Van der Waals phonon polariton microstructures for configurable infrared electromagnetic field localizations

Polar van der Waals (vdW) crystals that support phonon polaritons have recently attracted much attention because they can confine infrared and terahertz (THz) light to deeply subwavelength dimensions, allowing for the guiding and manipulation of light at the nanoscale. The practical applications of these crystals in devices rely strongly on deterministic engineering of their spatially localized electromagnetic field distributions, which has remained challenging. This study demonstrates that polariton interference can be enhanced and tailored by patterning the vdW crystal {\alpha}-MoO3 into microstructures that support highly in-plane anisotropic phonon polaritons. The orientation of the polaritonic in-plane isofrequency curve relative to the microstructure edges is a critical parameter governing the polariton interference, rendering the configuration of infrared electromagnetic field localizations by enabling the tuning of the microstructure size and shape and the excitation frequency. Thus, our study presents an effective rationale for engineering infrared light flow in planar photonic devices.


Introduction
Phonon polaritons (PhPs) are quasi-particles arising from the coupling of photons and optical phonons in polar crystals. [1][2] Compared to bulk crystals, the recently identified twodimensional (2D) vdW atomic crystals are attracting increasing attention due to the exotic PhPs they can support, which originate from their highly anisotropic, polar crystal structure. [3][4][5][6][7][8][9] Due to in-plane covalent bonding and out-of-plane vdW coupling, the permittivity tensors of 2D vdW crystals can exhibit opposite signs for the components parallel and perpendicular to the stacked planes, making the PhPs hyperbolic in nature. These hyperbolic PhPs (HPhPs) exhibit low losses and high quality factors, which enable the confinement and manipulation of electromagnetic fields at a deeply subwavelength scale and give rise to various important applications such as super-resolution imaging, [10,11] ultrasensitive sensing, [12] and thermal management. [13,14] In particular, recently explored 2D transition metal oxides, such as α-MoO3 [8,9,15,16] and α-V2O5, [17] represent natural biaxial hyperbolic crystals. The permittivities are highly anisotropic and even exhibit components with opposite signs along orthogonal in-plane directions over a broad spectral range, giving rise to PhPs with in-plane hyperbolicity. These HPhPs are observed as propagating wavefronts of concave shape, in striking contrast to those propagating in uniaxial crystals, such as hexagonal boron nitride (h-BN). [3,4] Such anisotropic characteristics open up new opportunities for confining and configuring electromagnetic waves at the nanoscale, especially in the mid-infrared to terahertz spectral range with limited photonic material candidates. For example, very recent studies have demonstrated that the stacking and twisting of two α-MoO3 layers result in intriguing planar light propagation and localization behaviors. [18][19][20][21] The practical utilizations of vdW HPhPs strongly rely on the configuration of their electromagnetic fields to overlap with the functional components of a device or photonic circuits. Several approaches have been proposed to engineer HPhPs, including controlling the dielectric environment, [22][23][24][25] carrier concentration, [26,27] material composition, [28,29] and geometry [30][31][32] of the crystals and devices. In particular, patterning the crystals into micro-or nanostructures can generate localized electromagnetic fields with configurable spatial distributions that are strongly dependent on geometric parameters. [30][31][32] Recent studies have successfully demonstrated rich and intriguing localized electromagnetic modes in micro-or nanostructured polaritonic crystals, such as graphene, [33] h-BN, [30][31][32] SiC, [34] and hybrids. [32,35] Most of the previous results focused on uniaxial crystals, where the in-plane polariton propagation is isotropic. Recently, subwavelength mid-infrared electromagnetic field localizations have been reported in circular and square α-MoO3 micro-disks. [8,9] Nevertheless, configuring the near-field distributions associated with the in-plane anisotropic, and critically hyperbolic polaritons, via patterning and the associated design principles is still incomplete.
As such developing these understandings of highly anisotropic polariton propagation and spatial confinement is anticipated to open a new degrees of freedom for sub-diffraction-limit light focusing and manipulation.
