Introduction

Van der Waals materials have emerged as a versatile platform for hosting polaritons, [1,2] which are hybrid light-matter quasiparticles that enable light confinement and control at the nanoscale. [1][2][3] This versatility arises in part because these materials feature exceptional intrinsic properties, including ultralow losses, high quality factors, and long polariton lifetimes. Along these lines, hexagonal boron nitride (hBN) [4][5][6][7][8][9][10][11][12][13][14][15] and alpha phase molybdenum trioxide (MoO 3 ) [14][15][16][17][18][19][20][21][22][23][24][25][26] have been shown to exhibit best-in-class polaritonic properties and have become two of the most studied polaritonic materials in the recent literature. This versatility also arises because van der Waals materials can be readily stacked to produce heterostructures with designed optical properties. [27] For the c-e) Calculated imaginary parts of the Fresnel reflection coefficient Im(r p ), which illustrates the dispersion of the polariton mode, for slabs of c) MoO 3 ,d)hBN,ande)aMoO 3 -hBN hetero-bicrystal. The MoO 3 and hBN slabs in each calculation are 100 nm thick. f-h) Numerical simulations of electric field strength in the cross-section of an MoO 3 -hBN bicrystal with slab thicknesses of 1 μm, at frequencies f) 799 cm -1 , g) 766 cm -1 , and h) 727 cm -1 . The field profile in (g) represents the bicrystal regime that supports a negative refraction cavity mode. The scale bar applies only to the horizontal direction.

examples of hBN and MoO 3 , stacking and twisting two flakes of the same material can produce a plethora of new properties and options for polaritonic tuning. [17,18,28,29] Single hetero-bicrystals comprising combinations of hBN and MoO 3 are a particularly rich system for the study of new polaritonic behavior. In an initial investigation, hBN-MoO 3 heterobicrystals were found to showcase optical phenomena such as negative polaritonic refraction, closed-loop light trajectories, polaritonic level repulsion, and avoided mode crossings. [15] One of the most striking features is the emergence of a new spectral gap in the polariton dispersion, which results from polaritonic level repulsion and strong mode coupling. However, systematic experimental studies of these heterostructures have proven to be elusive, calling for more efforts on the preparation of high-quality large-area samples and their broadband nanospectroscopic investigation, as well as the simplified but insightful physical interpretations.

In this report, we experimentally explore the evolution of polariton dispersion as a function of heterostructure geometry through the systematic fabrication and optical study of heterobicrystals comprising isotopically pure hBN and MoO 3 layers on an ultrasmooth gold substrate (Figure 1a,b). Our study leverages crystalline MoO 3 microstructures [22,26] that are chemically grown with high yield and a distribution of thicknesses, thereby facilitating the assembly of ensembles of high quality hetero-bicrystals.

Isotopically pure hBN crystals [10][11][12] are used because they feature exceptionally low loss and polaritonic modes that ideally overlap with those in MoO 3 . Experimental analysis is performed using synchrotron infrared nanospectroscopy (SINS), which utilizes the scattering of infrared radiation off an atomic force microscopy (AFM) tip to perform polariton interferometry [15][16][17][18][19][20][21][22][23][24][25][26][30][31][32][33][34][35][36][37][38] and enables direct visualization of polariton mode dispersion. We observe with SINS that the spectral position and width of the spectral gap in the polariton dispersion can be tuned via the thickness ratio of its constituting hBN and MoO 3 crystals, and these trends quantitatively match with analytic models. We further demonstrate that with our unique sample configuration, optical signatures of negative refraction cavity modes can be directly read out in the SINS hyperspectral line scans. We additionally explore the effects that are associated with the finite in-plane dimension of microscale bicrystals, such as the existence of polaritonic cavity modes and polaritonic propagation along "forbidden" directions.

