Hyperbolic Continuous Topological Transition in Real Space

Abstract: Hyperbolic topological transitions refer to the transformation of is 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, we investigate real-space hyperbolic continuous topological transitions using gradient-index (GRIN) lenses, exemplified by hyperbolic Luneburg lens. By introducing Wick rotations, we demonstrate how 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, we uncover the intrinsic relationship between the parameter E of hyperbolic Luneburg lens and its predominant topological behavior, whether hyperbolic Type I or Type II, 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.