Here, through the combination of theoretical calculations and real-space nano-imaging, we demonstrate the engineering of infrared electromagnetic field localization of HPhPs in microstructures constructed from biaxial α-MoO3 crystal. The polariton waves that propagate inside a specific microstructure will encounter and be reflected by boundaries, whereby polariton interference results, generating complex standing wave patterns. We show that such interference is governed by the orientation of the in-plane polaritonic isofrequency curve (IFC) relative to the microstructure edges. This is possible because the energy flow directions of the incident and reflected polariton waves are both normal to the IFC. Therefore, the electromagnetic field spatial distributions within a microstructure can be configured by tuning the size and shape, as well as the excitation frequency. The obtained localized fields are highly anisotropic, irrespective of the symmetric microstructure shapes that are supported within. Moreover, we show that the spatial near-field distributions of a wedge-shape microstructure are strongly dependent on the angles between the bisector of the vertex angle and the [001] axis of the α-MoO3 crystal. Such result further illustrates the combination of inplane anisotropy and boundary manipulation as a paradigm for configuring electromagnetic fields at the nanoscale.

Results
α-MoO3 supports anisotropic PhPs throughout the mid-infrared (545 to 1010 cm −1 ) and THz (267 to 400 cm −1 ) spectral range, [8,9,15,16] over much of which these modes are hyperbolic in nature. These hyperbolic spectral bands originate from the opposite signed permittivities along different crystalline directions: Re(εi)·Re(εj) < 0, with i and j representing the [100] (x-axis), [001] (y-axis), and [010] (z-axis) crystalline axes ( Figure S1a and S1b). [8,9] In the x−y plane, the HPhPs exhibit open hyperbolic-shaped IFCs and propagate with concave wavefronts inside Restrahlen Band 1 (545 to 851 cm −1 ) and Band 2 (820 to 972 cm −1 ) ( Figure   1a, 1b, 1d, and 1e, Figure S2b  where the IFC is closed due to both in-plane permittivities being positive, the polaritons remain anisotropic featuring elliptical wavefronts (Figure 1g, 1h and Figure S2d) as Re(εx) ≠ Re(εy). For ease of discussion, hereafter Band 1, 2, and 3 will be named as negative-εy-Band, negative-εx-Band, and elliptical band, respectively. Every polariton wave launched into a microstructure will propagate with the energy flow in direction orthogonal to the IFC. This will be reflected by the boundaries of the microstructure and subsequently interfere with the reflected wave, thereby generating standing-waves of various spatially localized electromagnetic field distributions. [33] Because the interference effects originate from the superpositions of the polariton wavefronts, the standing-waves are determined by the wavevector distributions, i.e., polariton IFC topology. Consequently, the anisotropic IFCs can bring a new degree of freedom for configuring the localized electromagnetic fields in α-MoO3 microstructures, which, otherwise is exclusively determined by the microstructure geometry fabricated from crystals with an isotropic IFCs (i.e., Re(εx) = Re(εy), Figure S2a, Figure   S3a−S3c). [3,4] Figure 1. HPhPs in biaxial vdW crystals. a, d, and g) Calculated 3D isofrequency contours in the α-MoO3 slab. The calculations were performed at 700 cm −1 (negative-εy-Band), 937 cm −1 (negative-εx-Band), and 980 cm −1 (elliptical band), respectively. b, e, and h) Calculated Re(Ez) above the α-MoO3 slab surface at the same three frequencies. Green arrows indicate the propagation directions of the polariton waves. c, f, and i) Isofrequency curves of the α-MoO3 slab at the same three frequencies. The Re(Ez) distributions were calculated by launching the HPhPs on the sample surfaces using z-polarized electric dipoles. The isofrequency curves shown in (c), (f), and (i) were obtained as Fourier transforms of (b), (e), and (h), respectively. The dashed lines represent the in-plane dispersions obtained from the analytical electromagnetic waveguide model calculations. The α-MoO3 slab is 170-nm thick.
An analytical model was first developed to calculate the in-plane IFC the HPhPs propagating inside a vdW slab of finite thickness (see note S1). [36,37] To simplify the discussion, ε denotes the real part of the permittivity in our following discussion.