Results

To understand and interpret the polaritonic dispersive response of an hBN-MoO 3 bicrystal on ultrasmooth gold, it is instructive to first examine the dispersion curves of the individual constituent materials and the bicrystal on gold. At mid-infrared wavelengths, the gold effectively functions as a perfect electrical conductor (PEC) and supports ideal image polaritons. [39] The polaritonic responses of van der Waals films on gold therefore follow those of the antisymmetric modes within van der Waals films with double the thickness and clad in air. Branches in polaritonic dispersion www.advmat.de can be realized by calculating the maxima of the imaginary part of the p-polarized Fresnel reflection coefficient, Im(r P ). [5,30] Figure 1c shows Im(r P ) for a 100 nm-thick MoO 3 flake for a broad range of incident frequencies 𝜔 and in-plane momenta q. The in-plane anisotropy [16] of MoO 3 is clear as the polariton branches differ significantly for q along the crystalline [100] direction (left in Figure 1c) versus the [001] direction (right in Figure 1c). Unlike MoO 3 , hBN supports in-plane isotropic polariton propagation and shows a single hyperbolic reststrahlen band [7] from around 760 to 820 cm -1 , as illustrated for a 100 nmthick hBN flake in Figure 1d. Notably, hBN supports negative dispersion (decreasing q with increasing frequency 𝜔) while MoO 3 supports positive dispersion (increasing q with increasing 𝜔)in this spectral range.

The combination of materials with positive and negative dispersion in the same spectral range, which is the case for the hBN-MoO 3 bicrystal, [15] is a key ingredient to realizing new polaritonic behavior. Figure 1e shows Im(r P ) for a 200 nm-thick bicrystal consisting of a 100 nm-thick MoO 3 slab on a 100 nm-thick hBN slab on the gold substrate. Within the spectral range where the polariton branches of hBN and MoO 3 overlap, we observe the hybridization of the hyperbolic phonon polaritons of both materials, that is, the formation of a new coupled polariton mode that is intrinsic to the hBN-MoO 3 heterostructure. [15] Notably, the hybridized polaritons show spectral regions with positive and negative dispersion separated by a spectral gap where no polariton propagation is allowed in the bicrystal. For this case where hBN and MoO 3 have the same thickness, the center of the gap (𝜔 0 )is at 766 cm -1 .

We further elucidate the underlying light-matter interactions near and in the polaritonic spectral gap by performing 2D fullwave simulations of an electric dipole excitation located above the bicrystal. This excitation condition follows our experimental configuration and is consistent with nanoscopic tip illumination in SINS. We consider 1 μm-thick van der Waals slabs in this analysis because polaritonic propagation follows ray-like behavior in such thick slabs and is more straightforward to interpret (see Experimental Section, Figures S3 and S4, Supporting Information). Figure 1f-h shows the electric field produced by the dipole within a cross-section of the bicrystal at frequencies of 727, 766, and 799 cm -1 , respectively. At 𝜔 = 727 cm -1 , the excitation frequency is outside the hBN reststrahlen band, and the polaritonic behavior is dominated by the MoO 3 response, with ray-like polariton propagation through the MoO 3 crystal. At 𝜔 = 799 cm -1 , the roles of hBN and MoO 3 are interchanged, and polaritonic dispersion, specifically the lateral displacement of the polaritonic rays, is dominated by propagation through the hBN crystal. The most intriguing behavior is observed at 𝜔 = 766 cm -1 , which corresponds to 𝜔 0 . There is a diamond-shaped closed-loop trajectory of the polaritons (Figure 1g), resulting from negative refraction of the polaritons at the hBN-MoO 3 interface. [15] Importantly, these simulations indicate that the negative refraction cavity mode within the bicrystal can be directly excited by and read out with the nanoscopic tip in SINS.

For our experimental efforts, sample preparation is performed by transferring microscale MoO 3 crystals onto hBN flakes, [10][11][12] which are initially exfoliated onto ultrasmooth gold. The MoO 3 crystals are grown using flame vapor deposition (FVD), from which densely packed microscale ribbons with a wide range of thicknesses are produced in a single batch. [22] These materials have been previously characterized to support polaritonic lifetimes matching those from single crystal exfoliated samples, and they have atomically sharp edge interfaces that are ideal for polariton interferometry. The isotopically pure h 11 BN flakes are relatively large, with dimensions on the order of 100 μmx 100 μm, such that multiple hetero-bicrystal structures can be readily formed by transferring multiple MoO 3 crystals onto a single h 11 BN flake. Our utilization of a gold substrate with atomic smoothness, prepared using template stripping, ensures that the interface between hBN and the substrate is ideal and that optical losses due to surface roughness are minimized.

