Abstract
Time-modulated photonic media have recently emerged as a powerful platform for controlling light-matter interactions, offering new degrees of freedom through the temporal modulation of material parameters such as permittivity. A central phenomenon in these systems is the appearance of momentum gaps, typically associated with complex-valued frequencies, which can result in non-Hermitian effects like amplification or attenuation. In this work, we derive a comprehensive analytical condition that enables the realization of momentum gaps in homogeneous, time-modulated photonic media while strictly preserving Hermiticity and thus avoiding unphysical energy gain or loss. By generalizing from binary-state to arbitrary n-state modulation schemes, we establish a unified framework using both discrete transfer matrix products and their continuous differential equation analogs. We provide explicit examples of modulation waveforms and demonstrate, both analytically and numerically, how the condition ensures a non-divergent, Hermitian dispersion relation, where denotes the imaginary part of the modulation frequency. Furthermore, we analyze the impact of simultaneous temporal and spatial (wavevector) dependencies, revealing a rich interplay between time evolution and dispersion engineering. Our findings not only deepen the theoretical understanding of time-modulated photonic systems but also pave the way for novel photonic devices with engineered band gaps, robust against non-Hermitian instabilities.
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