Rigid vibrations of a photonic crystal and induced interband transitions

We investigate the behavior of electromagnetic states associated with photonic crystals, which are undergoing rigid time-dependent translations in position space. It is shown, quite generally, that the Bloch wave vector q remains a conserved quantity and that an analogue of Bloch’s theorem for a time-dependent solution of the states can be formulated. Special attention is focussed on time-dependent translations involving harmonic rigid vibrations of the photonic crystal. Under these conditions it is shown how, and to what extent, inter-band transitions can be induced between the various bands in a photonic crystal in a microwave regime. In particular, a characteristic resonance transition time can be derived, which scales inversely with the amplitude of vibration and interband frequency. Finally, it is argued that given all parameters other than Bloch wave vector fixed, an interband transition time is minimized if the transition is made at a Bragg plane.