Perturbation theory for dispersion relations of spacetime-periodic materials

We consider Bloch states of weak spacetime-periodic perturbations of homogeneous materials in one spatial dimension. The interplay of space- and time-periodicity leads to an infinitely degenerate dispersion relation in the free case. We consider a general perturbation term, and, as a consequence of the infinite degeneracy, we show that the effective equations are given by the eigenvalue problem of an infinite matrix. Our method can be viewed as a time-modulated generalisation of the nearly-free electron model. Based on this result, we find that the infinite degeneracy may split into a family of non-degenerate bands. Our results are illustrated with numerical calculations, and we observe close agreement between the perturbation theory and the numerically computed full solution.