Rescaled Schwarz-Christoffel Transformations for Isotropic, Polygon, and Multiphysics Metamaterials

Conformal transformation is a powerful paradigm for designing isotropic metamaterials with significant practical value. However, conventional conformal transformation faces challenges in achieving precise multiphysics control, specifically for the concurrent regulation of thermal and electromagnetic fields. This difficulty arises because the interface matching mechanisms for dissipative thermal fields and nondissipative electromagnetic fields are fundamentally different. To overcome these challenges, we propose a common design paradigm for multiphysics-applicable interface matching, based on rescaled Schwarz-Christoffel transformation (RSCT). This framework leverages the Schwarz-Christoffel transformation-a conformal map between polygonal domains that intrinsically directs energy flow-and integrates it with rescaling techniques that enable the redistribution of energy flow density. By simultaneously controlling energy flow and energy flow density, two key quantities that can be manipulated in both thermal and electromagnetic fields, RSCT establishes a general platform for perfect interface matching using solely isotropic media. Employing RSCT, we design and experimentally validate three energy-flow regulators: expanders, guiders, and cloaks. Furthermore, the inherent capability of RSCT to form devices of arbitrary polygonal shapes makes it highly compatible with the layout of integrated circuits, holding great promise for coordinated signal and heat management in such systems.