3 transparency properties. It has been demonstrated with an increasing range of validity in [53], [54] and [7]. The expression is valid not only for lossless mirrors but also for lossy ones. In the latter case, it accounts for the ad- ditional fluctuations accompanying losses inside the mir- rors. It can thus be used for calculating the Casimir force between arbitrary mirrors, as soon as the reflection am- plitudes are specified. These amplitudes are commonly deduced from models of mirrors, the simplest of which is the well known Lifshitz model [34, 35] which corresponds to semi-infinite bulk mirrors characterized by a local di- electric response function ε ( ω ) and reflection amplitudes deduced from the Fresnel law. In the most general case, the optical response of the mirrors cannot be described by a local dielectric response function. The expression (8) of the free energy is still valid in this case with some reflection amplitudes to be determined from microscopic models of mirrors. Recent attempts in this direction can be found for example in [56–58]. Using the pole decomposition of the cotangent function and the analytic properties of ln d , this can finally be expressed as the Matsubara sum ( ∑ ′ m is the sum over positive integers m with m = 0 counted with a weight 1 $_{2}$) can be seen as resulting from the difference of radiation pressures exerted onto the inner and outer sides of the mirrors by the field fluctuations [53]. Using the analytic properties of the scattering amplitudes, the free energy may be written as the following expression after a Wick rotation ( ω = iξ are imaginary frequencies) This phaseshift formula can be given alternative inter- pretations. In particular, the Casimir force The same lines of reasoning can be followed when studying the geometry of two plane and parallel mir- rors aligned along the axis x and y . Due to the sym- metry of this configuration, the frequency ω , transverse vector k ≡ ( k$_{x}$, k$_{y}$ ) and polarization p = TE , TM are preserved by all scattering processes. The two mirrors are described by reflection and transmission amplitudes which depend on frequency, incidence angle and polar- ization p . We assume thermal equilibrium for the whole “cavity + fields” system, and calculate as in the sim- pler case of a 1-dimensional space. Care has however to be taken to account for the contribution of evanescent waves besides that of ordinary modes freely propagating outside and inside the cavity [7, 54]. The properties of the evanescent waves are described through an analyti- cal continuation of those of ordinary ones, using the well defined analytic behavior of the scattering amplitudes. At the end of this derivation, the free energy has the following form as a Matsubara sum [55] We now present a more general scattering formula al- lowing one to calculate the Casimir force between station- ary objects with arbitrary non planar shapes. The main generalization with respect to the already discussed cases is that the scattering matrix S is now a larger matrix ac- counting for non-specular reflection and mixing different wavevectors and polarizations while preserving frequency [7, 47]. Of course, the non-specular scattering formula is the generic one while specular reflection can only be an idealization. As previously, the Casimir free energy can be written as the sum of all the phaseshifts contained in the scat- tering matrix S The symbols det and Tr refer to determinant and trace over the modes of the matrix S . As previously, the for- mula can also be written after a Wick rotation as a Mat- subara sum The matrix D is the denominator containing all the res- onance properties of the cavity formed by the two ob- jects 1 and 2 here written for imaginary frequencies. It is expressed in terms of the matrices R$_{1}$ and R$_{2}$ which represent reflection on the two objects 1 and 2 and of This expression reproduces the Casimir ideal formula in the limits of perfect reflection r$_{1}$r$_{2}$ → 1 and null tem- perature T → 0. But it is valid and regular at ther- mal equilibrium at any temperature and for any optical model of mirrors obeying causality and high frequency ∑ k ≡ A ∫ d $^{2}$k 4 π 2 is the sum over transverse wavevectors with A the area of the plates, ∑ p the sum over polariza- tions and ∑ m$^{′}$ the Matsubara sum. This expression reproduces the Casimir ideal formula F = i ℏ ∫ ∞ 0 d ω 2 π N ( ω ) ln det S = i ℏ ∫ ∞ 0 d ω 2 π N ( ω )Tr ln S (9) F = k$_{B}$T ∑ m $^{′}$Tr ln D ( iξ$_{m}$ ) (10) D = 1 − R$_{1}$ exp −K $^{L}$R$_{2}$ exp −K L F = ∑ k ∑ p k$_{B}$T ∑ m $^{′}$ln d ( iξ$_{m}$, k , p ) (8) d ( iξ, k , p ) = 1 − r$_{1}$ ( iξ, k , p ) r$_{2}$ ( iξ, k , p ) exp − 2 κ L ξ$_{m}$ ≡ 2 πmk$_{B}$T ℏ , κ ≡ √ k 2 + ξ 2 c 2 F = k$_{B}$T ∑ m $^{′}$ln d ( iξ$_{m}$ ) , ξ$_{m}$ ≡ 2 πmk$_{B}$T ℏ (7) F = ℏ ∫ d ξ 2 π cot ( ℏ ξ 2 k$_{B}$T ) ln d ( iξ ) (6) F = ∂ F ( L, T ) ∂L (5) The non-specular scattering formula