content_image/332459.jpg content_image/332458.jpg content_image/332457.jpg content_image/332455.jpg content_image/332456.jpg
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: 

<img src='content_image/332459.jpg'>

The solutions of Maxwell equations should satisfy the quasi-periodicity conditions: 

<img src='content_image/332458.jpg'>

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 

<img src='content_image/332457.jpg'>

<img src='content_image/332455.jpg'>

<img src='content_image/332456.jpg'>