SINS analyses [8,22,[34][35][36][37]40] of two representative heterobicrystals are presented in Figure 2. Experimental details and data for two additional bicrystals are in the Experimental Section. AFM images of the bicrystal topographies (Figure 2a,d) show MoO 3 crystals with approximately rectangular shapes, homogeneous thicknesses, and clean crystal edges. The long axes of the MoO 3 crystals are aligned with their crystalline [001] direction, which is consistent with previous MoO 3 microribbon studies [22] and is confirmed by the nanospectroscopy analysis performed here. As the [001] direction of MoO 3 corresponds to the direction featuring the hBN-MoO 3 bicrystal polaritonic gap [15] (Figure 1e), we first perform hyperspectral line scans along this direction for both bicrystals (Figure 2a,d,arrow).The resulting hyperspectral data (Figure 2b,e)at𝜔<820 cm -1 show clear fringe patterns. As these stem from the interference of hybridized bicrystal polaritons launched by the tip and reflected back from the MoO 3 crystal edges, this is referred to as polariton interferometry. [15][16][17][18][19][20][21][22][23][24][25][26][30][31][32][33][34][35][36][37][38] The observation of clearly discernable fringes with high signal-to-noise across the mid-infrared band can be attributed to the high degree of crystalline sample quality, the high intensity of the light source, and the MoO 3 -on-hBN ordering of the bicrystal. As discussed in the Experimental Section, the placement of MoO 3 on hBN results in a more uniform spectral distribution of the bicrystal's polaritonic features compared to heterostructures comprising an inverted layer order.

Polariton dispersion can be directly extracted by quantifying fringe spacing variation as a function of frequency, as the fringe spacing corresponds to half the polariton wavelength. Experimental values for polariton dispersion are extracted by fitting a decaying sine wave of the form exp(ikx). The resulting Re(k) is represented by yellow dots in Figure 2c,f, together with dispersion curves calculated from Im(r p ). The experimental data and calculations quantitatively agree and show that the polariton momentum increases when approaching 𝜔 0 from both higher and lower frequencies for both samples. The reduced number of observable polariton fringes in Figure 2b,e indicates that the polariton quality factor, Q(𝜔), decreases as the frequency approaches the polaritonic gap.

Notably, we observe a pronounced SINS intensity peak in the hyperspectral scan of each bicrystal (Figure 2b,e, red arrow) in the spectral range around the polaritonic bandgap, which directly corresponds to the calculated 𝜔 0 's in Figure 2c,f (dashed line) and indicates that the polaritonic bandgap is tunable based on bicrystal geometry. The origins of these peaks can be understood by interpreting an individual bicrystal as a superlens [7,9,[41][42][43][44][45] that utilizes the negative refraction cavity mode to image a dipole source at the bicrystal's top surface (i.e., the physical MoO 3 -air interface) to its image plane at the bicrystal's bottom surface (i.e., the physical hBN-Au interface) with subwavelength resolution. The gold, serving as an ideal reflector in the PEC limit, reflects this radiation back through the superlens, which refocuses at the dipole source location. In this manner, the SINS AFM tip naturally serves as both the dipole source and detector for sensing the strong field intensities from the refocused radiation at 𝜔 0 . This double role of the tip is special to the case of a PEC substrate. We can generalize the superlens interpretation to arbitrary substrates, stating that at 𝜔 0 we are effectively probing the substrate underneath the bicrystal. For example, using SiO 2 substrate results in a reduced near-field amplitude close to 𝜔 0 , [15] while here we observe an increased near-field response around 𝜔 0 in Figure 2c,f due to the comparably large near-field response of gold.

The hyperspectral line scans also reveal polaritonic modes at frequencies outside of the hybridized bicrystal polariton frequency range. Fringes at 𝜔>950 cm -1 can be directly assigned to in-plane MoO 3 hyperbolic phonon polariton modes that are fully decoupled from the hBN reststrahlen band (Figure 1c,e). [16] Interestingly, we observe polaritonic fringes in the intermediate spectral range 820 cm -1 <𝜔<950 cm -1 , which cannot be assigned to polaritonic modes that naturally propagate along the [001] direction of MoO 3 or hBN. In fact, polaritonic modes in these materials only exist along the [100] direction in infinite inplane MoO 3 (Figure 1c,d). We attribute these modes to geometric confinement effects in our microribbon system, which have been recently studied in MoO 3 cavities [25] and can allow [001] waveguide modes to propagate along the [100] direction. Further numerical analysis of this phenomenon in our bicrystal system is provided later in this study.

