Figure 1

(Color online) A layer of moving magnetodielectric (iii) of thickness $d$ sandwiched in between two moving semi-infinite magnetodielectric layers (i) and (ii). The three layers slide along the same line which is parallel to the interfaces of the layers. The magnitudes and signs of the velocities $v_{1,2,3}$ are arbitrary. The Casimir-Lifshitz force is calculated from the reflection matrices ${\overline{\overline{R}}}_{1,2}$ defined at the interfaces of the layers.Reuse & Permissions

Figure 2

(Color online) Absolute values of the reflection coefficients $R_{\mathrm{abs}}^{\mathrm{ee}}=\left|R^{\mathrm{ee}}\right|$ and $R_{\mathrm{abs}}^{\mathrm{hh}}=\left|R^{\mathrm{hh}}\right|$ (the respective plots coincide and are shown with a single surface) at the real frequencies as functions of the normalized wave numbers $k_z/k_0$ and $k_y/k_0$ at an interface of a stationary medium with ${\varepsilon }_{\mathrm{r}}=2$, ${\mu }_{\mathrm{r}}=1$ and the same medium moving with velocity $v=0.6c$. The plotted surface is colored proportionally to the reflection amplitude, as indicated in the color bar on the left.Reuse & Permissions

Figure 3

(Color online) Absolute values of the normalized reflection coefficients $R_{\mathrm{abs}}^{\mathrm{eh}}=\left|R^{\mathrm{eh}}\right|\sqrt{{\varepsilon }_{\mathrm{r}}/{\mu }_{\mathrm{r}}}$ and $R_{\mathrm{abs}}^{\mathrm{he}}=\left|R^{\mathrm{he}}\right|\sqrt{{\mu }_{\mathrm{r}}/{\varepsilon }_{\mathrm{r}}}$ (the plots of these functions coincide). The parameters and the rest of the legend are the same as in Fig. 2.Reuse & Permissions

Figure 4

(Color online) Real and imaginary parts of the eigenvalues ${\lambda }_{1,2}$ of the matrix ${\overline{\overline{R}}}_1·{\overline{\overline{R}}}_2$ at imaginary frequencies as functions of the normalized wave numbers ${\kappa }_z$ and ${\kappa }_y$ (the plots for ${\lambda }_1$ and ${\lambda }_2$ coincide and are shown with a single surface). There are three layers of the same medium with ${\varepsilon }_{\mathrm{r}}=2$, ${\mu }_{\mathrm{r}}=1$. The middle layer is stationary and the two outer layers move with the velocity $v=0.6c$ along the positive direction of the $z$ axis.Reuse & Permissions

Figure 5

(Color online) Eigenvalues ${\lambda }_{1,2}$ at imaginary frequencies as functions of the normalized wave numbers ${\kappa }_z$ and ${\kappa }_y$ for the case when the outer layers move in opposite directions (the eigenvalues are purely real in this scenario). The absolute value of the velocity and the other parameters are the same as in Fig. 4.Reuse & Permissions

Figure 6

(Color online) Real and the imaginary parts of the integrand of Eq. (55) as functions of the normalized wave numbers ${\kappa }_z$ and ${\kappa }_y$ in the same scenario as in Fig. 4.Reuse & Permissions

Figure 7

(Color online) Integrand of Eq. (55) as a function of the normalized wave numbers ${\kappa }_z$ and ${\kappa }_y$ in the same scenario as in Fig. 5 (the integrand is purely real in this scenario).Reuse & Permissions

Figure 8

(Color online) Magnitude of the attractive and repulsive Casimir-Lifshitz forces in the triple-layered structures with ${\mu }_{\mathrm{r}}=1$ and ${\varepsilon }_{\mathrm{r}}$ indicated in the plot, as functions of the relative velocity $v/c$ of the outer layers (logarithmic scale). The force is normalized to the Casimir force between two perfect electric conductors (PECs) separated by the same distance as the thickness of the middle layer. The arrows in the plot indicate the directions of the movement of the outer layers that result in attraction and in repulsion.Reuse & Permissions

Figure 9

(Color online) Additional attractive force $\Delta F=F\left(v\right)-F(v=0)$ exerted on a dielectric with the relative permittivity ${\varepsilon }_{\mathrm{r}}$ moving with the relative velocity $v/c$ nearby a stationary dielectric of the same permittivity, for the three different values of the relative permittivity: 2, 4, and 8. The dielectrics are separated by a vacuum gap. The force is normalized to the Casimir force between two stationary PEC plates separated by the same gap. The brown solid lines: the force calculated with the theory of the present paper [Eq. (55)]. The blue dots: the same force calculated from Eq. (40) of Ref. [3]. The blue dashed line: the plot of the ${\beta }^2$-proportional term of Eq. (42) of Ref. [3] for ${\varepsilon }_{\mathrm{r}}=2$.Reuse & Permissions

Figure 10

(Color online) (a) PEC-backed structure used in calculation of the distance-dependent part of the zero-point energy. (b) Integration path $C$ in the complex plane of $\omega$.Reuse & Permissions