Minkowski-Space Modeling of Hyperbolic Lenses.

The extreme anisotropy of hyperbolic materials enables extreme wave confinement, but it is also associated with an inherent misalignment between phase and energy flow, which complicates device modeling and design. Here, we introduce a Minkowski-space approach to describe hyperbolic wave propagation, showing that this complexity is geometric rather than physical. By embedding anisotropy into an effective Lorentzian metric, we establish a rational design framework for hyperbolic interfaces and lenses, and analytically derive their transfer function and resolution limits, enabling ultralarge numerical apertures and deep subdiffraction focusing. We validate our theory with the design and full-wave modeling of a planar van der Waals polaritonic lens operating in the midinfrared frequency range.