SINS hyperspectral line scans along the [100] direction of the bicrystals reveal strong polaritonic Fabry-Pérot cavity modes that are enabled by the microscale width of the MoO 3 ribbons (Figure 3). Along this crystal direction, the polaritonic contributions from hBN and MoO 3 are in distinct spectral bands and do not hybridize. In the spectral range 𝜔>820 cm -1 ,M o O 3 supports polaritonic modes while hBN effectively functions as a passive dielectric substrate. The fringe patterns in Figure 3b,e can therefore be interpreted as MoO 3 polaritonic Fabry-Pérot cavity modes [19,[21][22][23]25,26,36,38,46,47] that are constrained to the finite-sized MoO 3 crystal. The overall polaritonic dispersion is well described via the calculated Im(r p )inFigure3c,f. However, as the analytical calculation of Im(r p ) implicitly assumes infinite in-plane sample dimensions, it does not explain the observed cavity modes. The latter are included in numerical simulations that we present later. In the spectral range 750 cm -1 <𝜔<820 cm -1 , hBN supports polaritonic modes while MoO 3 effectively functions as a passive dielectric. The fringes are attributed to hBN polariton Fabry-Pérot cavity modes in which the MoO 3 microribbon serves as an external Fabry-Pérot cavity that facilitates hBN polariton reflection [48] at the MoO 3 crystal edge.

The dispersion properties and tunability of the polariton gap for hybridized bicrystal polaritons along the MoO 3 [001] direction can be further analyzed with analytic modeling. We summarize here the key results and provide a detailed discussion of our approach in the Experimental Section. The dispersion relations can be extracted from the following transcendental expression: q is the in-plane wavevector, 𝜖 1 , 𝜖 M,i ,and𝜖 h,i represent the background permittivity, MoO 3 permittivity, and hBN permittivity, respectively (i ∈ ip, oop, where the oop denotes the out-of-plane direction and ip denotes the in-plane direction), d MoO 3 and d hBN refer to the respective thicknesses of the MoO 3 and hBN crystals, 𝜓 MoO 3 = √ -𝜀 M,oop 𝜀 M, ip ,and𝜓 hBN = √ -𝜀 h,oop 𝜀 h, ip

. The dispersion relations are computed by sweeping through l and real frequency to solve for q. Introducing material loss with the use of complex valued permittivities transforms q into a complex number and enables calculation of the polariton quality factor Q (𝜔) = Re(q)

Im(q) . In the approximation where polariton propagation is treated in the ray limit, [9,15] the expression for gap frequency 𝜔 0 can be simplified to:

𝜃 hBN (𝜔)and𝜃 MoO 3 (𝜔) refer to the polariton propagation angle with respect to the surface normal in each material. The dependence of 𝜃 on the frequency 𝜔 is provided by the anisotropic permittivity tensor of each material. [9] For a more rigorous treatment, 𝜔 0 also can be determined through the utilization of Equation (1) in the following way. First, the dispersion of the individual MoO 3 and hBN slabs can be solved by setting d MoO 3 or d hBN = 0i nE q u a t i o n( 1). Second, the strongest coupling frequency, that is, the gap frequency 𝜔 0 , can be identified by examining the resonance frequencies of these slabs at a fixed momentum. [49] We see from this analysis that the key parameter for 𝜔 0 tuning is the d MoO3 /d hBN thickness ratio. To support this observation, we calculate and plot the dispersion relations for different bicrystals consisting of 100 nm-thick MoO 3 and hBN with thick-nesses ranging from 20 nm ( 4a-d). Dispersion curves calculated using Im(r P ) (blue lines) and Equation (1) (green dots) match, indicating the consistency of these computational methods. The plots clearly indicate that 𝜔 0 , delineated by the red dashed lines in the plots, tunes from 815 to 765 cm -1 as the thickness ratio varies. These theoretical trends match well with experimental values for 𝜔 0 (Figure 4e), which are extracted from the SINS intensity peaks in the hyperspectral scans of four different bicrystals (the two samples in Figure 2 and the two samples in Figures S8 and S9, Supporting Information). The theoretical trends in quality factor (i.e., Q(𝜔) reduction for frequencies near the bandgap) also are consistent with experiment and reinforce the idea that polaritonic behavior in our bicrystals is well described by our analytic models.

