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.

The standard wave equation describing symmetrical wave propagation in all directions in three dimensions, was discovered by the French scientist d’Alembert, more than 250 years ago. In the 20th century it became important to search for ‘one-way’ versions of this equation in three dimensions– i.e., an equation describing wave propagation in one direction for all angles, and forbiting it in the opposite direction– for a variety of applications in compu tational and topological physics. Here, by borrowing techniques from relati vistic quantum field theory– in particular, from the Dirac equation–,and starting from Engquist and Majda’s seminal, approximative one-way wave equations, we report the discovery of theexactone-waywaveequationin three dimensions. Surprisingly, we find that this equation necessarily– simi larly to the innate emergence of spin in the Dirac equation– has a topological nature, giving rise to strong, spin-orbit coupling and locking, and non vanishing (integer) Chern numbers.

The index-near-zero (INZ) mode exhibits novel spatial phase invariance characteristics. Recent research has focused on exploring INZ-related phenomena using metamaterials, metasurfaces, and photonic crystal (PhC) structures. However, most currently proposed INZ modes lack flexible control and are challenging to implement. Additionally, INZ modes near the Dirac point in PhCs typically operate only at specific frequencies. In this study, Chern PhCs composed of simple magneto-optical materials are utilized to regulate topologically unidirectional INZ electromagnetic modes by adjusting the air thickness and varying the magnitude of an external magnetic field. Due to the unidirectional robustness and near-zero phase shift characteristics of the INZ mode, three application scenarios are proposed: a phase inverter, a perfect 50/50 splitter, and high-performance broadband sensors. This work provides a new platform and approach for optical communication and computing.

Complex-frequency excitations have recently attracted a lot of attention owing to their ability to solve a number of extraordinary challenges in photonics, such as overcoming losses without gain in metalenses and plasmonic waveguides and achieving virtual absorption. However, the totality of the works so far has been mainly computational or experimental, and a full theory of the complex dynamics enabled by these excitations is still missing. Here, we develop a fully analytical, exact time-domain theory for the dynamical scattering of these excitations by both sides of dielectric plates, which have been used to achieve virtual absorption. Our precise theoretical analysis confirms previous observations and, in addition, reveals a number of intriguing phenomena that were previously missed, such as discontinuities in the scattering of the outgoing electromagnetic field and release of the stored energy in distinct packets.

Unidirectional propagation based on surface magnetoplasmons (SMPs) has recently been realized at the interface of magnetized semiconductors. However, usually SMPs lose their unidirectionality due to nonlocal effects, especially in the lower trivial band gap of such structures. More recently, a truly unidirectional SMP (USMP)hasbeen demonstrated in the upper topological nontrivial band gap, but it supports only a single USMP, limiting its functionality. In this work, we present a fundamental physical model for multiple, robust, truly topological USMP modes at terahertz (THz) frequencies, realized in a semiconductor-dielectric-semiconductor (SDS) slab waveguide under opposing external magnetic fields. We analytically derive the dispersion properties of the SMPs and perform numerical analysis in both local and nonlocal models. Our results show that the SDS waveguide supports two truly (even and odd) USMP modes in the upper topological nontrivial band gap. Exploiting these two modes, we demonstrate unidirectional SMP multimode interference (USMMI), being highly robust and immune to backscattering, overcoming the back-reflection issue in conventional bidirectional waveguides. To demonstrate the usefulness of this approach, we numerically realize a frequency and magneti cally tunable arbitrary-ratio splitter based on this robust USMMI, enabling multimode conversion. We, further, identify a unique index-near-zero (INZ) odd USMP mode in the SDS waveguide, distinct from conventional semiconductor-dielectric-metal waveguides. Leveraging this INZ mode, we achieve phase modulation with a phase shift from −π to π. Our work expands the manipulation of topological waves and enriches the field of truly nonreciprocal topological physics for practical device applications.

Rainbow trapping is a wave localization phenomenon in which different frequencies are spatially separated and con f ined by engineering dispersion through structural gradients. Initially demonstrated in tapered metamaterial systems, this concept has since been extended to plasmonic, photonic, acoustic, and elastic platforms, where graded-index profiles, chirped periodicities, and tapered geometries are used to control the group velocity and localize wave components at distinct spatial positions. These implementations enable high resolution spectral manipulation and form the foundation for broadband wave control. More recently, topological rainbow trapping has emerged as a robust alternative, leveraging topo logically protected states to achieve disorder-immune fre quency localization. This approach offers enhanced resilience to fabrication imperfections and opens new possibilities for scalable, integrated wave-based devices. In this review, we examine the physical mechanisms, system-specific implemen tations, and recent advances in both conventional and topo logical rainbow trapping. We also highlight promising appli cations ranging from optical communication and wavelength multiplexing to acoustic wave manipulation and vibrational energy harvesting and discuss key challenges and future directions in this rapidly evolving field.

We develop a full-wave electromagnetic (EM) theory for calculating the multipole decomposition in two-dimensional (2-D) structures consisting of isolated, arbitrarily shaped, inhomogeneous, anisotropic cylinders or a collection of such. To derive the multipole decomposition, we first solve the scattering problem by expanding the scattered electric field in divergenceless cylindrical vector wave functions (CVWFs) with unknown expansion coefficients that characterize the multipole response. These expansion coefficients are then expressed via contour integrals of the vectorial components of the scattered electric field evaluated via an electric field volume integral equation (EFVIE). The kernels of the EFVIE are the products of the tensorial 2-D Green’s function (GF) expansion and the equivalent 2-D volumetric electric and magnetic current densities. We validate the theory using the commercial finite element solver COMSOL Multiphysics. In the validation, we compute the multipole decomposition of the fields scattered from various 2-D structures and compare the results with alternative formulations. Finally, we demonstrate the applicability of the theory to study an emerging photonics application on oligomer-based highly directional switching using active media. This analysis addresses a critical gap in the current literature, where multipole theories exist primarily for three-dimensional (3-D) particles of isotropic materials. Our work enhances the understanding and utilization of the optical properties of 2-D, inhomogeneous, and anisotropic cylindrical structures, contributing to advancements in photonic and meta-optics technologies.

We have theoretically investigated surface magnetoplasmons (SMPs) in an yttrium-iron-garnet (YIG) sandwiched waveguide. The dispersion demonstrated that this waveguide can support topological unidirectional SMPs. Based on unidirectional SMPs, magnetically controllable multimode interference (MMI) is verified in both symmetric and asymmetric waveguides. Due to the coupling between the modes along two YIG–air interfaces, the asymmetric waveguide supports a unidirectional even mode within a single-mode frequency range. Moreover, these modes are topologically protected when a disorder is introduced. Utilizing robust unidirectional SMP MMI (USMMI), tunable splitters have been achieved. It has been demonstrated that mode conversion between different modes can be realized. These results provide many degrees of freedom to manipulate topological waves.