  This study examines the electronic states of quantum materials influenced by
waves V(Qx-omega t) propagating through them. These waves can include laser
light, polaritons, acoustic waves, and sliding density waves, whose speed may
vary widely. Since these waves are periodic in both space and time, the
electrons in the materials form Floquet-Bloch states when influenced by the
waves. We investigate band engineering by these waves, using three-dimensional
(3D) Dirac electrons in circularly polarized laser wave as a case study. We
classify the fields by their speed into time, light and space-like waves, and
find that the electronic states are distinctive in the three cases. We
demonstrate phase transitions of the electronic bands and discover emergent
Floquet Weyl bands, which potentially become Type-II in the vicinity of the
transition. The effect of Floquet band engineering is enhanced by Lorentz
contraction, which is maximized when the wave speed approaches the Fermi
velocity. Additionally, we discuss the geometric nature of the response
currents.
