Linear nonreciprocal dynamics of coupled modulated systems

Waveguides subject to spatiotemporal modulations are known to exhibit nonreciprocal vibration transmission, whereby interchanging the locations of the source and receiver changes the end-to-end transmission characteristics. The scenario of typical interest is unidirectional transmission in long, weakly modulated systems: when transmission is possible in one direction only. Here, with a view toward expanding their potential application as devices, we explore the vibration characteristics of spatiotemporally modulated systems that are short and strongly modulated. Focusing on two coupled systems, we develop a methodology to investigate the nonreciprocal vibration characteristics of both weakly and strongly modulated systems. In particular, we highlight the contribution of phase to nonreciprocity, a feature that is often overlooked. We show that the difference between the transmitted phases is the main contributor to breaking reciprocity in short systems. We clarify the roles of primary and sideband resonances, and their overlaps, in breaking reciprocity. We discuss the influence of modulation amplitude and wavenumber on the resonances of the modulated system. We hope this work brings more attention to the dynamics of strongly modulated systems.