Hyperbolic materials have attracted significant interest for controlling light at the nanoscale [1–3], due to their intrinsic hyperbolic dispersion and strong electromagnetic field confinement [4]. They have been shown to support surface polaritons [5–13], with long-range and low-loss propagation of subwavelength information [8,11], guided waves [14–17], and negative refraction [4,5,18–20]. In these materials, hyperbolic dispersion results from the principal elements of the permeability or permittivity tensor possessing opposite signs [4,5,8,11,16,21–24]. This is typically a consequence of anisotropy due to elementary resonances in matter and can emerge across a vast range of frequencies depending on the mechanism (e.g., phonon resonances in the infrared [15,25–27] or magnetic resonances [28–32] in the GHz and THz frequency range). The role of anisotropy orientation with respect to the interface has recently received attention for creating direction-dependent optical devices [5,22,33–44] and has, more recently, led to the discovery of peculiar entities, namely ghost hyperbolic polaritons (GHPs) [8] and leaky hyperbolic polaritons (LHPs) [11].

GHPs come from the phenomenon of a “ghost wave” first theorized by Naraminov as an extension of the Dyakonov surface polariton [45]. Dyakonov surface polaritons are polaritons that propagate along the interface of transparent anisotropic media with its anisotropy axis aligned with the surface, or at the interface between two identical anisotropic crystals with different orientations [45–48]; they are highly directional, and confined to very narrow angular orientations of the anisotropy axis [49]. These ghost waves combine the properties of propagating and evanescent fields within anisotropic media. Coupling to these ghost waves in a hyperbolic material (i.e., the GHP) was first proposed in a metamaterial [50] with a tilted anisotropy axis and experimentally observed in tilted calcite [8], a natural hyperbolic material. GHPs have been characterized as virtual surface phonon-polariton modes [51,52], devoid of an electrostatic limit (with damped propagation even in the absence of material loss [53]) with characteristic high-resolution, ray-like propagation at the material surface over distances up to 20 μm [8]. These are a consequence of the anisotropy axis being neither parallel nor perpendicular to the sample surface; this, in turn, gives rise to phase wavefronts slanted away from the surface of the crystal, Poynting vectors parallel to the surface, and exponential decay away from the surface. Since their emergence, spatiotemporal ultrafast dynamics of GHP nanolight pulse propagation has also been studied [2] as well as planar junctions and anisotropic metasurfaces that can support “ghost line waves” that propagate unattenuated along the line interface, with phase oscillations combined with evanescent decay away from the interface [54]. In the visible light range, ghost modes have been observed in the bulk plasmonic material MoOCL2 [53]. Remarkably though, they do not require an interface or a tilted anisotropy axis away from the surface.

LHPs, on the other hand, are hybridizations of propagating ordinary bulk modes and extraordinary surface waves [11]. They originate from the physics of leaky waves, studied since the 1940s, which are guided modes that propagate along open wave-guiding structures [55,56]. Unlike conventional surface waves, leaky waves possess complex propagation constants, allowing energy radiation into free space as they travel [57–59]. Leaky wave antennas have utilized plasmonic metamaterials with epsilon-near-zero (ENZ) dielectric constants to enhance directionality [60]. Anisotropic ENZ materials with tunable transverse dielectric responses and low magnetic permeability materials with strong tangential magnetic fields have also been employed to control emission and directivity [61]. Hyperbolic materials are ideal candidates for supporting leaky waves, particularly around longitudinal optic phonon frequencies where dielectric tensor components are small [4]. These materials offer advantages over plasma layers, which only support leaky waves above their plasma frequency. Additionally, LHPs have been shown to exhibit tilted wavefronts into the bulk and Poynting vectors canted away from the interface [11].

These novel polariton modes have been, very recently, studied in-depth using s-SNOM and further supported by spectroscopy. Here, we demonstrate how GHPs and LHPs manifest through attenuated total reflection (ATR) spectroscopy in bulk hyperbolic crystals, specifically investigating their direction-dependent optical footprint across the crystal’s surface. By investigating how these polaritons appear in the far-field using more widely available infrared spectroscopy, we use ATR to investigate the conditions under which LHPs and GHPs may be supported in bulk hyperbolic media, using crystal quartz as the example material. This should provide valuable insight for integrating these materials into, as well as exploiting these properties for, novel optical devices.