According to the model calculations, the electromagnetic waves in the hyperbolic vdW slab are dominated by the TM polariton modes with a dispersion relation:  (1) where ; d is the slab thickness; k0 is the free-space wavevector; θ denotes the angle of the propagation direction relative to the x-axis, M represents the order of the different TM modes, and εc and εs are the dielectric constants for air and the substrate, respectively. Using the permittivity of α-MoO3 as the input parameter (see note S1, table S1), the in-plane IFCs of the slabs were calculated according to Equation (1) at representative frequencies, which agree well with the numerical results (Figure 1c, 1f, and 1i). Specifically, for an α-MoO3 slab, the IFCs in negative-εx-Band is hyperbolas opening toward the x-axis bisected by x-axis. As a result, polaritons within the Restrahlen band are forbidden to propagate along y-axis. Similar calculation results can be obtained for the negative-εy-Band (see note S2, Figure S4a). In elliptical band, the IFCs are ellipses ( Figure   2b), resulting in polaritons spreading along all directions in the x−y plane, but with orientation-dependent wavevectors. Having established the IFC, the interference patterns of α-MoO3 microstructures can be readily calculated using the phenomenological cavity model (see note S3). [15,20,36,38] Only the boundary perpendicular to the polariton wavevector is considered in our calculations to simplify the discussion and correlate the model calculations directly with near-field measurements performed using scattering-type scanning near-field optical microscopy (s-SNOM). [8] In other words, the interference effects are dominated by those between a polariton wave and its antiparallel back-reflected wave. As will be shown in the following discussion, results. [8,9] For the polaritons in elliptical band, the interference fringes are a series of ellipses In addition to the fringe shapes, the anisotropic IFCs can modulate the interference patterns in three aspects. First, in negative-εx-Band, the IFCs exhibit a larger φ at a smaller ω ( Figure 2a and Figure S4b). Therefore, as ω is reduced, interference fringes contributed by polaritons with wavevectors close to the IFC asymptotes will extend toward the two disk end points along the y-axis (Figure 2c The results indicate that unlike other polaritons generating isotropic concentric patterns, [33,39] the localized fields in the micro-disk clearly reveal highly anisotropic frequency-dependent spatial distributions. Crossshape fringes that stem from the directional polariton propagation associated with the in-plane hyperbolic response of the α-MoO3 can be observed near the disk center at ω = 900 cm −1 (negative-εx-Band) (Figure 3a and Figure 1e). Meanwhile, elongated concentric fringes that originate from the in-plane elliptical response can be observed for excitation at 994 cm −1 (elliptical band) (Figure 3d and Figure 1h). These anisotropic HPhPs will be reflected by the circular edge and generate deformed fringes close to the edge (Figure 3a and 3d). The phase difference between two adjacent bright fringes is approximate π, which is a typical characteristic of a wave interference. Accordingly, the polariton wavelengths can be readily quantified by measuring the separations between the near-field maxima and minima (as  Figure S8a, and S8b). It should be noted that the polariton field decays faster at 994 cm −1 outside the disk than that at 900 cm −1 (Figure 3a, 3b, 3d, and 3e). Such a difference can be understood from two aspects. First, polaritons excited by 900 cm −1 exhibit in-plane hyperbolicity. The energy density within the IFC cone at 900 cm −1 will be larger than that at 994 cm −1 . This can lead to stronger polariton reflection and transmission at the disk edge, which will result in longer decay length of the polariton field outside the disk. Second, the diameter of the disk excited at 900 cm −1 (1.5 μm) is smaller than that excited by 994 cm −1 (2.0 μm). A larger disk diameter will lead to a longer polariton propagation length and therefore a stronger damping. This will also make the polariton field decay fast outside the disk at 994 cm −1 .  wave patterns formed by isotropic polariton superpositions, [33,39] the in-plane anisotropic HPhPs will lead to abnormal asymmetric interference patterns in each microstructure.  The interference patterns become even more asymmetrical for the excitation frequency in negative-εx-Band (915 cm −1 ). Both the square and regular hexagon microstructures only exhibit fringes parallel to the y-axis, with the two strongest located next to the edges ( Figure   5d, 5j, 5f, and 5l). This result can be understood by considering that the IFC at 915 cm −1 is a hyperbola opening toward the x-axis. The opening angle between the two asymptotes is φ = 108°. Therefore, for polaritons with wavevectors confined by the two asymptotes, only the two boundaries parallel to the y-axis can reflect the polariton waves, leading to the observed fringes along the y-axis. The polariton wavelengths extracted from the near-field profiles of these two microstructures are 475 nm and 480 nm (Figure 5p and 5r), respectively. These values match well with λp = 2π/q[100] = 480 nm, where q[100] is the wavevector determined from the IFC for the HPhPs propagating along the x-axis. However, for a regular pentagon microstructure, the angle between the two edges intersecting with the x-axis is 108°. Two polariton waves can always be back-reflected by these two edges because π−108°= 72°< φ.
