The interaction of light with a semiinfinite dielectric as a phonon problem; the generalized Snellius law and Fresnel formulae

Using Ewald's formulae it is possible to evaluate the electromagnetic field generated by a mechanical plane wave propagating through a semiinfinite dipole crystal. The problem of refraction and reflection of an external electromagnetic wave on a semiinfinite dipole crystal is thus reduced to the evaluation of the forced vibrations of this crystal excited by the incident electromagnetic wave. Applying this approach, the generalized Snellius law and generalized Fresnel formulae can be deduced. In the limit of long waves these general formulae change into the usual classical expressions. The Ewald extinction theorem follows directly from the lattice dynamics of our system.