Lateral Casimir Force beyond the Proximity-Force Approximation

Abstract

We argue that the appropriate variable to study a non trivial geometry dependence of the Casimir force is the lateral component of the Casimir force, which we evaluate between two corrugated metallic plates outside the validity of the Proximity Force Approximation (PFA). The metallic plates are described by the plasma model, with arbitrary values for the plasma wavelength, the plate separation and the corrugation period, the corrugation amplitude remaining the smallest length scale. Our analysis shows that in realistic experimental situations the Proximity Force Approximation overestimates the force by up to 30%.Considerable experimental progress has been achieved [1] in the measurement of the Casimir force, opening the way for various applications in nano-science [2], particularly in the development of nano-or micro-electromechanical systems (NEMS or MEMS). Calculations are much simpler in the original Casimir geometry of two plane plates [3] which obeys a symmetry with respect to lateral translations and thus allows to derive the expression of the Casimir force from the reflection amplitudes which describe specular scattering on the plates [4].More general geometries open a far richer physics with a variety of extremely interesting theoretical predictions [5]. Up to now the experimental studies of the effect of geometry have been restricted to simple configurations which can be calculated with the help of the Proximity Force Approximation (PFA). This approximation is essentially equivalent to an averaging over plane-plane geometries and its result can be deduced from the force known in this geometry [6]. For example, it allows to evaluate the force between a plane and a sphere [7] provided the radius R of the sphere is much larger than the mirror separation R ≫ L. It is also valid for the description of the effect of roughness when the wavelengths associated with the plate deformations are large enough [8]. However PFA relies heavily on assuming some additivity of Casimir forces which is known to be generally not valid except for very smooth geometrical perturbations [9].The aim of the present paper is to study a configuration allowing a new test of QED theoretical predictions outside the PFA domain and independent of those already performed in the plane-plane geometry. The idea is to look for the lateral component of the Casimir force which appears, besides the usual normal component, when periodic corrugations with the same period are imprinted on the two metallic plates. This configuration contrasts with other ones, for example the normal Casimir force in the plane-sphere geometry or roughness corrections to it. There PFA can also be invalid, but this leads only to small corrections of the dominant normal Casimir force, which do not seem accessible experimentally at the moment. The lateral component of the Casimir force has recently been measured and analyzed within the PFA [10,11]. We find for experimentally realizable parameters that PFA overestimates the force by as much as 30%, which should allow for a ...

Casimir torque between nanostructured plates

“…These expressions, specifying the angular mismatch dependence of E Cas , have allowed evaluating induced torques up to second order in the corrugation amplitudes [27]. This result contains the pure lateral force as the θ = 0 special case and the dependence on the material properties and corrugation period λ C of both lateral forces and torques are determined by identical response functions in the perturbative regime [10].…”

“…This lateral force stems from the translational symmetry breaking along the plates by the imprint of parallel and periodic corrugations on one or both plates. It has been evaluated in the past for perfect boundaries using the path integral approach [8,9] as well as for more realistic dielectric materials [10,11] within the scattering approach [12,13]. The latter allows to compute Casimir forces between arbitrarily shaped bodies taking into account the material properties.…”

Computation and Visualization of Casimir Forces in Arbitrary Geometries: Nonmonotonic Lateral-Wall Forces and the Failure of Proximity-Force Approximations

“…In order to handle more arbitrary geometries, two avenues have been pursued. First, one can employ approximations derived from limits such as that of parallel plates; these methods include the proximity-force approximation (PFA) and its refinements [4], renormalized Casimir-Polder [5] or semi-classical interactions [6], multiple-scattering expansions [7], classical ray optics [8], and various perturbative techniques [9,10]. Such methods, however, involve uncontrolled approximations when applied to arbitrary geometries outside their range of applicability, and have even been observed to give qualitatively incorrect results [11,12].…”

Calculating Casimir interactions for periodic surface-relief gratings using theCmethod

“…It is important to note that the scattering method has been previously used to calculate the Casimir interactions for surface relief gratings using a small amplitude perturbation series for the reflection coefficients [21][22][23]. We emphasize that in all previous work, in order to find the perturbative corrections to the reflection coefficients the Rayleigh hypothesis was assumed.…”

“…The first theoretical calculation of the Casimir force between geometrically patterned surfaces not using the PFA was reported in 2001 [19,20], which described the normal and lateral forces between two aligned corrugated surfaces in terms of a perturbative expansion in the profile height. Since then, this perturbative approach has been expanded to include real material properties and unaligned corrugations [21][22][23][24][25]. More recently the scattering method has been used to obtain the Casimir forces in periodic systems [26][27][28].…”