Asymmetric band structure calculations using the plane wave expansion method with time-modulated permittivity

In this work we show how the plane wave expansion method for calculating the band structure of materials with periodic electric permittivity may be extended to calculate the band structure of periodic materials that also possess a sinusoidal time-modulation. The numerical technique is applied to a structure possessing a synthetic linear momentum which causes unidirectional bandgaps to appear in the band structure. Such devices could be of use for tunable magnet-free isolators in integrated photonics platforms.