We review the theory of the Casimir effect using scattering techniques. After years of theoretical efforts, this formalism is now largely mastered so that the accuracy of theory-experiment comparisons is determined by the level of precision and pertinence of the description of experimental conditions. Due to an imperfect knowledge of the optical properties of real mirrors used in the experiment, the effect of imperfect reflection remains a source of uncertainty in theory-experiment comparisons. For the same reason, the temperature dependence of the Casimir force between dissipative mirrors remains a matter of debate. We also emphasize that real mirrors do not obey exactly the assumption of specular reflection, which is used in nearly all calculations of material and temperature corrections. This difficulty may be solved by using a more general scattering formalism accounting for nonspecular reflection with wavevectors and field polarizations mixed. This general formalism has already been fruitfully used for evaluating the effect of roughness on the Casimir force as well as the lateral Casimir force appearing between corrugated surfaces. The commonly used 'proximity force approximation' turns out to lead to inaccuracies in the description of these two effects.

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 ...

Theory of quantized fields. PACS. 68.35.Ct -Interface structure and roughness.Abstract. -We calculate the roughness correction to the Casimir effect in the parallel plates geometry, for metallic plates described by the plasma model. The calculation is perturbative in the roughness amplitude, with arbitrary values for the plasma wavelength, the plate separation and the roughness correlation length. The correction is found to be always larger than the result obtained in the Proximity Force Approximation.

Theory of quantized fields. PACS. 68.35.Ct -Interface structure and roughness.Abstract. -The Casimir force between two metallic plates is affected by their roughness state. This effect is usually calculated through the so-called 'proximity force approximation' which is only valid for small enough wavevectors in the spectrum of the roughness profile. We introduce here a more general description with a wavevector-dependent roughness sensitivity of the Casimir effect. Since the proximity force approximation underestimates the effect, a measurement of the roughness spectrum is needed to achieve the desired level of accuracy in the theory-experiment comparison.

The thermal Casimir force between two metallic plates is known to depend on the description of material properties. For large separations the dissipative Drude model leads to a force a factor of 2 smaller than the lossless plasma model. Here we show that the plane-sphere geometry, in which current experiments are performed, decreases this ratio to a factor of 3/2, as revealed by exact numerical and large-distance analytical calculations. For perfect reflectors, we find a repulsive contribution of thermal photons to the force and negative entropy values at intermediate distances.

We give an exact series expansion of the Casimir force between plane and spherical metallic surfaces in the non trivial situation where the sphere radius R, the plane-sphere distance L and the plasma wavelength λP have arbitrary relative values. We then present numerical evaluation of this expansion for not too small values of L/R. For metallic nanospheres where R, L and λP have comparable values, we interpret our results in terms of a correlation between the effects of geometry beyond the proximity force approximation (PFA) and of finite reflectivity due to material properties. We also discuss the interest of our results for the current Casimir experiments performed with spheres of large radius R ≫ L.The Casimir force is a striking macroscopic effect of quantum vacuum fluctuations which has been seen in a number of dedicated experiments in the last decade (see for example [1,2] and references therein). One aim of the Casimir force experiments is to investigate the presence of hypothetical weak forces predicted by unification models through a careful comparison of the measurements with quantum electrodynamics predictions. This aim can only be reached if theoretical computations are able to take into account a realistic and reliable modeling of the experimental conditions. Among the effects to be taken into account are the material properties and the surface geometry, these effects being also able to produce phenomena of interest in nanosystems [3,4].A number of Casimir measurements have been performed with gold-covered plane and spherical surfaces separated by distances L of the order of the plasma wavelength (λ P ≃ 136nm for gold), making material properties important in their analysis [5]. As those measurements use spheres with a radius R ≫ L, they are commonly analyzed through the Proximity Force Approximation (PFA) [6], which amounts to a trivial integration over the sphere-plate distances. An exception is the Purdue experiment dedicated to the investigation of the accuracy of PFA in the sphere-plate geometry [7], the result of which will be given as a precise statement below.In the present letter, we give for the first time an exact series expansion of the Casimir force between a plane and a sphere in electromagnetic vacuum, taking into account the material properties via the plasma model (see Fig. 1). We present numerical evaluation of this expansion which are limited to not too small values of L/R, because of the multipolar nature of the series. We show below that these new results lead to a striking correlation between the effects of geometry and imperfect reflection when evaluated for nanospheres, with R, L and λ P having comparable values. In the end of this letter, we also discuss the interest of these results for the Casimir experiments performed with large spheres R ≫ L [7].Our starting point is a general scattering formula for the Casimir energy [8]. Using suitable plane-wave and multipole bases, we deduce the Casimir energy E PS be- tween a plane and a spherical metallic surface in electromagnetic...

