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

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• Photonic crystals
• Time crystals
• Metamaterials
• Perturbative methods