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A river flows in one direction, but a wave on its surface travels in the opposite direction. The same splitting happens in hyperbolic materials: the energy of a light wave and its fronts diverge, preventing a sharp lens from being created. Physicists tamed this chaos by applying the geometry of spacetime from special relativity, created by Einstein based on the work of Lorentz. The material's properties are transformed into a metric, where the speed of light plays the role of a fundamental constant, and everything falls into place.

Scientists derived universal formulas that allow designing lenses that focus light with enormous precision, bypassing the natural blur inherent to any lens. Such optics distinguish details hundreds of times finer than the wavelength — for example, a magnifying glass capable of reading text on an atom. In the experiment, a flat lens was simulated for the mid-infrared range, using polaritons — quantum oscillations of the crystal lattice.

Original abstract

as written by the authors arXiv:2603.14166

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 ultra-large numerical apertures and deep sub-diffraction focusing. We validate our theory with the design and full-wave modeling of a planar van der Waals polaritonic lens operating in the mid-infrared frequency range.