Transformation optics (TO) facilitates flexible designs of spatial modulation of optical materials via coordinate transformations, thus, enabling on-demand manipulations of electromagnetic waves. However, the application of TO theory in control of hyperbolic waves remains elusive due to the spatial metric signature transition from ( + , + ) to ( − , + ) of a two-dimensional hyperbolic geometry. Here, we proposed a distinct Pythagorean theorem, which leads to establishing an anisotropic Fermat’s principle. It helps to construct anisotropic geometries and is a powerful tool for manipulating hyperbolic waves at the nanoscale and polaritons. Making use of absolute instruments, the excellent collimating and focusing behaviors of naturally in-plane hyperbolic polaritons in van der Waals α – MoO 3 layers are demonstrated, which opens up a new way for polaritons manipulation.

Natural in-plane hyperbolic crystals (such as α-MoO3) and natural monoclinic crystals (such as β-Ga2O3) have recently drawn great research focus. Despite their obvious similarities, however, these two kinds of materials are usually studied as separate topics. In this Letter, we explore the intrinsic relationship between materials like α-MoO3 and β-Ga2O3 under the framework of transformation optics, providing another perspective to understand the asymmetry of hyperbolic shear polaritons. It is worth mentioning that we demonstrate this novel, to the best of our knowledge, method from theoretical analysis and numerical simulations, which maintain a high degree of consistency. Our work not only combines natural hyperbolic materials with the theory of classical transformation optics, but also opens new avenues for future studies of various natural materials.

Hyperbolic materials have attracted considerable interest for their unique open hyperbolic dispersion properties. These materials support high-momentum propagating modes and strong light confinement, leading to a wide range of applications including super-resolution technologies, negative refraction and long-life propagation. Even with these wonderful optical properties, hyperbolic materials, however, cause problems when applying perfectly matched layer (PML) boundary conditions in numerical simulation software such as COMSOL Multiphysics. Due to the unfit embedded attenuation function, the built-in PML of simulation software would result in a mass of reflections in the computational domain when the background medium is hyperbolic materials. Here, we take advantage of an imaginary coordinate mapping and the complex coordinate stretching of transformation optics theory to design a PML for biaxial hyperbolic materials, which avoids any reflections and can be tuned flexibly. The proposed recipe can provide antidote and new insights for hyperbolic material studies.