Hyperbolic Continuous Topological Transition in Real Space

Abstract Hyperbolic topological transitions refer to the transformation of isofrequency contours in hyperbolic materials from one topology (e.g., hyperbolic) to another (e.g., elliptical or a different hyperbolic topology). However, current research remains limited to investigating topological transitions in momentum space, thereby hindering the simultaneous real‐space observation of distinct hyperbolic states and their associated topological transitions within a single system. In this work, real‐space hyperbolic continuous topological transitions are investigated using gradient‐index (GRIN) lenses, exemplified by a hyperbolic Luneburg lens. By introducing Wick rotations, it is demonstrated that spatially modulated refractive indices, mediated by variations in out‐of‐plane permittivity, drive continuous transitions between hyperbolic Type I and Type II topologies. Furthermore, using a harmonic oscillator model, the intrinsic relationship between the parameter of a hyperbolic Luneburg lens and its predominant topological behavior, whether hyperbolic Type I or Type II is uncovered, and extend this concept to a broader framework of Morse lenses. This work provides a theoretical foundation for designing materials with tunable topological properties, advancing applications in photonics, metamaterials, and beyond.