Fig. 1. (a) Experimental setup geometry, where a polarizer is used to generate a TM-polarized beam, which is incident at the surface of crystal quartz at an incident angle of θ=45° from a dielectric prism (εp=5.5). (b) Real part of the principal components of the dielectric function of quartz in the frequency range 410–610cm−1. (c) Theoretical (dashed line) and experimental (solid line) reflectance spectra [using the setup shown in (a)] for a crystal quartz sample whose anisotropy lies along the z axis and an air gap of d=2.0μm.

Fig. 2. (a) Geometry of the dielectric components with respect to the laboratory axis when introducing the angle φ. (b) Geometry of the dielectric components with respect to the laboratory axis when introducing the angle β.

Fig. 3. The dependence of ATR in crystal quartz on the azimuthal angle β. (a) φ=45°, (b) φ=90°, with kxk0=1.66. εp=5.5 and d=2μm. (c) ATR of quartz at ω/2πc=490cm−1 when φ=90°, with the radius corresponding to kx where εp=5.5, and the azimuthal angle corresponding to the angle β, with d=2μm.

Fig. 4. ATR spectra for crystal quartz at ω/2πc=460cm−1, with the radius corresponding to kx where εp=50, and the azimuthal angle corresponding to the angle β. The white circle denotes where kx/k0=1. In (a) there is no air gap (d=0μm) and φ=90°. In (b) the anisotropy orientation is unchanged (φ=90°) and an air gap is introduced to study the GHP, with d=0.1μm. In (c), the air gap is removed (d=0μm) and the anisotropy is rotated to φ=60° to alter the hyperbolic dispersion. In (d), the anisotropy orientation is unchanged (φ=60°) and an air gap is introduced again to study the GHP, with d=0.1μm.

Fig. 5. The dependence of ATR in quartz on the azimuthal angle β, with kxk0=5 and εp=50. An air gap is included (d=0.1μm) to probe the GHP. In (a), the anisotropy is aligned with the interface (φ=90°). In (b), the anisotropy is rotated below the interface where φ=60°.

Fig. 6. ATR spectra for crystal quartz at ω/2πc=545cm−1, with the radius corresponding to kx where εp=2.2, and the azimuthal angle corresponding to the angle β. In (a) there is no air gap (d=0μm) and φ=90°. In (b) the anisotropy orientation is unchanged (φ=90°) and an air gap is introduced to study the LHP, with d=10μm. In (c), the air gap is removed (d=0μm) and the anisotropy is rotated to φ=70° to alter the hyperbolic dispersion. In (d), the anisotropy orientation is unchanged (φ=70°) and an air gap is introduced again to study the LHP, with d=10μm. The white, solid lines denote the light line (i.e., kx/k0=ky/k0=1.0).

Fig. 7. The dependence of ATR in quartz on the azimuthal angle β, with kx/k0=1.05 and εp=5.5. An air gap is included (d=10μm) to probe the LHP. In (a), the anisotropy is aligned with the interface (φ=90°). In (b), the anisotropy is rotated below the interface such that φ=70°.

Fig. 8. The dependence of ATR in crystal quartz on the azimuthal angle β when φ=90° and kxk0=1.66. We show our experimental work alongside two different theoretical air gap sizes, where in (a) d=1.5μm and in (b) d=2.5μm.

Fig. 9. The cross polarization conversion (Rps) induced by the GHP in quartz at ω/2πc=460cm−1. This quantity is the amount of s-polarized light reflected off the surface from incident p-polarized radiation. The radius corresponds to kx where εp=50, and the azimuthal angle corresponds to the angle β. The white circle denotes where kx/k0=1. In (a) there is no air gap (d=0μm) and φ=90°. In (b) the anisotropy orientation is unchanged (φ=90°) and an air gap is introduced to study the GHP, with d=0.1μm. In (c), the air gap is removed (d=0μm) and the anisotropy is rotated to φ=60° to alter the hyperbolic dispersion. In (d), the anisotropy orientation is unchanged (φ=60°) and an air gap is introduced again to study the GHP, with d=0.1μm.

Fig. 10. The dependency of the ATR response of quartz on the air gap thickness d at constant frequency ω/2πc=465cm−1 and the anisotropy axis aligned with the surface (φ=90°), with the radius corresponding to kx where εp=80, and the azimuthal angle corresponding to the angle β. The white circle denotes where kx/k0=1. In (a) d=0.1μm. In (b) d=0.15μm. In (c) d=0.2μm. In (d) d=0.25μm.

Fig. 11. The frequency-dependent ATR response of quartz with a constant air gap d=0.1μm and the anisotropy axis aligned with the surface (φ=90°), with the radius corresponding to kx where εp=80, and the azimuthal angle corresponding to the angle β. The white circle denotes where kx/k0=1. In (a) ω/2πc=455cm−1. In (b) ω/2πc=460cm−1. In (c) ω/2πc=465cm−1. In (d) ω/2πc=470cm−1.

Fig. 12. The frequency-dependent ATR response of calcite with a constant air gap d=0.5μm and the anisotropy axis aligned with the surface (φ=90°), with the radius corresponding to kx where εp=18, and the azimuthal angle corresponding to the angle β. The white circle denotes where kx/k0=1. In (a) ω2πc=1460cm−1. In (b) ω2πc=1470cm−1. In (c) ω2πc=1480cm−1. In (d) ω2πc=1490cm−1.

Fig. 13. The cross polarization conversion (Rps) induced by the LHP in quartz at ω2πc=545cm−1. This quantity is the amount of s-polarized light reflected off the surface from incident p-polarized radiation. The radius corresponds to kx where εp=4.5, and the azimuthal angle corresponds to the angle β. The white circle denotes where kx/k0=1. In (a) there is no air gap (d=0μm) and φ=90°. In (b) the anisotropy orientation is unchanged (φ=90°) and an air gap is introduced to study the LHP, with d=10μm. In (c), the air gap is removed (d=0μm) and the anisotropy is rotated to φ=70° to alter the hyperbolic dispersion. In (d), the anisotropy orientation is unchanged (φ=70°) and an air gap is introduced again to study the LHP, with d=10μm.

Fig. 14. The frequency-dependent ATR response of quartz with a constant air gap d=10μm and the anisotropy axis aligned with the surface (φ=90°), with the radius corresponding to kx where εp=4.5, and the azimuthal angle corresponding to the angle β. The white circle denotes where kx/k0=1. In (a) ω/2πc=540cm−1. In (b) ω/2πc=545cm−1. In (c) ω/2πc=550cm−1. In (d) ω/2πc=555cm−1.