δ R$_{1}$ and δ R$_{2}$ are the first-order variation of the reflec- tion matrices R$_{1}$ and R$_{2}$ induced by the corrugations; D$_{0}$ is the matrix D evaluated at zeroth order in the cor- rugation; it is diagonal on the basis of plane waves and commutes with K . Explicit calculations of (14) have been done for the simplest case of experimental interest, with two corru- gated metallic plates described by the plasma dielectric function. These calculations have led to the following expression of the lateral energy propagation factors exp −K $^{L}$. Note that the matrices D , R$_{1}$ and R$_{2}$ , which were diagonal on the basis of plane waves when they described specular scattering, are no longer diagonal in the general case of non specular scat- tering. The propagation factors remain diagonal in this basis with their diagonal values written as in (8). Clearly the expression (10) does not depend on the choice of a specific basis. Remark also that (10) takes a simpler form at the limit of null temperature (note the change of no- tation from the free energy F to the ordinary energy E ) Formula (11) has been used to evaluate the effect of roughness or corrugation of the mirrors [47–49] in a per- turbative manner with respect to the roughness or cor- rugation amplitudes (see the next section). It has clearly a larger domain of applicability, not limited to the per- turbative regime, as soon as techniques are available for computing the large matrices involved in its evaluation. It has also been used in the past years by different groups using different notations [39–41, 59, 60]. The relation be- tween these approaches is reviewed for example in [61]. The variation of this ratio ρ$_{C}$ with the parameters of interest has been described in a detailed manner in [48, 49]. Curves are drawn as examples in the Fig. 1 of [48] with λ$_{P}$ = 137nm chosen to fit the case of gold covered plates. An important feature is that ρ$_{C}$ is smaller than unity as soon as k$_{C}$ significantly deviates from 0. For large values of k$_{C}$ , it even decays exponentially to zero, leading to an extreme deviation from the PFA. Other situations of interest have also been studied. When the corrugation plates are rotated with respect to each other, a torque appears to be induced by vacuum fluctuations, tending to align the corrugation directions [62]. In contrast with the similar torque appearing be- tween misaligned birefringent plates [63], the torque is here coupled to the lateral force. The advantage of the configuration with corrugated plates is that the torque has a larger magnitude. Another case of interest may be designed by using the possibilities offered by cold atoms techniques. Non trivial effects of geometry should be vis- ible in particular when using a Bose-Einstein condensate as a local probe of vacuum above a nano-grooved plate [64, 65]. As already stated, the lateral Casimir force between corrugated plates is a topic of particular interest. This configuration is more favorable to theory/experiment comparison than that met when studying the normal Casimir force. It could thus allow for a new test of Quantum ElectroDynamics, through the dependence of the lateral force to the corrugation wavevector [48, 49]. Here, we consider two plane mirrors, M1 and M2, with corrugated surfaces described by uniaxial sinusoidal pro- files (see Fig. 1 in [49]). We denote h$_{1}$ and h$_{2}$ the local heights with respect to mean planes z$_{1}$ = 0 and z$_{2}$ = L h$_{1}$ and h$_{2}$ have null spatial averages and L is the mean distance between the two surfaces; h$_{1}$ and h$_{2}$ are both counted as positive when they correspond to separation decreases; λ$_{C}$ is the corrugation wavelength, k$_{C}$ = 2 π/λ$_{C}$ the corresponding wave vector, and b the spatial mis- match between the corrugation crests. These results suggested that non trivial effects of ge- ometry, i.e. effects beyond the PFA, could be observed with dedicated lateral force experiments. It was however difficult to achieve this goal with corrugation amplitudes a$_{1}$, a$_{2}$ meeting the conditions of validity of the pertur- bative expansion. As already stated, recent experiments have been able to probe the beyond-PFA regime with deep corrugations [50, 51] and it also became possible to calculate the forces between real mirrors without the perturbative assumption. In particular, an exact expres- At lowest order in the corrugation amplitudes, when a$_{1}$, a$_{2}$ ≪ λ$_{C}$, λ$_{P}$, L , the Casimir energy may be obtained by expanding up to second order the general formula (11). The part of the Casimir energy able to produce a lateral force is thus found to be h$_{1}$ = a$_{1}$ cos( k$_{C}$x ) , h$_{2}$ = a$_{2}$ cos ( k$_{C}$ ( x − b )) (12) F lat = − ∂δ E ∂b (13) δ E = − ℏ ∫ ∞ 0 d ξ 2 π Tr ( δ R$_{1}$ exp −K L D$_{0}$ δ R$_{2}$ exp −K L D$_{0}$ ) ρ$_{C}$ ( k$_{C}$ ) = G$_{C}$ ( k$_{C}$ ) G$_{C}$ (0) (15) δE = A 2 G$_{C}$ ( k$_{C}$ ) a$_{1}$a$_{2}$ cos( k$_{C}$b ) (14) F = d E d L , E = ℏ ∫ ∞ 0 d ξ 2 π ln det D ( iξ ) (11) with the function G$_{C}$ ( k$_{C}$ ) given in [49]. It has also been shown that the PFA was recovered for long corrugation wavelengths, when G$_{C}$ ( k$_{C}$ ) is replaced by G$_{C}$ (0) in (14). This important argument can be considered as a prop- erly formulated “Proximity Force Theorem” [49]. It has to be distinguished from the approximation (PFA) which consists in an identification of G$_{C}$ ( k$_{C}$ ) with its limit G$_{C}$ (0). For arbitrary corrugation wavevectors, the de- viation from the PFA is described by the ratio The lateral Casimir force between corrugated plates 4