Floquet modes in quasi-periodic temporal photonic crystals based on the Kolakoski sequence

We present a theoretical framework for quasi-periodic temporal photonic crystals (PCs) based on four types of deterministic Kolakoski sequences. The proposed system consists of two temporally modulated dielectric responses, where the refractive index contrast generates a momentum gap in the frequency-momentum space. Unlike conventional temporal PCs, the deterministic aperiodic ordering of the Kolakoski sequence creates multiple momentum gaps and resonant states over the momentum space. Using the transfer matrix method, we systematically analyze the influence of the permittivity contrast and temporal layer duration contrast on the transmission characteristics in both lossless and lossy cases. Results show that aperiodic order enhance the momentum gaps due to the increased number of temporal layers and stronger Floquet modes. Increasing permittivity contrast increasing width in the momentum gaps, while varying layer durations, causes larger momentum shifts in the momentum spectrum in both systems. In the lossy regime, material loss slightly enhance the amplification without significantly altering the position of the momentum gaps. These findings highlight the potential of Kolakoski-based temporal PCs for ultrafast, dynamically tunable photonic devices.