The dispersion curves also provide a quantitative assessment of the gap width, Δ𝜔, which is an important parameter for characterizing polaritonic coupling. A rigorous definition of the gap width is provided in the Experimental Section. Figure 4f shows Δ𝜔 as a function of

, from which we find a maximum gap width when the thickness ratio is around 0.3. The presence of an ideal thickness ratio for polaritonic coupling can be intuitively understood in the context of polariton momentum matching in both materials. For MoO 3 and hBN of the same thickness, polaritons are not momentum matched, as the MoO 3 polariton momentum is about a factor of three higher than that of hBN (Figure 1c,d). However, due to the approximate inverse scaling of hyperbolic phonon polariton momentum with material thickness, [4,7,9] polariton momenta in both materials are matched when the MoO 3 crystal is ≈3× thicker than the hBN crystal. Qualitatively, momentum-matched polaritons interact more strongly with each other, thereby introducing a larger gap width. Quantitatively, momentum-matched polaritons yield mode overlap integrals with maximal values in expressions for mode coupling. In the final part of our analysis, we perform full 3D fullwave modeling of a finite-sized bicrystal, which we use to rigorously capture all observed polaritonic behavior in our SINS experiments. We employ 3D full-wave finite-difference timedomain (3D FDTD) simulations using the fast numerical solver Tidy3D, and we capture the SINS experimental flow by scanning a broadband dipole excitation source across the bicrystal. The simulated bicrystal consists of a rectangular 1.5 μmwide, 8.0 μm-long MoO 3 microribbon on a larger hBN crystal on a gold substrate, which is close to the experimental bicrystal geometry featured in Figure 2a. The dipole source is located 200 nm above the MoO 3 crystal and we monitor the out-ofplane component of the electric field 40 nm above the crystal. The hyperspectral line scans are taken along the reflection symmetry planes of the bicrystal along the MoO 3 [001] and [100] directions and are shown in Figure 5a,b, respectively. Overall, there is excellent agreement between our numerical and experimental line scans, including the presence of hybridized bicrystal polaritons at 𝜔<820 cm -1 , enhanced near-field intensity at the polaritonic bandgap at 𝜔 0 = 766 cm -1 (further supporting our interpretation of the bicrystal as a superlens at 𝜔 0 ), and forbidden guided modes at 750 cm -1 ≲ 𝜔 ≲ 820 cm -1 along the [001] direction. We also observe clear signatures of MoO 3 and hBN Fabry-Pérot cavity modes along the [100] direction. The plots additionally show that "allowed" polariton modes along [100] support positive dispersion while the dispersion is inverted for the forbidden polariton modes along the [001] direction, which agrees with the experiments and further confirms the ability of these simulations to capture subtle polaritonic behavior.

To visualize polaritonic propagation more directly in our finitesized bicrystals, we utilize 3D FDTD simulations to produce electric field profiles for dipole excitations with a fixed position and varying frequency. False color field plots of the real part of the electric field along the out-of-plane direction, Re(E z ), are summarized in Figure 5c. We observe the characteristic wavefronts associated with MoO 3 hyperbolic polaritons, [16,20,21,24] that is, in-plane elliptical wavefronts around 1000 cm -1 and in-plane hyperbolic concave wavefronts around 600 cm -1 . In spectral regions where the MoO 3 dimensions are small compared to the polariton decay length (e.g., around 860 cm -1 ), there are more complex standing wave patterns that are formed not only along the direction where hyperbolic polariton propagation is allowed (e.g., [100] direction at 860 cm -1 , refer to Figure 1e), but also along the orthogonal direction where propagation is forbidden (e.g., [001] direction at 860 cm -1 ). These standing wave patterns illustrate how geometrical confinement can enable polariton propagation in the forbidden direction. [19,25]