As a result, additional fringes parallel to the two edges can be observed (Figure 5e and 5k).
Moreover, due to the forbidden of polariton propagation, no fringes emanating from the rotated edges can be observed in all of the three microstructures (Figure 5d−5f and 5j−5l).
Most interestingly, according to the IFC, the q associated with the polariton waves propagating toward the two oblique edges is much larger than q[100] (Figure 2a), thereby rendering much shorter polariton wavelengths (λp = 190 nm) (Figure 5q) and indicating much stronger electric field confinements near the oblique edges. Another feature should be noted is that slightly deformed fringes can be observed experimentally at corners of square microstructure, while they are absent in the calculated image (Figure 5d and 5j). These deformed fringes are due to interference involve high-wavevector polaritons, which are similar to those observed in the circular micro-disk (Figure 2c−2j). As mentioned before, in our calculations only the boundary perpendicular to the polariton wavevector is considered.
Therefore, in the square microstructure, the contributions from polaritons with high wavevectors are neglected, giving rise to discrepancy of interference fringes at the corners between calculation and experimental results.
The results above unambiguously demonstrate that the electromagnetic field localizations can be configured by controlling the shape and the size of an α-MoO3 microstructure, and the excitation frequency. We further demonstrate the configurability by fabricating and imaging a set of α-MoO3 micro-wedges with fixed vertex angles at 30°and varied skew angles (β) from 0 to 90°, which is defined by the angle between the wedge bisector and the y-axis (upper panel, Figure 6a). The near-field distributions were first examined at an excitation frequency of 990 cm −1 , where the IFC is an ellipse (Figure 2b). Two polariton waves that can be back-reflected by the two edges always exist, leading to the interference fringes parallel to the wedge boundaries (Figure 6a). In particular, the orientations of the two edges relative to the major axes of the IFC will be adjusted by rotating the wedge around the z-axis (i.e., changing the β). Therefore, the two reflected polariton waves will exhibit different wavevectors, enabling modifications of their interference fields (see note S4 and Figure S10). For ease of discussion, the edge above (below) the x-axis is labeled as Edge I (Edge II) (as indicated in Figure 6a). For β = 0 and 90°, the wedges exhibit lateral symmetries relative to the y-and x-axes, respectively. The polariton waves perpendicularly reflected by the two edges exhibit the same q ( Figure S10), making the two sets of fringes close to the edges equivalent ( Figure S11). The q of the polariton wave reflected by Edge I will first be reduced and then increased as β is steadily decreased from 90°.
This is opposite to that of the polariton wave reflected by Edge II ( Figure S11). Consequently, the corresponding two sets of fringes will differ from each other in terms of their spacings and linewidths ( Figure S11). Specifically, λp associated with Edge I (Edge II) will first increase (decrease) and then decrease (increase), as confirmed by the experimental measurements   Figure   S12a). No interference fringes can be observed parallel to the two edges ( Figure S13a). As β is progressively decreased, the two normal directions will successively sweep the two asymptotes of the IFC ( Figure S12a and S12b), enabling polariton reflection and interference at the edges. Specifically, when the β is reduced by more than 21°(β < 69°), the normal direction of Edge II will intersect with the IFC, while the normal direction of Edge I is still outside the IFC. As a result, polariton fringes parallel to Edge II appear, while those parallel to Edge I are still absent ( Figure S13a). When β is further reduced, polariton wave vector qII will be reduced and then increased again as β is smaller than 15°( Figure S12b and S12c).
