region is equal to h for the bottom grating. The meaning of the lateral displacement x$_{0}$ becomes obvious from the comparison of Fig. 11 with Fig. 12 and Fig. 13. We suppose that the space between the two gratings is vacuum with ε = µ = 1 and we assume µ = 1 inside the medium. The physical problem is time and y invariant, so that the particular solutions for the electric and magnetic fields can be written in the form: The solutions of Maxwell equations should satisfy the quasi-periodicity conditions: In every Casimir problem one needs to determine the complete basis of solutions. Let us suppose that the top grating is absent. We consider a generalized conical diffraction problem ( k$_{y}$ = 0) on the lower grating for the incident wave with the x -component of the wave vector equal to some fixed value which we denote ˜ k$_{x}$ . The longitudinal components of the electromagnetic field outside the corrugated region ( z ≥ h ) may be written by making use of the Rayleigh expansion [72] for an incident monochromatic wave: Here with 27 k$_{x}$ = ˜ k$_{x}$ − 2 π Λ [ Λ ˜ k$_{x}$ 2 π ] ≡ ˜ k$_{x}$ − 2 π Λ p, (46) α$_{n}$ = k$_{x}$ + 2 πn/ Λ (45) E$_{y}$ ( x, z ) = I ( e ) ˜ k x exp( i ˜ k$_{x}$x − i ˜ β $^{(1)}$z ) + ∞ ∑ n = −∞ R ( e ) np exp( iα$_{n}$x + iβ (1) n z ) , (44) H$_{y}$ ( x, z ) = I ( h ) ˜ k x exp( i ˜ k$_{x}$x − i ˜ β $^{(1)}$z ) + ∞ ∑ n = −∞ R ( h ) np exp( iα$_{n}$x + iβ (1) n z ) . E$_{i}$ ( x + Λ , z ) = e ik x $^{Λ}$E$_{i}$ ( x, z ) , (43) H$_{i}$ ( x + Λ , z ) = e ik x $^{Λ}$H$_{i}$ ( x, z ) . E$_{i}$ ( x, y, z, t ) = E$_{i}$ ( x, z ) exp( ik$_{y}$y − iωt ) , H$_{i}$ ( x, y, z, t ) = H$_{i}$ ( x, z ) exp( ik$_{y}$y − iωt ) . (42)