We investigate in detail the focusing of a circularly polarized Laguerre-Gaussian laser beam ( ℓ orbital angular momentum per photon; σ = 1/ − 1 for left/right-handed polarization) by a high numerical aperture objective. The diffraction-limited focused beam has unexpected properties, resulting from a strong interplay between the angular spatial structure and the local polarization in the non-paraxial regime. In the region near the beam axis, and provided that |ℓ| ≥ 2 and ℓ and σ have opposite signs, the energy locally counter-propagates and the projection of the electric field onto the focal plane counter-rotates with respect to the circular polarization of the incident beam. We explicitly show that the total angular momentum flux per unit power is conserved after focusing, as expected by rotational symmetry, but the spin and orbital separate contributions change.

We derive a partial-wave (Mie) expansion of the axial force exerted on a transparent sphere by a laser beam focused through a high numerical aperture objective. The results hold throughout the range of interest for practical applications. The ray optics limit is shown to follow from the Mie expansion by size averaging. Numerical plots show large deviations from ray optics near the focal region and oscillatory behavior (explained in terms of a simple interferometer picture) of the force as a function of the size parameter. Available experimental data favor the present model over previous ones. 87.80.Cc, 42.50.Vk, 42.25.Fx Optical tweezers are single-beam laser traps for neutral particles that have a wide range of applications in physics and biology [1]. Dielectric microspheres are trapped and employed as handles in most of the quantitative applications. The gradient trapping force is applied by bringing the laser beam to a diffraction limited focal spot through a large numerical aperture microscope objective.Typical size parameters β = ka (a = microsphere radius, k = laser wavenumber) range in order of magnitude from values < 1 to a few times 10 1 . A theory of the trapping force based on geometrical optics (GO) [2] should not work in this range. Other proposals (cf.[1]), based on Mie theory, have employed unrealistic near-paraxial models for the transverse laser beam structure near the focus, incompatible with its large angular aperture.We take for the incident beam before the objective, propagating along the positive z axis, the usual Gaussian (TEM) 00 transverse laser mode profile, with beam waist w 0 at the input aperture, where kw 0 ≫ 1. We employ the Richards and Wolf [3] representation for the corresponding strongly focused beam beyond the objective, with a large opening angle θ 0 (no paraxial assumption), taking due account of the Abbe sine condition. This should be a more realistic representation.The microsphere, with real refractive index n 2 (we neglect absorption), is immersed in a homogeneous medium with refractive index n 1 . We consider here the simplest situation, in which the sphere center is aligned with the laser beam axis, so that we evaluate the axial trapping force. With origin at the sphere center, we denote by r = −qẑ the focal point position. The fraction A of total beam power that enters the lens aperture iswhere γ is the ratio of the objective focal length to the beam waist w 0 . By axial symmetry, the trapping force in this situation is independent of input beam polarization: we take circular polarization. The electric field of the strongly focused beam (we omit the time factor exp(−iωt)) has the Debye-type [3] integral representationwhere k = |k(θ, φ)| = n 1 ω/c,ǫ(θ, φ) =x ′ + iŷ ′ , and the unit vectorsx ′ andŷ ′ are obtained fromx andŷ, respectively, by rotation with Euler angles α = φ, β = θ, γ = −φ. The factor √ cos θ arises from the Abbe sine condition.For each plane wave exp(ik · r) in the superposition (2), the corresponding scattered field is given by the well-known Mie par...