Accordingly, polariton wavelength λp associated with Edge II will first increase and then decrease against β (Figure 6d and Figure S13a). On the other hand, the normal direction of Edge I can intersect with the IFC only when β is smaller than 39°. Therefore, the fringes parallel to Edge I will not appear unless β < 39° (Figure 6c and Figure S12c). Afterwards, the wave vector qI is reduced against β, giving rise to monotonically increased λp at Edge I ( Figure 6d). These experimental findings are well corroborated with the model calculations ( Figure 6d and Figure S13b).

Conclusion
The patterning of photonic materials into micro-and nanostructures has been well established for controlling electromagnetic waves at the nanoscale. Our study demonstrates that by combining in-plane anisotropic polaritons with high spatial confinement, that polariton interference effects inside an α-MoO3 microstructures can be controlled by tuning the excitation frequency (IFC topology), shape (number and type of the boundary), and size (polariton propagation length) of the microstructure. Thus, the configurability of the nanoscale electromagnetic fields can be extended in terms of anisotropic spatial distributions and orientation dependence, which, otherwise, usually requires the design and patterning of complex architectures in isotropic counterparts. It should be noted that very recently two inplane THz polaritonic bands have been discovered in α-MoO3. [16] The results obtained in the current study can therefore also be applied to the THz spectral region, which can help design of nanophotonic devices for THz applications. In principle, this rationale can also be During submission of our manuscript, we became aware of a very recent publication reporting similar studies. [40] Our work was conducted independently. In addition, in our present study, we further demonstrated configuring the electromagnetic field localizations in α-MoO3 microstructures of different sizes and shapes.

Experimental Section
Fabrication of α-MoO3 Microstructures: α-MoO3 single crystal slabs were prepared using thermal physical vapor deposition method. [41] The crystals were directly synthesized on silicon substrate grown with 300-nm thick SiO2 layer. The various α-MoO3 microstructures were fabricated using the focused ion beam (FIB) technique. Specifically, Ga + ions were used as ion sources in our FIB etching system (AURIGA, Zeiss). The acceleration voltage and current of Ga 3+ beam were respectively set as 30 kV and 10 pA, with a Ga 3+ beam dose of 1 nC/μm 2 . The dwell time was 0.5 μs. After the microstructures were constructed, they were annealed at 300°C for 2 h in O2 ambient condition to eliminate the Ga 3+ embedded in the α-

MoO3.
Numerical Simulations: Numerical simulations were performed using the finite element method (FEM, Comsol). To generate the near-field spatial distributions above the various 2D slabs and microstructures, HPhPs were launched using a z-polarized electric dipole source.
Specifically, in each simulation, the dipole was located 200-nm above the sample surface. The near-field distributions, Re(Ez), were obtained on the plane 20-nm above the sample surface.
The dipole was scanned across the sample at a step of 5 nm. The thicknesses of the slabs and α-MoO3 microstructures were set as 170 nm. Permittivities of the samples were modeled by fitting their respective experimental data using Lorentzian dielectric models. [3,15] Nanoimaging of Various α-MoO3 Microstructures: Real-space nanoimaging was performed using an s-SNOM (NeaSNOM, Neaspec GmbH), which was built based on an atomic force microscope (AFM). In a specific measurement, the metal-coated tip (Arrow-IrPt, NanoWorld) was illuminated by a mid-infrared laser source with tunable frequencies from 900 to 1240 cm −1 (quantum cascade laser, Daylight Solutions). The tip was vibrated at a frequency of 270 kHz. The back-scattered light from the tip was directed to an MCT detector (HgCdTe, Kolmar Technologies). The near-field signal was then extracted using a pseudo-heterodyne interferometric method, and the detected signal was demodulated at a third harmonic of the tip vibration frequency.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author. By taking advantage of the in-plane hyperbolic phonon polariton and spatial confinement effects, we demonstrate the nanoscale tailoring of mid-infrared electromagnetic field localizations in van der Waals α-MoO3 microstructures of different shapes and sizes. We further show that orientation of the in-plane isofrequency curve relative to the microstructure edge is a critical parameter governing the localized electromagnetic field distributions.