On the Fourier transform of the hyperbola and its role in hyperbolic photonics

TL;DR

The paper analyzes emission from a localized source in hyperbolic media by deriving the 2D inverse Fourier transform of hyperbolic dispersion, revealing a piecewise real-space field with distinct major and minor regions separated by a separatrix. It interprets the results through a pitch–wavevector correspondence, extends Huygens' principle to hyperbolic wavefronts, and demonstrates negative refraction and lensing within this framework. The work also discusses image aliasing as a practical artifact and provides analytical tools for modeling polariton propagation in materials with extreme anisotropy, with potential broad applicability to metamaterials and wavefield synthesis. Overall, the study offers a rigorous, geometry-driven view of how hyperbolic dispersion shapes radiation patterns and wave propagation.

Abstract

Motivated by recent breakthrough studies of wave hyperbolicity in extremely anisotropic natural materials and artificial composites, we investigate the radiation pattern of a localized emitter in a hyperbolic medium. Since the emission of a point source is associated with the Fourier transform of the iso-frequency contours of a medium, we derive and analyze the properties of the Fourier transform of hyperbolic dispersion, which sheds light into the emission properties in the presence of hyperbolic bands. Our analysis leads to a generalized form of Huygens' principle for hyperbolic waves, connecting to the emergence of negative refraction and focusing with hyperbolic media. We also highlight the occurrence of aliasing artifacts in polariton imaging. More broadly, our findings provide analytical tools to model polariton propagation in materials with extreme anisotropy, and may be applied to several other physical platforms featuring hyperbolic responses, from astrophysics to seismology.

Figures (13)

• Figure 1: Fourier transform of a hyperbola obtained (a) experimentally via Fourier imaging of a pair of truncated hyperbolic slits; (b) analytically for an infinite hyperbola using our formulation. In both cases, rectangular hyperbolas with $\epsilon_y = -\epsilon_x = 1$ were considered (left panels), and the corresponding Fourier patterns exhibit distinct hyperbolic fringes (right panels).
• Figure 2: Fourier transform of a hyperbola in the separatrix is characterized by a pair of straight lines where its functional value is unbounded. For a given hyperbola in reciprocal space (a), each of its two asymptotes is perpendicular to one of the separatrix lines in the corresponding transform (b). This orthogonality relation is a natural consequence of the transform holding true in the limiting case of $\kappa \rightarrow 0$.
• Figure 3: Fourier transform of a hyperbola in the minor region shows sharp decay from the separatrix lines with hyperbolic contours. For the hyperbola in Fig. \ref{['Fig:regIII']}a, this region contains the upper and lower quadrants of Fig. \ref{['Fig:regIII']}b. The transform contours are given by a family of coasymptotic hyperbolas with three such hyperbolas shown in (a) for different values of the hyper-radius $\Lambda, \Lambda'$ and $\Lambda"$. As the transform value decays quickly with the hyper-radius (inset of (a)), we have the resulting transform in this region (b) exhibit sharp decay with hyperbolic contours.
• Figure 4: Fourier transform of a hyperbola in the major region shows hyperbolic fringes. For the hyperbola in Fig. \ref{['Fig:regIII']}a, this region contains the left and right quadrants of Fig. \ref{['Fig:regIII']}b. The transform contours are given by a family of coasymptotic hyperbolas with three such hyperbolas shown in (a) for different values of the hyper-radius $\Lambda, \Lambda'$ and $\Lambda"$. As the transform value shows oscillations in addition to a slow decay with the hyper-radius (inset of (a)), we have the resulting transform in this region (b) exhibit hyperbolic fringes decaying slowly from the separatrix lines.
• Figure 5: With the transform pairs shown for a circle (a)--(b) and a hyperbola (c)--(d), the respective Fourier patterns can be interpreted by the "pitch--wavevector correspondence" which states that a narrow sector (highlighted slices) in the real space, with its fringes approximated by a series of parallel tangents (dashed lines) accommodating a certain pitch, provides us with a wavevector (red arrows) of the right magnitude and right direction, so that it corresponds to a valid momentum state in the reciprocal space. For the purpose of illustration, we have not shown the wavevectors in the opposite direction coming from the same sector.