Numerical regularization of electromagnetic quantum fluctuations in inhomogeneous dielectric media

Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as microelectromechanical systems embedded in dielectric media, Casimir induced friction in nanomachinery, microfluidics, and molecular electronics. Computation of such stresses based on cavity QED generally requires numerical analysis based on a regularization process. The scheme described below has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating numerically the Casimir stress on opposite faces of a pair of perfectly conducting planes separated by a vacuum and the change in this result when the region between the plates is filled with an incompressible inhomogeneous nondispersive dielectric.

Figure 1

For data based on (3), the algorithm determines $\mathcal{N}=-4$ and a collection of $[x,y]$ points $[{\widehat{n}}_2,{\widehat{\mathcal{C}}}_0(\mathcal{N},{\widehat{n}}_2,n_2)]$ with $1\le n_2\le 8$. The eight curves obtained by joining these points are indistinguishable in this figure and the average of ${\tilde{c}}_0(\mathcal{N},{\widehat{n}}_2,n_2)$ at their turning points yields a Casimir coefficient $c_0\left(\mathcal{N}\right)=0.27281$. The horizontal dotted line indicates the value of the Casimir coefficient that determines the observed Casimir attractive pressure between perfectly conducting plates separated by the vacuum.

Figure 2

For data based on (4) with $\sigma =8/27$ the algorithm determines $\mathcal{N}=-4$ and a collection of $[x,y]$ points $[{\widehat{n}}_2,{\widehat{\mathcal{C}}}_0^{\mathrm{TE}}(\mathcal{N},{\widehat{n}}_2,n_2)]$ and $[{\widehat{n}}_2,{\widehat{\mathcal{C}}}_0^{\mathrm{TM}}(\mathcal{N},{\widehat{n}}_2,n_2)]$ with $1\le n_2\le 8$. The TE curves obtained by joining the TE points are indistinguishable in this figure and the average of ${\tilde{c}}_0^{\mathrm{TE}}(\mathcal{N},{\widehat{n}}_2,n_2)$ at their turning points yields the Casimir coefficient $c_0^{\mathrm{TE}}\left(\mathcal{N}\right)=0.19744$. The same is true for the TM points with $c_0^{\mathrm{TM}}\left(\mathcal{N}\right)=0.20231$. Each such Casimir coefficient contributes to the total regularized force difference ${\mathcal{F}}_0\left[c_0^{\mathrm{TE}}\left(\mathcal{N}\right)+c_0^{\mathrm{TM}}\left(\mathcal{N}\right)\right]$, between opposite $z$ faces of a pair of perfectly conducting plates separated by an inhomogeneous dielectric with permittivity $\epsilon (x,y,z)={\epsilon }_0\exp (\alpha z/L_z)$, where $\sigma =\exp (\alpha /2)$.