We employ ATR in the Otto configuration [37,62,63], as shown in Fig. 1(a), where reflection occurs at the boundary between a prism and the crystal supporting surface polaritons. In turn, the prism with a high dielectric constant, εp, generates a high in-plane momentum with wavevector kx=k0εpsinθ, where k0=ω/c and θ is the incident angle of the infrared waves with angular frequency ω that couple to polariton modes. Note that if an air gap d is included between the prism and the crystal and radiation is incident at an angle higher than the critical angle of the prism/air interface, total internal reflection occurs and only evanescent modes travel across the air gap.

The natural hyperbolic material used here, crystal quartz, supports hyperbolic polaritons due to infrared-active phonon resonances. This makes the crystal anisotropic with a diagonal dielectric tensor. For instance, in the frequency range between 410 and 610cm−1, corresponding to free-space wavelengths between 16 and 25 μm, there are two regions where the principal components of the dielectric tensor possess opposite signs, as shown in Fig. 1(b). All parameters used for calculating the dielectric tensor components were taken from Ref. [25] as detailed in Section 5. Therefore, hyperbolic behavior is observed: a Type I region, i.e., the extraordinary component ε∥ of the dielectric tensor is negative while the ordinary component ε⊥ is positive, between 510 and 550cm−1, and a Type II region, ε∥>0 and ε⊥<0, between 450 and 480cm−1.

To build context, let us look at the case where ε∥ is perpendicular to the surface of the crystal (along z) and the crystal axes are aligned with the laboratory axes. SPhPs typically propagate along the crystal’s surface with an exponential decay of amplitude away from the surface. In the case shown in Fig. 1(a), SPhPs couple with TM-polarized infrared radiation. Assuming propagation along x and decay along z, given by the parameters α0 (air) and α (hyperbolic material), plane wave solutions take the following form in air: H(x,z,t)=y^Hei(kxx−ωt)eα0z,(1)and in the hyperbolic material: H(x,z,t)=y^Hei(kxx−ωt)e−αz.(2)Using Maxwell’s equations, similarly to Ref. [28], and matching its solutions inside and outside the hyperbolic material, the boundary conditions allow us to find the following dispersion relation for the surface polaritons: 0=α0+αεxx.(3)Since α0 and α are positive (to ensure decay away from the surface) and the anisotropy axis is oriented along z such that εxx=ε⊥, Eq. (3) can only be satisfied when ε⊥<0. The corresponding ATR response is shown in Fig. 1(c) where a distinct minimum in the reflection is observed between hyperbolic regions, corresponding to where the condition set out above is met. In this case, evanescent waves in air, generated by the total internal reflection in the prism, couple to SPhPs in the hyperbolic material. Here, we used a coupling prism with dielectric constant εp=5.5 and an air gap of d=1.5μm at a fixed incident angle of θ=45°, corresponding to an in-plane momentum of kx/k0=1.66 to induce total internal reflection.

While the behavior of SPhPs detailed above is well known, it can be drastically modified by controlling the orientation of the anisotropy. One way to achieve this is by tilting the anisotropy axis by an angle φ away from the z axis, as shown in Fig. 2(a), so that ε∥ is neither perpendicular nor parallel to the interface. When discussing this angle, a few special configurations are worth noting, namely, φ=0° and φ=90°, which correspond to ε∥ aligning perpendicular and parallel to the surface, respectively. Altering φ will effectively rotate the hyperbolic dispersion [5], introducing off-diagonal components to the dielectric tensor [5,34]. Another way to modify the behavior of SPhPs is by introducing an azimuthal rotation around the z axis by an angle β with respect to the incidence plane (here taken as x−z) as shown in Fig. 2(b), which mimics the effect of an in-plane wavevector ky. Therefore, a special case includes φ=90° and β=90° making ε⊥ and ε∥ aligned along the x and y axes, respectively. Another special case is when φ=0°. In this case, the components on the dielectric tensor aligned with x and y are both equal to ε⊥, as shown in Fig. 2(b). Therefore, there is no azimuthal dependence of the spectrum shown in Fig. 1(c).