Conclusion

In summary, we have performed a systematic theoretical and experimental analysis of the polaritonic properties of finite-sized hBN-MoO 3 hetero-bicrystals. We demonstrate that the thickness ratio of the van der Waals heterostructure can be used to explicitly tune the polaritonic gap position and width, and we experimentally measure signatures of the negative refraction cavity mode within the polaritonic bandgap that tunes with thickness ratio. Compared to alternative schemes for negative refraction in van der Waals materials based on graphene plasmon polaritons, [24] our work does not rely on doping-dependent plasmonic phenomena but employs only low-loss phonon polariton modes. We also characterize forbidden polaritonic modes [25] and polaritonic Fabry-Pérot cavity modes [19,[21][22][23]25,26,36,38,46,47] that arise from finite-size effects of our bicrystal system. All of our experimental findings are corroborated by a combination of analytic and full 3D numerical calculations, which capture real-space phenomena that are not covered via simpler analyses such as the calcul a t i o no fI m ( r p ). Future research directions include the explicit characterization of the bicrystal as a tunable superlens and the development of new flame vapor-deposited van der Waals recipes, which can enable the high throughput assembly and characterization of new classes of hetero-bicrystals.

Experimental Section

Sample Preparation: hBN-MoO 3 bicrystals consisting of a smaller MoO 3 flake on top of a larger (≈100 μm x 100 μm), isotopically pure h 11 BN crystal were studied. The bicrystal was placed on top of an ultrasmooth gold (Au) substrate.

Growth of MoO 3 : The synthesis of 𝛼-MoO 3 micro-and nanostructures was based on the previously reported FVD method. The flame synthesis setup in this work has a 6 cm diameter premixed flat-flame burner (McKenna burner, Holthuis & Associates) with CH 4 as the fuel and air as the oxidizer. The fuel-to-air equivalence ratio of the flame was fixed as 0.88 with a CH 4 flowrateof1.80standardliterperminute(SLPM)and an airflow rate of 19.48 SLPM. The equivalence ratio was defined as the ratio of actual fuel/oxidizer molar ratio to the stoichiometric fuel/oxidizer molar ratio. A 4 cm × 4 cm Mo mesh prepared from Mo wires (0.203 mm in diameter, 99.9%, Alfa Aesar) was placed over the premixed flame as the solid Mo source. The area density of this Mo source was about 18 mg cm -2 . 𝛼-MoO 3 structures were grown on Si wafer, which was placed above Mo source. The temperature of the Mo source was tuned by adding steel cooling meshes between Mo mesh and burner, and the temperature of the growth substrate was controlled by adjusting the distance between Mo mesh and Si substrate. A K-type thermocouple was used to measure the temperature of Mo mesh and substrate. 𝛼-MoO 3 with different morphologies were prepared by tuning the area density of Mo source and the temperatures of Mo source and Si substrate. A higher Mo source temperature (792 °C) facilitates Mo vapor generation, thus resulting in 𝛼-MoO 3 microplates. Lower Mo source temperatures lead to smaller nanoribbons. To demonstrate the growth of 𝛼-MoO 3 nanoribbons with a higher aspect ratio, the area density of the Mo source was halved (9 mg cm -2 ), which reduces the partial pressure of the MoO 3 vapor. The substrate temperature was also increased. The driving force for atom attachment was thus lowered and the growth occurs preferentially along the axial [001] direction due to smaller surface energy cost. As a result, the aspect ratio was increased and 𝛼-MoO 3 nanoribbons were formed. All the 𝛼-MoO 3 samples were prepared by a 5 min deposition. Under such conditions, more than 1 cm 2 of the substrate can be uniformly covered by the flamegrown 𝛼-MoO 3 structures.

Reasons for Choosing Isotopically Pure h 11 BN and Its Growth: Three different isotopic variations were commonly used for polaritonic applications, [10,11] namely naturally abundant hBN, isotopically pure h 10 BN, and isotopically pure h 11 BN. As expected, heavier isotopes lead to a lower resonance frequency for the phonons. As such h 11 BN shows a lower resonance frequency compared to h 10 BN. Due to this frequency shift, the hyperbolic phonon polaritons resulting from the phonon resonance of h 11 BN show a better overlap with the relevant phonon polaritons of MoO 3 , which here was the main motivation for choosing h 11 BN instead of h 10 BN. Compared to the naturally abundant hBN, the isotopically pure h 11 BN shows reduced losses, [10,11] which provides another advantage for polariton studies. The monoisotopic h 11 BN crystals were grown from a metal solution at atmospheric pressure as has been previously described in reference. [12] Sample Exfoliation and Choice of Substrate: The hBN bulk samples were thinned down and transferred to ultrasmooth Au substrate using low-adhesion cleanroom tape. The ultrasmooth Au substrate was template stripped from a Si wafer (Platypus Technologies). The as-grown 𝛼-MoO 3 structures were then transferred onto hBN on gold sample using low-adhesion cleanroom tape for optical characterizations.

