Casimir force between $\delta -{\delta }^{\prime }$ mirrors transparent at high frequencies

We investigate, in the context of a real massless scalar field in $1+1$ dimensions, models of partially reflecting mirrors simulated by Dirac $\delta -{\delta }^{\prime }$ point interactions. In the literature, these models do not exhibit full transparency at high frequencies. In order to provide a more realistic feature for these models, we propose a modified $\delta -{\delta }^{\prime }$ point interaction that enables full transparency in the limit of high frequencies. Taking this modified $\delta -{\delta }^{\prime }$ model into account, we investigate the Casimir force, comparing our results with those found in the literature.

Figure 1

$|r_{+}(\omega )|$ (vertical axis), for $\mu =1$, $\lambda =3$, and several values of $\beta$. The solid line ($\beta =0$), shows the reflection coefficient correspondent to the $\delta -{\delta }^{\prime }$ model given by Eq. (5). The dashed ($\beta =0.1$), dot-dashed ($\beta =5.0$), and space-dashed ($\beta =1.0$) lines correspond to $|r_{+}(\omega )|$ for the modified $\delta -{\delta }^{\prime }$ model given by Eq. (22). The behavior of $|r_{-}(\omega )|$ is similar.

Figure 2

$F{|}_{\beta \ne 0}/F{|}_{\beta =0}$ as a function of $q$, considering the parameters ${\mu }_1=1$, ${\mu }_2=3$, ${\lambda }_1=3$, ${\lambda }_2=-2$ and ${\beta }_1={\beta }_2=\beta$.

Figure 3

Casimir force as function of $\mu ={\mu }_1={\mu }_2$ and $\lambda ={\lambda }_1={\lambda }_2$, considering $q=1$ and two different values for $\beta ={\beta }_1={\beta }_2$. Figure 3a shows $F$ with $\beta =0$ [recovering the results for the model given by Eq. (5)], and Fig. 3 exhibits $F$ with $\beta =1$. The solid lines are the level curves, and the dashed lines show where the Casimir force is null.

Figure 4

Casimir force as function of $\lambda$ and $\beta$, considering ${\lambda }_1={\lambda }_2=\lambda$, ${\beta }_1={\beta }_2=\beta$, ${\mu }_1={\mu }_2=2$ and $q=1$.