On the other hand, an azimuthal rotation will introduce significant change to this spectrum if φ≠0 as shown in Fig. 3 where both theoretical and experimental results for φ=45° [Fig. 3(a)] and φ=90° [Fig. 3(b)] are given. Clear agreement is shown between theory and experiment and, in both cases, we can see a sharp dip in reflectivity between the two hyperbolic regions, corresponding to the existence of a surface polariton. When φ=45° as shown in Fig. 3(a), the surface wave displays a sinusoidal azimuthal dependence. The excitation frequency reaches its maximum value of ω/2πc=500cm−1 at azimuthal angles β=0° and 180°, while the minimum value of ω/2πc=480cm−1 occurs at β=90° and 270°. We note that the surface wave exists entirely within a region where bulk propagation is not allowed, shown by the rectangular shape of high reflectivity around the surface wave.

When we rotate the anisotropy further (φ=90°), as shown in Fig. 3(b), the dependence on azimuthal orientation now takes a rectified sinusoidal shape. This frequency reaches its maximum value of ω/2πc=505cm−1 at azimuthal angles β=0° and 180°, while the minimum value of 475cm−1 occurs at β=90° and 270°.

So far, we have focused on the azimuthal dependence of the ATR spectra at a fixed kx/k0 value for a range of frequencies; we will now turn to the case of a fixed frequency between the two hyperbolic regions where the surface polariton exists. This is shown in Fig. 3(c), in which the behavior of varying kx/k0 is given for φ=90° and at 490cm−1. We can see that the surface polariton displays an elliptical profile with respect to the azimuthal angle β, which is to be expected where all dielectric tensor components are negative [23,64]. We can see that a smaller kx/k0 is needed when β=0°, and a larger value is needed towards β=90°. This is because ε∥ has a much larger negative value than ε⊥, so less in-plane momentum is required to couple with the surface polariton when the anisotropy is aligned with the x axis.

Now that we have shown how the orientation of the anisotropy axis can greatly impact the existence of surface polaritons in crystal quartz, we can redirect our analysis towards the reflective traits of GHPs and LHPs.

We have seen how surface waves behave in anisotropic materials, namely, their elliptical behavior with respect to β (or in-plane wavevector). Now, let us change the frequency region of interest; instead of looking at regions where no propagation is allowed, and thus surface waves can emerge, we turn to hyperbolic regions where propagation is allowed, beginning with the Type II hyperbolic region. In Type II hyperbolic regions we again focus on two main scenarios: (i) the anisotropy perpendicular to the crystal surface (here along z), and (ii) the case when the anisotropy aligns with the crystal’s surface, i.e., a geometry that allows for hyperbolic Dyakonov polaritons [49]. The first case would be similar to the classic surface polariton, which we have discussed in detail, as both components of the dielectric tensor along the surface are negative and despite εzz being positive, the condition in Eq. (3) could still be met for any β. The second case, on the other hand, is far more complex. For instance, when φ=90° and β=0° (ε∥ lies along x and ε⊥ is along y), Eq. (3) can thus only be satisfied if β=90°, resulting in ε∥ aligning with y and ε⊥ along x. In this orientation, and considering the incidence plane to be the x−z plane, crystal quartz behaves entirely like a metal (εxx=εzz=ε⊥).

To illustrate this we calculate the ATR behavior at ω/2πc=460cm−1, shown in Fig. 4. First we look at the case where φ=90° but no air gap (d=0μm). This is shown in Fig. 4(a), where we can see the clear bulk hyperbolic dispersion traced by propagating radiation (i.e., continuous dips in reflectance). Propagation is mostly supported when β=0° or 180°, reducing in intensity for higher kx values. On the other hand, propagation is completely forbidden when β=90° due to the quartz behaving entirely like a metal. In Fig. 4(b), we show the ATR when an air gap of d=0.1μm is present so that evanescent waves can couple to surface waves. We observe a hyperbola-shaped drop in reflectivity, indicative of Dyakonov surface waves, in the region where bulk propagation is forbidden (i.e., upper region of the plot). As expected, it requires smaller kx values where β=90°, requiring more in-plane momentum when azimuthal rotation is introduced, until the surface wave is no longer supported at β=45° and β=135° where the bulk hyperbolic band exists. This highly angular dispersion mirrors that found in the original s-SNOM results of the original GHP discovery [8].