As Au provides a spectrally flat optical response in the infrared spectral range, using Au substrate allows a more direct interpretation of the measured spectral sample response since one does not need to consider resonant substrate excitations, for example, phonon modes. Also, bare Au substrate regions provide an ideal spectroscopic reference that here was used to normalize the measured nanospectroscopic sample response. Finally, in ultrasmooth gold, losses were reduced compared to more common evaporated Au substrates. [22] Synchrotron-Based Infrared Nanospectroscopy: Scattering-type scanning near-field optical microscopy (s-SNOM) employs the field enhancement at the tip of an atomic force microscope to provide optical and infrared information at a wavelength-independent spatial resolution of a few 10s of nm. [50] When combined with broadband illumination and Fouriertransform infrared (FTIR) spectroscopy, s-SNOM allows one to acquire broadband infrared spectra at the nanoscale. [51] Here, SINS [34,40] was performed with a commercial s-SNOM setup (NeaSNOM by attocube) at the MET beamline of the National Synchrotron Light Source II (NSLS-II) at Brookhaven National Laboratory, Upton, USA. The synchrotron radiation was predominantly linearly polarized in the orbital plane of the electron storage ring (horizontal). [52] The optical system rotates the predominant polarization to vertical upon reaching the s-SNOM optical table and as it was incident onto the off-axis parabolic (OAP) mirror that focuses the light onto the s-SNOM tip. The OAP mirror reflects the light at a somewhat downward angle such that the light polarization includes a horizontal component as it was incident on the tip. However, the polarization remains predominantly with the E-field parallel to the tip axis. For many experimental situations, aligning the light polarization with the tip axis increases the near-field response due to the tip's increased polarizability along its axis. [53,54] While the ultrabroadband synchrotron radiation of NSLS-II enables near-field spectroscopy in the full infrared spectral range down to <175 cm -1 , [34] here the spectral range was limited to ≳600 cm -1 by the sensitivity of the liquid-nitrogen-cooled mercury-cadmium telluride detector. As SINS measurements were performed at ambient conditions, the accessible spectral range was further limited by infrared absorption in air, explaining a lower signal-to-noise level, for example, in the spectral range ≳1200 cm -1 (not shown in the manuscript). Nevertheless, the usage of accelerator-based infrared nanospectroscopy [21][22][23][34][35][36][37]40,44,[55][56][57] here allows to access the bicrystal's spectral response in a much broader spectral range than usually accessible via more common table-top nanospectroscopy setups. For separation of the near-field signal from far-field background, the optical signal was demodulated at higher harmonics nΩ of the mechanical tip oscillation frequency Ω. [50,58] All near-field spectra shown in the present publication were demodulated at n = 2; they were normal-ized to reference spectra obtained on the ultrasmooth gold substrate (refer to sample fabrication).

Analytical Description of the Polariton Gap Frequency: The ray-like propagation of the polaritons allows a straightforward analytical description of the displacement 𝛿 that a polariton ray experiences when traveling through a single hyperbolic material of thickness d: [9] 𝛿 (𝜔) = 2d tan 𝜃 (𝜔)

where the polaritonic propagation angle 𝜃(𝜔) with respect to the surface normal is given by the ratio of the dielectric permittivities 𝜖 ip and 𝜖 oop along the in-plane and out-of-plane direction as tan 𝜃(𝜔) = i

.

Note that 𝛿 as described in ref. [9] will generally be a complex quantity; in the following, interest lies in the real part of the ray displacement Re(𝛿(𝜔)).

