Analog optical computing with hyperbolic phonon polaritons

Analog optical computing presents a promising alternative to conventional electronic methods, offering advantages in speed and energy efficiency. However, its practical deployment has been hindered by device size, fabrication complexity, and limited functionality. In this work, we demonstrate a compact and efficient analog computing device that leverages deep-subwavelength hyperbolic phonon polaritons (HPhPs) in hexagonal boron nitride (hBN). These polaritons are excited by a pixelated gold (Au) launcher pattern and observed directly using scattering-type scanning near-field optical microscopy (s-SNOM) in the mid-infrared. By mapping the measured near-field intensity and phase distributions, we reconstruct a scattering matrix that characterizes the device’s spatial response. We then employ a Rayleigh-Ritz discretization method to convert differential equations into stiffness matrices, which we compare to the experimentally obtained scattering matrix to identify the specific differential operator realized by the structure. Our results represent a step toward practical, miniaturized analog computing devices based on phonon polariton platforms, offering low-loss, broadband operation and straightforward fabrication.