Earlier, we have also seen how combining a tilted anisotropy axis (with respect to the surface) with in-plane wavevectors can yield an extra degree of control of the surface polariton behavior. Since tilting the anisotropy in such a way has been used as a route to introducing GHPs [8], we now look at the effect of tilting the anisotropy away from the surface with φ=60°, shown in Figs. 4(c) and 4(d), again at ω/2πc=460cm−1. For the first case, in Fig. 4(c) we again look at no air gap (d=0μm). The propagating behavior tracing the hyperbolic dispersion is very similar to Fig. 4(a), except that the reflectivity dip occurs at a slightly larger range of azimuthal angles. Introducing an air gap of d=0.1μm, shown in Fig. 4(d), clearly displays the impact of rotating the anisotropy in the form of a GHP. Qualitatively, the shape of the ATR spectra is very similar to the untilted case. However, the surface wave component is now supported by a narrower range of azimuthal angles, centered at β=90°. The drop in reflectivity is less intense, showing the GHP moving away from the bulk hyperbolic dispersion. Moreover, tilting the anisotropy away from the surface means that the positive extraordinary dielectric tensor component will now provide propagation characteristics to the polariton (instead of a “true” surface polariton). This has been previously explained as tilted wavefronts into the bulk of the material and as long-distance propagation along the surface [8]. This anisotropy tilting introduces more pronounced optical effects associated with large anisotropy such as asymmetric cross polarization conversion [34], where details are shown in Appendix D.

In order to obtain the results above we needed to modify a few parameters, namely, increase the prism dielectric constant compared to the experimental case shown in earlier sections. This is necessary to be able to generate high enough in-plane momentum to probe a large portion of the hyperbolic dispersion. As a consequence of the larger in-plane momentum, we also needed to reduce the air gap thickness to support critical coupling (see Appendix E for details on the effect of varying d). While we illustrate this behavior at a single frequency, in order to address behavior observed in s-SNOM, the frequency-dependent behavior is also vastly different to what we previously showed for true surface polaritons. For more information on the ATR response at specific frequencies, see Appendix F. In Fig. 5 we show the ATR spectra over the frequency range 430–500cm−1 at a fixed kx/k0=5. For φ=90°, in Fig. 5(a), the elliptical surface polariton examined previously splits into segments at 490cm−1, positioned above the bulk propagation linked with Type II hyperbolic dispersion. A distinct hyperbolic Dyakonov surface polariton emerges between 450 and 470cm−1, peaking around β=90°, 270°, where ε⊥ aligns on both x and z axes. The polariton’s frequency-β relationship resembles an upward arrow, segregating bulk-propagation zones in the Type II hyperbolic region. At φ=60°, shown in Fig. 5(b), the GHP’s intensity reduces. The elliptical surface wave, rather than being separated, now connects via a slight reflectivity dip at 485cm−1 near β=90°, 270°, also increasing the range of β angles for which there is propagation within the Type II region. Increasing the range of angles for which there is propagation inside the Type II regions seems to also alter the GHP, decreasing its reflectance intensity, consistent with true surface waves being more pronounced where bulk propagation is absent [5].

Similar to GHPs, leaky polaritons have recently been studied using s-SNOM where the direction of the anistropy with respect to the crystal surface has been shown to play a crucial role [11]. To understand how LHPs manifest in ATR spectra, we now shift our focus to Type I hyperbolic regions where these waves have been shown to emerge in calcite [11]. To allow radiation leakage in an ATR setup, an air gap must be introduced to include the coupling of evanescent waves with the leaky wave. Leaky waves experience an exponential increase in air [58], which in the past has led them to be known as “mathematically improper” [65], meaning that quite a large air gap (as used here) will be necessary to allow coupling. As leaky waves appear at low wavevectors, we only observe surface wave-like to be supported slightly beyond the critical angle, close to kx/k0=1. Figure 6 shows the ATR behavior of leaky polaritons at 545cm−1 in crystal quartz.