For the hBN-MoO 3 hetero-bicrystal, the total ray displacement 𝛿 tot is simply given by the sum of the ray displacement in its constituents, that is, 𝛿 tot (𝜔) = Re(𝛿 hBN (𝜔)) + Re(𝛿 MoO 3 (𝜔)). As shown in Figure 1f, at the gap frequency 𝜔 0 = 766 cm -1 , a closed polariton trajectory with no effective ray displacement was observed: 𝛿 tot (𝜔 0 ) = 0. Using the equations above, this condition can also be stated as -

, that is, the gap frequency 𝜔 0 depends on the thickness ratio

.

Analytical Dispersion of Air-MoO 3 -hBN-Gold Hetero-Bicrystal: By employing source-free Maxwell equations and incorporating boundary conditions, the dispersion of the Air-MoO 3 -hBN-Gold hetero-bicrystal can be expressed through a 4 × 4 matrix for highly confined polaritonic modes: | Here, 𝜖 1 , 𝜖 M,i ,and𝜖 h,i (i ∈ ip, oop) represent the background permittivity, MoO 3 permittivity, and hBN permittivity, respectively. The in-plane wavevector q, along with q z1 = q 𝜓 MoO 3

, contribute to the intricate structure of the equation. After the straightforward simplification, this equation can be simplified as follows: [26] q = 𝜓 MoO 3

In the dispersion calculation from this equation, considering the dispersion of materials, it was imperative to sweep through frequencies (in real numbers), and l in the specified range to solve for q. Introducing the consideration of material loss transforms q into a complex number, leading to the definition of the quality factor Q (𝜔) = Re(q) Im(q) . It is important to note that in the calculation process, q is a complex number obtained by sweeping each 𝜔. In regions where the band was nearly flat, the authors were constrained to derive only a single solution from the analytical equations, despite the existence of multiple q values corresponding to 𝜔. This limitation results in the absence of Q factors in certain regions of Figure 4. For the calculation of the gap width, it was crucial to identify the gap range near the intersection point of single MoO 3 polariton modes and single hBN polariton modes (depicted in Figure 1c,d). Additionally, the intersection points of the single MoO 3 polariton mode and the hetero-bicrystals polariton mode were key parameters. The initial step involves solving for the first intersection point between the upper band (above the gap) of hetero-bicrystals and the band of single MoO 3 , denoted as (q 0 ,𝜔 u ). Subsequently, by fixing q = q 0 , the frequency 𝜔 d of the lower band of hetero-bicrystals (below the gap) was determined. The gap width Δ𝜔 = 𝜔 u -𝜔 d was then defined. In specific regions, precise determination of 𝜔 d was challenging due to its proximity to the resonance frequency of hBN at 𝜔 hBN TO,oop = 755 cm -1 . To elucidate this phenomenon, the thickness of the MoO 3 slab was maintained at 100 nm while altering the thickness of the continuous hBN layer. The resulting variations in 𝜔 d , 𝜔 c ,and 𝜔 u are illustrated in Figure S1, Supporting Information. Figure S2,S u pporting Information illustrates the impact of different MoO 3 thicknesses, with all samples supporting the assertions in the main text.

Numerical Simulations: The simulation results were obtained using a well-established full-wave simulation method demonstrated in multiple previous publications. [16,59] In the simulations, the illuminated s-SNOM tip was modeled as a vertically oriented point dipole source located 200 nm above the top surface of the MoO 3 microcrystal. The cross-sectional electric field distributions shown in Figure 1f,g as well as Figures S3 and S4, Supporting Information were obtained from 2D FDTD simulations on a stack of MoO 3 /hBN/gold. The in-plane dimensions of the layers were assumed to be infinitely large and their thicknesses were varied.

3D FDTD simulations were used to simulate the hyperspectral images and the electric field distributions in Figure 5. The dimensions of the MoO 3 microcrystal were precisely based on the AFM measurement. To approximate the scattering s-SNOM signal from the tip, the z-component of the electric field 150 nm below the dipole was considered. The simulations of the hyperspectral were rather computationally demanding since the dipole needs to emit broadband light and the position of the dipole needs to be changed in a fine resolution (20 nm) to mimic the tip scanning in the experiment, resulting in a large number (71 for Figure 5b and 36 for Figure 5c) of FDTD runs. The fast FDTD solver Tidy3D was used, where the simulations were hardware-accelerated and parallelized to significantly reduce the solving time.