Topological rainbow trapping

Topological rainbow trapping (TRT) arises from the interplay between topological states and frequency-dependent slow-wave effects. Waves first slow down, then become spatially separated by frequency and are ultimately trapped at distinct locations. TRT designs have been primarily explored in the context of photonic crystals and subsequently extended to acoustic and elastic systems. This emerging TRT concept enables robust, frequency-selective localization beyond conventional rainbow trapping, supporting compact, multi-wavelength, topologically protected platforms for extreme wave manipulation. In this Review, we elucidate the fundamental principles of TRT, emphasizing the physical mechanisms that create near-zero group velocity points with robust frequency-dependent localization. We highlight three key TRT mechanisms: graded index profiles, which gradually vary material parameters to reshape dispersion and induce slow-wave effects; higher-order topological corner modes, which exploit localized corner states for robust frequency-specific wave confinement; and synthetic dimensions, which expand the parameter space of the system to engineer stable interface states at distinct frequencies. Furthermore, we address key challenges in TRT, such as energy dissipation and tunability, while highlighting its broad range of potential applications. Finally, we discuss emerging research directions for TRT. Topological rainbow trapping combines slow-wave effects with topological robustness to spatially separate wave frequencies. This Review highlights its physical principles, implementation in different waves-based systems and potential technological impacts.