a) Temporal profile of the current density, as defined in Eq. \ref{['eq:currDens_time']} for $s=4$, expressed in units of $j_0^E = \epsilon_0 E_0 \ell_{0}^2 \mathrm{c}$; (b) Scattered electric field $E_{\parallel}^{(\parallel)}$ at $\mathbf{r}_{0}=(0,0,L/2)$ in units of $E_0$; (c) Kernel function in units of $g_0^E = \mathrm{c}^3 / \ell_{0}^3 j_0^E$, with a logarithmic scale inset showing the behavior at zero (blue) and room temperature (red). For details on the asymptotic behavior refer to Appendix \ref{['app:asym']}; (d) Time integrand $I_{\parallel, T}^{E}(\tau)=\operatorname{Im}\{g_T^E(-\tau)\} \epsilon_{0}E_{\parallel}^{(\parallel)}(\mathbf{r}_{0}, \tau)$ in units of $I_0 = g_0^E \epsilon_{0}E_0 = \mathrm{c}^2 / \ell_0^5$, with an inset focusing on the long-time behavior; (e) Numerically computed cumulative integral as a function of the simulation time $t_{\mathrm{max}}$, normalized to the reference value $\bar{f}_{\parallel, T}^{E}$, obtained via complex frequency-domain integration of the corresponding Green's tensor component, cf. Appendix \ref{['app:RefValue']}. The dashed lines correspond to the result incorporating long-time contributions. Right column: Same as the left column, but for a perpendicularly oriented magnetic dipole. All quantities are expressed in the following units: (a) $j_0^H = E_0 \ell_{0}^2$; (b) $H_0 = E_0\sqrt{\epsilon_0 / \mu_0}$; (c) $g_0^H = \mathrm{c}^3 / \ell_{0}^3 j_0^H$; (d) $I_0 = \mathrm{c}^2 / \ell_0^5$.
• Figure 3: Casimir force between two half-spaces as a function of their separation $L$. The force is normalized to the nonretarded asymptotic value, cf. Eq. \ref{['eq:asympLifshitzFormula']}. Symbols denote numerical results, while solid lines represent semi-analytical reference calculations based on the Lifshitz formula in Eq. \ref{['eq:LifshitzFormula']}. At short separations, the numerical results recover the $\sim -1 / L^{3}$ behavior predicted by Eq. \ref{['eq:asympLifshitzFormula']} and are independent of temperature, whereas at large separations they reproduce the $\sim -1 / L^{4}$ and $\sim -1 / L^{3}$ dependencies expected from Eqs. \ref{['eq:CasimirP']} and \ref{['eq:asympthermal']} at zero and finite temperature, respectively. The inset shows a representative mesh of the plane--plane geometry generated with Gmsh Geuzaine2009.
• Figure 4: Relative error in the component-wise Casimir force per unit area at $L=100\,\textrm{nm}$, averaged over $22$ nominally identical simulations with total duration $\gamma t_\textrm{max}=30$, as a function of the polynomial order $p$ for different mesh resolutions. The mesh size $h$ is defined as the average edge length in the central region of the computational domain. Results are presented for both zero (blue) and room temperature (red). Individual contributions from the electromagnetic field components are shown separately: (a) and (b) correspond to the parallel ($\parallel$) and perpendicular ($\perp$) electric-field components, respectively, while (c) and (d) show the corresponding magnetic-field components.
• Figure 5: Relative error of the component-wise Casimir force per unit area as a function of the simulation time $t_\mathrm{max}$, for $L = 100\,\mathrm{nm}$, polynomial order $p = 5$, and mesh size $h = 16.67\,\mathrm{nm}$. Solid lines represent direct quadrature of the time-domain response over $[0,t_\mathrm{max}]$, while dashed lines include analytically integrated long-time contributions from $t_\mathrm{max}$ to $\tau\to\infty$, using the modal parameters obtained via harmonic inversion over $[t_0,t_\mathrm{max}]$, which are added to the quadrature result according to Eq. \ref{['eq:f_time-HI']}, effectively extending the integration to $t_\mathrm{max}\to\infty$. Results are shown for both zero (blue) and room temperature (red). The shaded region represents the deviation of the relative error from $22$ nominally identical simulations with computational domains where the roundtrip time exceeded the maximum $t_\mathrm{max}$, ensuring no spurious back-reflections within the simulation window. Panels (a) and (b) show the parallel ($\parallel$) and perpendicular ($\perp$) electric-field components, respectively, while panels (c) and (d) show the corresponding magnetic-field components.