Because the Casimir force depends on the optical properties ε of the materials — in particular ε at imaginary frequencies — an accurate knowledge of Im(ε) is necessary over a wide spectral range. Even though these data are available for metals such as gold, they do not correspond to any single gold film but rather have been assembled from different authors who have per- formed measurements on different samples using different depo- sition processes to cover a large frequency range.
[...]Only a few artificial non-planar geometries admit semi-analytical solutions, such as the hollow perfect-metal spherical shell, or the ‘Casimir piston’, which consists of two perfect-metal blocks sliding between perfect-metal walls. Self-energies were computed semi-analytically for the perfect-metal box6 and sphere configurations and seemed to predict a repulsive (expansion-inducing) self-force, but these predictions turned out to be problematic: the repulsion disappears if the object is cut in half or is cutoff-dependent, or if the box expansion is replaced by the rigid ‘piston’ motion of one wall.
[...]Over the past ten years, a number of techniques have been demonstrated that can accurately predict Casimir interactions for arbitrary geometries and materials, limited only by the available computational power. These techniques began with pioneering results for corrugated surfaces and cylinder–plate geometries, and have recently led to solutions for a plethora of complex three-dimensional structures.
[...]
Recent theoretical predictions
Although Casimir calculations tend to be more computationally intensive than classical photonics simulations owing to the large number of classical scattering problems that must be solved to compute a single force, recent work has demonstrated that a wide variety of highly non-planar geometries can be modelled exactly, starting from early solutions for corrugated plate, cylinder–plate, eccentric cylinders, sphere–plate and sphere–sphere geometries, extending to piston-like suspended-waveguide geometries, corrugated dielectrics, and even cones and fluid- suspended objects, with realistic permeable materials. The goal of much of this recent theoretical work has been to identify new geometries in which Casimir interactions behave in ways that differ qualitatively from the 1948 monotonic power-law attraction between parallel plates and that differ substantially from the PFA picture of pairwise surface–surface attractions. A small sampling of recent work that exploits the generality of these new numerical developments, including non-additive or unusual Casimir phenomena, is shown in Fig. 3. By breaking translation symmetry with corrugated surfaces, one can induce lateral forces. For two waveguides sandwiched between parallel plates or suspended above a single plate, there is a non-additive effect in which the presence of the plate(s) non-monotonically alters the attraction between the waveguides as a function of plate–waveguide separation.
[...]However, it is still possible to change the sign of the force merely by changing the geometry; repulsion was recently demonstrated between a needle- like metallic particle and a metal plate with a hole. Repulsion can also arise in circumstances involving interleaved objects because of the trivial competition between pairwise attractive interactions (lateral forces) between surfaces. In addition to forces, quantum fluctuations can also induce torques on objects that are free to rotate. This possibility was first studied theoretically in geometries consisting of planar objects with anisotropic materials102–104, and recently in more complicated geometries involving corrugated metallic surfaces, dilute rectangular objects suspended above plates, and eccentric metallic waveguides. [somehow this got me thinking of Leedskalnin]
Controlling film thickness.
One of the simplest ways of tailoring the Casimir force is to use films of varying thickness. At submicrometre distances, the Casimir force depends on the reflectivity of the interacting surfaces for wavelengths in the ultraviolet to the far-infrared. The attraction between transparent materials is expected to be smaller than that between highly reflective mirrors because of the lower effective confinement of electromagnetic modes inside the optical cavity (as is the case for ITO compared with gold). A thin metallic film can be transparent to electromagnetic waves that would otherwise be reflected by the bulk metal, particularly when the film thickness is much smaller than the material skin depth. Consequently, the Casimir force on a metallic film is significantly reduced when its thickness is smaller than the skin depth of the bulk metal at ultraviolet to infrared wavelengths. For most common metals, this condition is reached when the layer thickness is around 10 nm.
Demonstrating the skin-depth effect requires the thickness and surface roughness of the film to be carefully controlled. The experiment described in ref. 8 involved coating a sphere with a 9.23-nm-thick film of palladium. The sphere was imaged with an optical profiler to determine its roughness. After Casimir force measurements between the sphere and a metal-coated flat surface had been made, the sphere was removed from the experimental apparatus, coated with an additional 200 nm of palladium and analysed with the optical profiler. Repeated measurements showed that the Casimir force was larger with the thicker palladium film, by an amount that was in good agreement with the Lifshitz theory.
[...]Two plates made from the same material will always attract, regardless of the choice of intermediate material (typically a fluid or vacuum). However, the force between slabs of different materials (here labelled ‘1’ and ‘2’) can become repulsive by suitable choice of the intermediate liquid (labelled ‘3’). [...]In a recent experiment, the long-range repulsive Casimir force between a gold-coated sphere and a silica plate immersed in bro- mobenzene was measured. The silica plate was then replaced by a thick gold film, and the measurements were repeated35. The results show (Fig. 5a) that the force is attractive for a gold film but repulsive for a silica plate, which is in agreement with theoretical predictions.
[...]Repulsive Casimir forces could also be of significant technological interest, such as for the development of ultrasensitive force and torque sensors that levitate objects above surfaces without disturbing electric or magnetic interactions and with virtually no static friction to rotation or translation.
Concluding remarks
Many interesting theoretical and experimental avenues remain to be pursued in this fascinating field. For instance, many of the recent theoretical predictions of unusual Casimir physics outlined above, including non-monotonic force dependencies, repulsive forces, large temperature effects, fluid suspensions and orientation transitions arising from fluid dispersion or geometry, have yet to be observed experimentally. There are interesting theoretical predictions concerning anisotropic crystals that are awaiting experimental verification, such as the orientation-dependent Casimir force arising from highly anisotropic crystals and the quantum electro-dynamical torque between birefringent materials. An important question is whether the Casimir force can undergo a significant change near a suitable phase transition. Interesting candidates for this effect are materials undergoing a metal–insulator transition, as these experience large variations in the plasma frequency.
