We present a method to compute the optical functions of superlattices in the excitonic energy region including the effect of the coherence between the electron-hole pair and the electromagnetic field. The electron-hole screened Coulomb potential is adopted and the valence-band structure is taken into account in the cylindrical approximation, thus separating light- and heavy-hole motions. The calculated optical functions have poles in correspondence to the polariton eigenvalues for a multiplicity of excitonic states. We also calculate the amplitudes of higher polariton branches and the line shapes of the optical functions. Numerical examples appropriate to GaAs/${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As superlattices are given and the effect of coherence is displayed in the line shape of the excitation spectrum. \textcopyright{} 1996 The American Physical Society.