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# THE SCATTERING APPROACH TO THE CASIMIR FORCE 

S. REYNAUD $^{∗}$, A. CANAGUIER-DURAND, R. MESSINA, A. LAMBRECHT 1 and and P.A. MAIA NETO 2 

$^{1}$Laboratoire Kastler Brossel, ENS, UPMC, CNRS, Jussieu, 75252 Paris, France $^{∗}$E-mail: serge.reynaud@upmc.fr 

$^{2}$Instituto de F´ısica, UFRJ, CP 68528, Rio de Janeiro, RJ, 21941-972, Brazil 

We present the scattering approach which is nowadays the best tool for describing the Casimir force in realistic experimental configurations. After reminders on the simple geometries of 1d space and specular scatterers in 3d space, we discuss the case of stationary arbitrarily shaped mirrors in electromagnetic vacuum. We then review specific calculations based on the scattering approach, dealing for example with the forces or torques between nanostructured surfaces and with the force between a plane and a sphere. In these various cases, we account for the material dependence of the forces, and show that the geometry dependence goes beyond the trivial Proximity Force Approximation often used for discussing experiments. 

The Casimir effect [1] is a jewel with many facets. First, it is an observable effect of vacuum fluctuations in the mesoscopic world, which deserves careful attention as a crucial prediction of quantum field theory [2–7]. 

Then, it is also a fascinating interface between quan- tum field theory and other important aspects of funda- mental physics. It has connections with the puzzles of gravitational physics through the problem of vacuum en- ergy [8, 9] as well as with the principle of relativity of mo- tion through the dynamical Casimir-like effects [10–12]. Effects beyond the Proximity Force Approximation also make apparent the extremely rich interplay of vacuum energy with geometry (references and more discussions below). 

Casimir physics also plays an important role in the tests of gravity at sub-millimeter ranges [13, 14]. Strong constraints have been obtained in short range Cavendish- like experiments [15] : Should an hypothetical new force have a Yukawa-like form, its strength could not be larger than that of gravity if the range is larger than 56 µ m. For scales of the order of the micrometer, similar tests are performed by comparing with theory the results of Casimir force measurements [16, 17]. At even shorter scales, the same can be done with atomic [18] or nuclear [19] force measurements. 

Finally, the Casimir force and closely related Van der Waals force are dominant at micron or sub-micron distances, which entails that they have strong connec- tions with various important domains, such as atomic and molecular physics, condensed matter and surface physics, chemical and biological physics, micro- and nano-technology [20]. 

## Comparison of the Casimir force measurements with theory 

In short-range gravity tests, the new force would ap- pear as a difference between the experimental result F$_{exp}$ and theoretical prediction F$_{th}$ . This implies that F$_{th}$ and 

## The many facets of the Casimir effect 

F$_{exp}$ have to be assessed independently from each other and should forbid anyone to use theory-experiment com- parison for proving (or disproving) some specific experi- mental result or theoretical model. 

Casimir calculated the force between a pair of perfectly smooth, flat and parallel plates in the limit of zero tem- perature and perfect reflection. He found universal ex- pressions for the force F$_{Cas}$ and energy E$_{Cas}$ 

with L the distance, A the area, c the speed of light and ℏ the Planck constant. This universality is explained by the saturation of the optical response of perfect mirrors which reflect 100% (no less, no more) of the incoming fields. Clearly, this idealization does not correspond to any real mirror. In fact, the effect of imperfect reflection is large in most experiments, and a precise knowledge of its frequency dependence is essential for obtaining a reliable theoretical prediction for the Casimir force [21]. 

The most precise experiments are performed with metallic mirrors which are good reflectors only at fre- quencies smaller than their plasma frequency ω$_{P}$ . Their optical response is described by a reduced dielectric func- tion usually written at imaginary frequencies ω = iξ as 

The function ˆ ε [ iξ ] represents the contribution of inter- band transitions and it is regular at the limit ξ → 0. Meanwhile σ [ iξ ] is the reduced conductivity ( σ is mea- sured as a frequency and the SI conductivity is ϵ$_{0}$σ ) which describes the contribution of the conduction electrons. 

A simplified description corresponds to the lossless limit γ → 0 often called the plasma model. As γ is much smaller than ω$_{P}$ for a metal such as Gold, this simple model captures the main effect of imperfect reflection. However it cannot be considered as an accurate descrip- tion since a much better fit of tabulated optical data is obtained with a non null value of γ [21]. Furthermore, the Drude model meets the important property of ordinary metals which have a finite static conductivity σ$_{0}$ = ω 2 P γ , 

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