Minkowski-Space Modeling of Hyperbolic Lenses

TL;DR

This paper introduces a Minkowski-space geometric approach to model hyperbolic lenses, enabling rational design and analysis of their focusing capabilities and resolution limits.

Contribution

It presents a novel Minkowski-space framework that simplifies hyperbolic wave modeling and guides the design of ultra-large numerical aperture lenses.

Findings

Derived transfer function and resolution limits for hyperbolic lenses.

Validated the theory with a mid-infrared van der Waals polaritonic lens.

Enabled deep sub-diffraction focusing with high numerical aperture.

Abstract

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.