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Surface and interface optical waves in superlattices: transverse electric localized and resonant modes

We investigate the existence and behaviour of localized and resonant optical waves associated with the surface of a semi-infinite superlattice or its interface with a substrate, considering the case of transverse electric modes. In this paper, we present an analytic determination of the response function (Green function) for a semi-infinite superlattice with or without a cap layer, and for a superlattice in contact with a substrate. This calculation enables us to obtain both local and total densities of states as functions of the frequency and the wavevector (parallel to the interfaces). Besides the surface and interface localized modes which occur as delta peaks inside the gaps, one can also obtain resonant states as well defined peaks of the density of states inside the bulk bands of the superlattice. In particular, we emphasize the different types of mode induced by a cap layer. In addition, we show that the creation of a semi-infinite superlattice gives rise to delta peaks of weight -1/4 in the density of states at every edge of the superlattice bulk bands.