In Fig. 6(a), where φ=90° and d=0μm, we can see that the ATR spectra trace the hyperbolic dispersion (left- and right-hand side) due to the in-plane anisotropy, with two extraordinary cones on either side of the bulk ordinary cone. Outside of the ordinary cone, propagation is mostly supported when β=0° or 180°, reducing in intensity for higher kx values. In Fig. 6(b), we introduce an air gap d=10μm, which induces the emergence of leaky polaritons. We observe a lenticular-shaped drop in reflectivity, indicative of the leaky polariton [11]. What is interesting is that this drop in reflectivity is only at a small kx interval larger than the critical angle, but it is within the bulk ordinary circle bounds of Fig. 6(a). The leaky polariton is supported mostly when β=0°, where metallic contribution from the negative dielectric tensor component is at its maximum. At β=90°, only non-metallic (positive) dielectric tensor components contribute; therefore no surface-like wave phenomena are supported.

The effect of tilting the anisotropy away from the surface (φ=70°) is shown in Figs. 6(c) and 6(d) for both no air gap and d=10μm. When probing bulk waves (no air gap) we can see that rotating the anisotropy causes the extraordinary cones to move inward, overlapping with the bulk ordinary cone, facilitating radiation leakage [11]. Notable asymmetry is observed due to cross polarization conversion, which has been studied previously, albeit not using ATR [34]. On the other hand, introducing an air gap causes the lenticular shape of the leaky polariton to move inwards too. The reflectivity drop is still shown slightly beyond the critical angle, so that evanescent coupling is still occurring, but within the bulk ordinary cone. We also note the stark asymmetry within the critical angle due to much larger cross polarization conversion than what was observed with no air gap (see Appendix G for details on cross polarization conversion due to LHPs).

In Fig. 7(a), we see the azimuthal dependence of leaky polaritons in crystal quartz when the anisotropy is aligned with the surface, and an incident angle of θ=26.6° (corresponding to kx/k0=1.05 for a prism of εp=5.5). The reflectivity dip takes a downward-arrow shape, centered at azimuthal angles of β=0° and 180°, which correspond to the anisotropy aligning with the x axis and separating into two branches that widen as the frequency increases. This behavior is directly linked to the isofrequency contours of dispersion with lenticular shape observed in near-field imaging of LHPs in calcite [11]; namely, at the bottom of the downward-arrow shape, the lenticular shape is connected at β=0° and thus waves at the surface propagate along x (and all) directions, whereas at higher frequencies, the splitting into two branches indicates that waves at the surface of the crystal will only propagate in some directions. Notably, this is consistent with our understanding that larger negative extraordinary permittivity components reduce the directionality associated with ENZ ordinary components (see details of this in Appendix H, in the form of polar plots showing how the LHPs’ lenticular shape changes with frequency).

When we tilt the anisotropy into the bulk at an angle φ=70° in Fig. 7(b), The intensity of the leaky polariton decreases slightly, and its minimum supporting frequency shifts above 540cm−1. This shift is accompanied by an increase in bulk hyperbolic dispersion below 540cm−1, illustrating the sensitive dependence of leaky polaritons on the orientation of the crystal’s anisotropy.

Our investigation of elliptical surface polaritons, GHPs, and LHPs in hyperbolic media using ATR spectroscopy reveals several key insights into their optical behavior. We have demonstrated that anisotropy orientation significantly influences the optical properties of hyperbolic polaritons, namely, their ATR spectra.

As such, crystallographic orientation could be of significant importance in designing nanophotonic devices utilizing these phenomena, and it offers a new degree of freedom for manipulating light at the nanoscale.

