Divergence of Casimir stress in inhomogeneous media

We examine the local behavior of the regularized stress tensor commonly used in calculations of the Casimir force for a dielectric medium inhomogeneous in one direction. It is shown that the usual expression for the stress tensor is not finite anywhere within the medium, whatever the temporal dispersion or index profile, and that this divergence is unlikely to be removed through a simple modification to the regularization procedure. Our analytic argument is illustrated numerically for a medium approximated as a series of homogeneous strips, as the width of these strips is taken to zero. The findings hold for all magnetodielectric media.

Figure 1

The medium is assumed inhomogeneous along $x$ and is divided into $N$ homogeneous slices of width $a$. The local value of the regularized stress tensor (1) is then investigated within the medium in the limit as $a\to 0$. For the purposes of illustration only the permittivity $\epsilon$ is shown here. Our analysis holds for both inhomogeneous permittivity and permeability.

Figure 2

The continuous refractive index profile of the system and a piecewise approximation using 20 homogeneous slices.

Figure 3

The medium, inhomogeneous between $x=0$ and $x=\ln \left(3\right)$, is divided into 100, 200, 400, and 800 homogeneous slices. The local absolute value of the regularized stress tensor (13)—normalized in units of $\hslash c/{\pi }^2$—is plotted for each case at a given position $x$. The stress increases as the number of divisions increases.

Figure 4

The integrand of the stress (13) ${\sigma }^{\prime }$ (normalized in the same units as Fig. 3) is plotted for $\kappa =1$ at the center of the system, with $k_{\parallel }$ varying from $k_{\parallel }=1$ to $k_{\parallel }=6000$ (horizontal axis), and $N$ ranging from $t=10$ to $t=6000$ (depth axis). As the number of slices $N$ is increased, the integrand falls off less rapidly with $k_{\parallel }$, and thus the integral of the stress converges less rapidly.