Casimir forces in inhomogeneous media: Renormalization and the principle of virtual work

We calculate the Casimir forces in two configurations, namely, three parallel dielectric slabs and a dielectric slab between two perfectly conducting plates, where the dielectric materials are dispersive and inhomogeneous in the direction perpendicular to the interfaces. A renormalization scheme is proposed consisting of subtracting the effect of one interface separating two inhomogeneous media. Some examples are worked out to illustrate this scheme. Our method always gives finite results and is consistent with the principle of virtual work; it extends the Dzyaloshinskii-Lifshitz-Pitaeveskii force to inhomogeneous media.

Figure 1

(a) The generalized Lifshitz configuration, where the permittivities and permeabilities of the three parallel dielectric slabs are ${{ϵ}_i}_{,}{\mu }_i,i=1$, 2, 3. (b) The reference configuration of (a) for the $z=b$ interface.

Figure 2

The TE and TM contributions to Casimir force ratios, denoted as ${\sigma }^E$ and ${\sigma }^H$, in the GCC, where the permittivity and permeability of the intervening medium are $ϵ(z)=\lambda /(c-z{)}^2=\tilde{\lambda }/(1-d_z{)}^2,\tilde{\lambda }=\lambda /(c-a{)}^2,d_z=(z-a)/(c-a)$ and $\mu =1$ respectively, with $\tilde{\lambda }=1$. Those Casimir force ratios are defined as ${\sigma }^E={\mathcal{F}}^E/{\mathcal{F}}^{\mathrm{HE}}$ and ${\sigma }^H={\mathcal{F}}^H/{\mathcal{F}}^{\mathrm{HE}}$, where ${\mathcal{F}}^E$ and ${\mathcal{F}}^H$ are TE and TM contributions to the Casimir forces in Eq. (f5) and ${\mathcal{F}}^{\mathrm{HE}}$ is the homogeneous Casimir force as shown in Eq. (11) with permittivity and permeability being $ϵ(a)$ and 1 respectively.

Figure 3

The Casimir forces in the GLC, where the permittivity and permeability of the intervening medium are the same as those defined in Fig. 2, and for the lower and upper dielectric slabs ${ϵ}_1=2$, ${\mu }_1=1$ and ${ϵ}_3=3$, ${\mu }_3=1$. (a) The TE and TM contributions to the scaled Casimir forces, defined as ${\eta }^E={10}^3(c-a{)}^4{\mathcal{F}}^E$ and ${\eta }^H={10}^2(c-a{)}^4{\mathcal{F}}^H$ respectively, in which ${\mathcal{F}}^E$ and ${\mathcal{F}}^H$ are given in Eq. (f4). (b) The Casimir force ratios, defined as ${\sigma }^E={\mathcal{F}}^E/{\mathcal{F}}^{\mathrm{HE}}$ and ${\sigma }^H={\mathcal{F}}^H/{\mathcal{F}}^{\mathrm{HM}}$, where the homogeneous Casimir forces ${\mathcal{F}}^{\mathrm{HE}}$ and ${\mathcal{F}}^{\mathrm{HM}}$ are given in Eq. (10) with permittivity and permeability being $ϵ(a)$ and 1 respectively.

Figure 4

Consider the same GLC configuration in Fig. 3 with the same parameters, except for $\lambda$ and $b$. The TE and TM contributions to the scaled Casimir forces, ${\eta }^E$ and ${\eta }^H$ as defined in Fig. 3 and ${\rho }^E=(c-a{)}^4{\mathcal{F}}^E$ and ${\rho }^H={10}^{-1}(c-a{)}^4{\mathcal{F}}^H$, as functions of $\lambda$ are shown with their dependence on $\tilde{\lambda }=\lambda /(c-a{)}^2$. Here the $d=(b-a)/(c-a)=0.5$ case is plotted with solid lines (${\eta }^E$ and ${\eta }^H$) and the $d=(b-a)/(c-a)=0.05$ case is plotted with dotted lines (${\rho }^E$ and ${\rho }^H$).

Figure 5

The separation dependence of the relative Casimir force ${\sigma }^E={\sigma }^H={\mathcal{F}}^E/{\mathcal{F}}^{\mathrm{HE}}$ in the GCC with permittivity and permeability $ϵ=e^{\lambda (z-c{)}^2}$ and $\mu =e^{-\lambda (z-c{)}^2}$, where $\sqrt{\lambda }(c-a)=2^{-3/2}$. ${\mathcal{F}}^E$ is given in Eq. (f9) and ${\mathcal{F}}^{\mathrm{HE}}$ is the TE Casimir force of a homogeneous GCC satisfying $ϵ\mu =1$.