The ATR spectra of GHPs clearly trace the hyperbolic dispersion, outside of the bulk bands, when the anisotropy aligns with the surface, but reflectance reduces in intensity when the anisotropy is tilted with respect to the surface, due to an increase in cross polarization conversion. This strong direction-dependent behavior could be used in the directional control of electromagnetic waves. Moreover, GHPs respond to very high in-plane momenta, which could be utilized in receiving signals from a wide field of view. On the other hand, we find that LHPs are limited to a smaller range of kx values. However, the large asymmetry due to cross polarization conversion when the anisotropy is oriented away from the surface could be used in novel devices, such as potentially enabling one way communications. In this work, we investigate the LHP in the Type I region, and the GHP in Type II dispersion to more deeply understand the original experimental observation. However, ghost modes were observed in the visible range in the material MoOCL2 [53], which featured Type I hyperbolic dispersion. We note that, when investigating Type I hyperbolic dispersion, we could not obtain a reflective footprint similar to the GHP behavior observed in Type II regions in quartz and calcite. Therefore, more work is needed to understand how individual dielectric tensor elements, including off-diagonals, real components, and imaginary components, affect ghost-like features manifesting in the material.

Finally, similar methods to what we explored here could be directly translated to investigating other unusual hyperbolic behavior such as the newly studied hyperbolic shear polariton in materials with larger asymmetry in their crystal structure [3,21,23,66,67]. In addition, this could also be a way to investigate the impact of twisted-layer structures [42,67–73] on these phenomena, particularly as this could be a way to introduce further control over optical behavior.

To calculate the reflectivity coefficients, the 4×4 transfer matrix method was used [36,37,74–78] with full details given in Appendices A and B.

The parameters used to model the optical behavior of crystal quartz were obtained from the parameters given by Estevam et al. [25], built on the work of Gervais and Piraeus [79]. These parameters include the high-frequency permittivity ε∞, frequencies of the transverse optical (TO) phonons and of the longitudinal optical (LO) phonons, and the appropriate damping parameters responsible for absorption around the phonon frequencies.

A method for calculating the wavevector solutions to be calculated using the dielectric tensor is also included in Appendix A. These solutions then allow for the fields at the interface to be calculated, which can then be matched with that of the air gap (and other layers) and ultimately trace the frequency of specific polariton modes, taking into account all layers. For understanding these mathematical solutions (for both the GHP and the LHP), similar to our context building within Eq. (3), we refer the reader to the supplementary material in Ref. [11], along with Refs. [50–52].

The experimental data was collected via reflectance measurements using Fourier-transform infrared spectroscopy with a Bruker Vertex 70 spectrometer. The spectra featured a resolution of 4cm−1, and each spectrum was averaged 15 times.

A KRS-5 polarizer was placed in the path of the incident beam to obtain spectra for p-polarized light.

To obtain ATR spectra, a diamond coupling prism was used with a dielectric constant εp=5.5 and at a fixed incident angle of 45°. To introduce an appropriate air gap, 1.5 μm silica spacers were thinly coated over the ATR stage. Throughout the azimuthal rotation, the silica spacers were partially moved, so extra care was taken in order to maintain a consistent air gap thickness. There is significant margin for error, but results show clear agreement with theory despite this. Experimental results align with a theoretical air gap distance of 2 μm, with variation between 1.5 and 3 μm (see Appendix C for details of the error associated with the air gap and its effect on reflectance).

The crystal quartz samples used were provided by Boston Piezo Optics Inc. The anisotropy orientation was determined with respect to the crystal’s surface using single-crystal X-ray diffraction, with the crystal mounted on a goniometer for precise positioning at selected orientations. These samples were then cut into flat slabs of chemically polished faces of 20 mm in diameter and 10 mm in thickness. The faces were cut at specific angles with respect to the anisotropy; one of our samples had anisotropy set at φ=45° and another at φ=90°. Figure 3(a) contains experimental results for the φ=45° sample and Fig. 3(b) contains experimental results for the φ=90° sample.

Acknowledgment. M.C.C. acknowledges funding from the EPSRC. We thank N. Parry, C. A. McEleney, M. Smith, A. Joseph, K. Stevens, A. MacGruer, and S. Mekhail for useful insights, discussion, and helpful comments on the manuscript.