Semiclassical quantization rules and numerical calculations are applied to study polariton modes of materials whose permittivity tensor has principal values of opposite sign (so-called hyperbolic materials). The spectra of volume-and surface-confined polaritons are computed for spheroidal nanogranules of hexagonal boron nitride, a natural hyperbolic crystal. The field distribution created by polaritons excited by an external dipole source is predicted to exhibit raylike patterns due to classical periodic orbits. Near-field infrared imaging and Purcell-factor measurements are suggested to test these predictions.
KEYWORDS:Hyperbolic materials, boron nitride, polariton, nanoresonator, Purcell's factor, semiclassical quantization R ecently much interest has been attracted to a class of uniaxial materials whose axial ε z and tangential ε ⊥ permittivities have opposite signs. These hyperbolic materials (HM) possess extraordinary rays with unusual properties. In this Letter, we focus on polar dielectric HM 1−5 where the extraordinary rays are phonon-polariton collective modes. Our results may also apply to other HM, including ferromagnets, 6 magnetized plasmas, 7 artificial metamaterials, 8 layered superconductors, 9,10 and liquid crystals.
11The basic properties of hyperbolic polaritons are as follows. Their isofrequency surfaces ω(p) = const in momentum space p = (p x , p y , p z ) are hyperboloids. In a broad range of |p| from the free-space photon momentum ω/c to an upper cutoff imposed by microscopic structure, these hyperboloids can be approximated by cones (Figure 1a)The group velocity v(p) = ∂ p ω is always orthogonal to the isofrequency surface. Hence, within the conical approximation it has a fixed angle α = tan/(ε z ) 1/2 ]} with respect to the optical axis. Such a strictly directional propagation of polaritons may be used for subdiffractional focusing 12,13 and super-resolution imaging known as "hyperlensing".
8,14−16Because the high-momenta polaritons remain immune to evanescent decay, volume-confinement of polaritons inside nanogranules 3,17,18 is possible. Several experimental observations of such modes in hexagonal boron nitride (hBN), a natural mid-infrared HM, have been reported. 2,3,12,13,17,19 (This layered insulator is also known to be a premier substrate 20 or a spacer for van der Waals heterostructures. 21,22 ) In the far-field spectroscopy, 3 the polariton modes of hBN nanogranules show up as discrete resonances. Remarkably, the spectrum of such resonances was found to depend primarily on the aspect ratio of the granules rather than their size or precise shape. Exact solutions 1,6,10 for spheroidal or spherical shapes enable one to compute such spectra but they do not elucidate the underlying physical picture.In this Letter, we further develop an alternative ray optics method 23 that makes connection to the Einstein−Brillouin− Keller (EBK) quantization 24,25 of a classical particle inside a cavity having the same shape as the granule. The indefinite permittivity tensor of the HM maps o...