[en] This paper examines the band structure and optical selection rules in superlattices with a sinusoidal potential profile. The analysis is motivated by the recent successful fabrication of high quality ZnSe_1-xTe_x superlattices in which the composition x varies sinusoidally along the growth direction. Although the band alignment in the ZnSe_1-xTe_x sinusoidal superlattices is staggered (type II), they exhibit unexpectedly strong photoluminescence, thus suggesting interesting optical behavior. The band structure of such sinusoidal superlattices is formulated in terms of the nearly-free-electron (NFE) approximation, in which the superlattice potential is treated as a perturbation. The resulting band structure is unique, characterized by a single minigap separating two wide, free-electron-like subbands for both electrons and holes. Interband selection rules are derived for optical transitions involving conduction and valence-band states at the superlattice Brillouin-zone center, and at the zone edge. A number of transitions are predicted due to wave-function mixing of different subband states. It should be noted that the zone-center and zone-edge transitions are especially easy to distinguish in these superlattices because of the large width of the respective subbands. The results of the NFE approximation are shown to hold surprisingly well over a wide range of parameters, particularly when the period of the superlattice is short. (c) 2000 The American Physical Society