An exact numerical diagonalization technique is used to calculate the energy eigenstates of a Landau-quantized two-dimensional electron gas under a one-dimensional periodic delta -modulation. On the basis of these calculations, we develop a self-consistent field theory for the absorption coefficient in the mid-infrared frequency regime. An extensive investigation is carried out in which we find that the plasmon mode changes from a cyclotron mode when the modulation is weak to tunnelling-split modes for the intermediate modulation and finally to edge and 1D lattice magnetoplasmon modes when the modulation is strong. Our numerical results show that as the magnetic field decreases, there exists electron tunnelling. However, for strong magnetic fields, electron tunnelling is suppressed. This suppression of electron tunnelling is signalled by the crossover of two tunnelling-split modes into a cyclotron mode at strong magnetic fields. The eigenenergy of the edge mode oscillates with electron density, related to soft or hard potential walls from the modulation for which the electron stays in extended and localized states, respectively.