Casimir-Lifshitz force out of thermal equilibrium between dielectric gratings

We calculate the Casimir-Lifshitz pressure in a system consisting of two different one-dimensional dielectric lamellar gratings having two different temperatures and immersed in an environment having a third temperature. The calculation of the pressure is based on the knowledge of the scattering operators, deduced using the Fourier modal method. The behavior of the pressure is characterized in detail as a function of the three temperatures of the system as well as the geometrical parameters of the two gratings. We show that the interplay between nonequilibrium effects and geometrical periodicity offers a rich scenario for the manipulation of the force. In particular, we find regimes where the force can be strongly reduced for large ranges of temperatures. Moreover, a repulsive pressure can be obtained, whose features can be tuned by controlling the degrees of freedom of the system. Remarkably, the transition distance between attraction and repulsion can be decreased with respect to the case of two slabs, implying an experimental interest for the observation of repulsion.

Figure 1

Geometry of the system. Two gratings, labeled with 1 and 2, at a distance $d$, always assumed to be positive. The gratings, in general made of different materials, are infinite in the $xy$ plane, and periodic in the $x$ direction with the same period $D$. They have corrugation depths $h_i$ ($i=1,2$), thicknesses ${\delta }_i$, and lengths of the upper part of the grating $l_i$. This defines the filling factors $f_i=l_i/D$. The two gratings have different constant temperatures $T_1$ and $T_2$, and are immersed in an environment having a third temperature $T_{\mathrm{e}}$.

Figure 2

Geometry of the FMM calculation. We consider one grating with interface $z=0$, corrugation depth $h$, and underlying thickness $\delta$. This defines four zones (see text) with four (in general different) dielectric permittivities. The period is $D$ and the filling factor is defined as $f=l/D$.

Figure 3

Pressure acting on grating 1 (made of fused silica, having $h_1=1\mu \mathrm{m}$, ${\delta }_1=10\mu \mathrm{m}$, $D=1\mu \mathrm{m}$, and $f_1=0.5$) in front of grating 2 (made of silicon, having $h_2=1\mu \mathrm{m}$, infinite thickness, $D=1\mu \mathrm{m}$, and $f_2=0.5$) as a function of distance $d$. The four curves correspond to different choices of the three temperatures $(T_1,T_2,T_{\mathrm{e}})$ (see legend).

Figure 4

Nonequilibrium (OTE) pressure [$(T_1,T_2,T_{\mathrm{e}})=(200,400,10)\mathrm{K}$, solid lines] compared to equilibrium pressure ($T=200\mathrm{K}$, dashed lines) for two gratings (black squares), and two slab-slab configurations corresponding to filled gratings ($f=1$, green circles) and empty ones ($f=0$, red triangles).

Figure 5

Ratio between the exact pressure and the PFA counterpart [see Eq. (57)], for the same distances and choices of temperatures of Fig. 3.

Figure 6

Variation of the pressure between two gratings at $d=4\mu \mathrm{m}$ [temperatures $(T_1,T_2,T_{\mathrm{e}})=(200,400,10)\mathrm{K]}$ as a function of the geometrical parameters. The reference point (black circle) corresponds to the set of parameters $f_1=f_2=0.5$, $h_1=h_2=1\mu \mathrm{m}$, ${\delta }_1=10\mu \mathrm{m}$, infinite ${\delta }_2$, $D=1\mu \mathrm{m}$. The three curves show the variation of pressure when changing one parameter at a time (red diamonds for the filling factor, green triangles for the period, blue squares for the corrugation depth). On the $y$ axis, the pressures are normalized with respect to the reference one, while on the $x$ axis each varying parameter is normalized with respect to its reference value ($f_0=0.5$, $D_0=1\mu \mathrm{m}$, $h_0=1\mu \mathrm{m}$). Note that the plot on the right side continues the one on the left with a modified $x$ scale.

Figure 7

Pressure on grating 1 as a function of distance [temperatures $(T_1,T_2,T_{\mathrm{e}})=(200,400,10)\mathrm{K]}$ for three different values of filling factor, all the other geometrical parameters being the reference ones.

Figure 8

Spectral density of the OTE contribution to the force [defined in Eq. (58)] at $d=4\mu \mathrm{m}$ [temperatures $(T_1,T_2,T_{\mathrm{e}})=(200,400,10)\mathrm{K]}$. The solid black line corresponds to filled gratings ($f=1$), the dot-dot-dashed red line to empty ones ($f=0$), the dotted blue line to our reference gratings, having $f=0.5$. In the other curves we vary the geometrical parameters one by one with respect to our reference case: dot-dashed violet line for $f=0.75$, short-dashed green line for $D=4\mu \mathrm{m}$, long-dashed brown line for $h=2\mu \mathrm{m}$.

Figure 9

Pressure acting on grating 1 for $d=4\mu \mathrm{m}$ and $T_1=T_2=T_{\mathrm{b}}$ as a function of $T_{\mathrm{b}}$ and $T_{\mathrm{e}}$. The solid line corresponds to zero pressure, while the dashed lines to the other contour lines shown in legend.

Figure 10

Pressure acting on grating 1 for $d=4\mu \mathrm{m}$ as a function of $T_2$ and $T_{\mathrm{e}}$ for three different values of $T_1$. Same convention of Fig. 9 for contour lines.

Figure 11

Distance $d_0$ of attractive-repulsive transition of the pressure as a function of $T_{\mathrm{e}}$ for three different values of $T_1=T_2=T_{\mathrm{b}}$.