Surface excitons in the presence of a transition layer

In the preceding derivation of the frequencies of surface polaritons and surface excitons the boundary conditions were applied at a sharp boundary without surface currents and charges. In this simplest version of the theory the so-called transition subsurface layer has been ignored however, this layer is always present at the interface between two media, and its dielectric properties differ from the dielectric properties of the bulk. Transition layers may be of various origins, even created artificially, e.g. by means of particular treatment of surfaces or by deposition of thin films of thickness d A. Nevertheless, within the framework of the phenomenological theory it is rather easy to take account of their effects on surface wave spectra in an approximation linear in k (15). 

It follows from eqn (12.4) for E and D, when the transition layer is taken into account, that the tangential component Et and the normal component Dn are discontinuous. Within the framework of the linear theory, at an interface with 

Now substituting (12.7) into (12.18) and taking into account that for Coulomb surface excitons = k2 = k (see eqn 12.11), we obtain a system of two equations for the quantities C and C2  

The condition that the determinant of this system be equal to zero has the form 

The quantity a for real fi, 7 and e(iv) represents the group velocity of a surface exciton. Since it follows from (12.13) that 

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