Twist-tunable in-plane anisotropic polaritonic crystals

1 Introduction

The discovery of polaritons supported by van der Waals (vdW) materials [1], [2] has sparked significant interest due to their potential for manipulating light on the nanoscale [3]. In particular, the propagation of in-plane anisotropic phonon polaritons (PhPs) is supported in certain crystal layers, such as calcite [4], α-MoO₃ [5], V₂O₅ [6], or bGO [7] among others. These materials exhibit PhPs with hyperbolic dispersion across specific frequency ranges, leading to exotic optical phenomena such as light canalization in twisted crystal bilayers [8], [9], [10], [11] and trilayers [12], [13], or twist-tunable nanoresonators [14]. These findings open possibilities for an active tuning of the polariton propagation through the twist angle between layers, which is the key feature of the emerging field known as twistoptics. Another interesting option for controlling polaritons on the nanoscale consists of constructing polaritonic crystals (PCs) – lattices composed of elements periodically spaced at distances comparable to the wavelength of polaritons – made in vdW materials, such as, e.g., h-BN [15], α-MoO₃ [16], [17], [18], monolayer graphene [19], [20], or twisted bilayer graphene [21]. PCs support ultra-confined Bloch modes [22], [23], [24] that can favor a topological funneling of PhPs [25]. Recently, PCs composed of hole arrays (HAs) in α-MoO₃ slabs have been suggested and studied, both theoretically [16] and experimentally [17], [18]. In these PCs, emerging Bragg resonances depend on the orientation of the lattice vectors with respect to the crystallographic axes of the vdW crystal layer. However, these HAs were etched directly in the anisotropic vdW layer, so that their geometry is not actively reconfigurable. Moreover, the fabrication of holes within the α-MoO₃ increases the optical losses due to severe scattering processes [22], [26]. Consequently, although the rotational dependence of the Bragg resonances has been showcased, achieving postfabrication tunability remains a challenge.

In this work, we introduce a novel PC concept based on the fabrication of twistable α-MoO₃/metal heterostructures that overcomes the above limitations, enabling low-loss optical tuning. As depicted in Figure 1, it comprises a twisted pristine layer of α-MoO₃ on top of a periodic HA made in a gold layer. Remarkably, the α-MoO₃ has been chosen due to its long polaritonic lifetimes, reaching up to 20 ps [5]. However, conceptually, it could be replaced by any other in-plane anisotropic material, such as V₂O₅ [6] or bGO [7], which provide similar twist-dependent properties. The lattice period matches the wavelengths of the PhPs in α-MoO₃ crystal on top of gold. This configuration allows for active tunability of the angle between the lattice vectors and crystallographic directions by rotating the α-MoO₃ layer. It is important to note that the parameters of our periodic HA are chosen to ensure the presence of “collective” lattice effects, in contrast to the periodic set of individual twist-tuneable Fabry–Perot resonators previously reported [14]. Employing a theoretical approach, which incorporates full-wave simulations and an analytical approximation, we describe the formation of PhP band structure and excitation of Bragg resonances. Furthermore, we conduct near-field measurements for different rotation angles, managing to disentangle the contribution of individual PhP Bloch modes emerging in this configuration. With our combined theoretical approach, we also reconstruct the PhP bands from the near-field data, by extracting the wavelength of the PhPs along different directions in plane.

Figure 1:

Schematic of a twist-tunable polaritonic crystal: an in-plane anisotropic α-MoO₃ layer twisted on top of a HA made on gold. The periodic lattice comprises a HA drilled in a gold layer with thickness d _m , hole radius a, and periodicity L. The anisotropic crystal layer, α-MoO₃, with thickness d, has its crystallographic axes twisted by an angle ϕ with respect to the lattice vectors. To visualize the polaritons excited in the PC, we modeled the distribution of the electric field (z-component), generated by a vertically oriented point dipole placed above the α-MoO₃ layer.

2 Results

In Figure 1, we show the schematics of our heterostructure representing a twisted PC. A pristine layer of α-MoO₃ (of thickness d) lies on a metallic film (of thickness d _m ) with a periodic array of holes (of radius a and filled by a material with permittivity ɛ _h ) etched in it. Although our concept is valid for any periodic lattice with arbitrary lattice vectors L ₁ and L ₂ , here for simplicity we focus on a square lattice ( L ₁ and L ₂ are orthogonal and L 1 = L 2 = L ). The α-MoO₃ layer is twisted by an angle ϕ with respect to the L ₁ lattice vector of the HA. Notice that the thickness of the gold film is assumed to be larger than the skin depth, so that the PhPs in the vdW layer are not sensitive to d _m.

Crystal layers of α-MoO₃ support PhPs within three different Reststrahlen bands [27] (RBs, range of frequencies defined between the longitudinal and transversal optical phonons frequency, LO and TO, respectively): 544.6 cm⁻¹ – 850.1 cm⁻¹ for RB1, 821.4 cm⁻¹ – 963 cm⁻¹ for RB2, and 956.7 cm⁻¹ – 1,006.9 cm⁻¹ for RB3. In these RBs, the isofrequency curves (IFC) – describing available in-plane momenta at a fixed frequency – can take different shapes. Thus, in RB1 and RB2 IFCs have hyperbola-like shapes, with vertices lying on the [001] and [100] crystallographic directions, respectively. In contrast, PhPs with elliptical IFCs are supported in RB3. These highly anisotropic PhPs evolve into a series of modes, commonly designated as Mn, where n ∈ N indicates the quantization of the mode in the transversal (perpendicular to the slab) direction, displaying a lower propagation length and wavelength when increasing n. As in the mid-IR frequency range gold behaves as a perfect electric conductor (a “mirror”), only symmetrical modes could be excited in an α-MoO₃ layer on top of a gold film, so that the mode with the longest polaritonic wavelength is M1 [14].

To study and illustrate the emergence of tunable Bragg resonances in our twisted heterostructure, we first consider illumination by a plane wave, i.e., the excitation of PhPs from the far-field. To that end, we calculate the field amplitudes of different diffraction orders (see Supplementary Material, Section 1 for details). The parameters of the lattice (period L = 250 nm, and the hole radius a = 55 nm, where the holes are filled by air, thus ɛ _h = 1) have been chosen to match the available wavelengths of the PhP mode in an α-MoO₃ slab of thickness d = 70 nm. This selection guarantees the presence of Bragg PhP resonances within the hyperbolic RB2. The hole radius is large enough to ensure strong collective Bragg resonances, yet small enough to prevent the formation of Fabry–Perot resonances that could interfere with collective Bragg modes. For simplicity, but without loss of generality, we assume that both the substrate and superstrate are air, while the thickness of the metal film is set to d _m = 30 nm. The light propagating through the PC is scattered into various diffraction orders (plane waves), which we labeled as (n ₁, n ₂). Their field amplitudes, R σ n 1 n 2 , (with σ = s, and σ = p staying for s- and p-polarizations, respectively), can be individually determined from a linear system of equations (see Supplementary Material, Section 1, Eq. (77)). In Figure 2b, f, and j, we show the spectra of the normalized reflection coefficient, ΔR (solid black line), for three values of the twisting angle ϕ, as illustrated by the schematics in Figure 2a, e, and i. Here ΔR = R _b − R, with R _b being the field amplitude of the (0, 0) order mode of an α-MoO₃ layer on top of an unstructured gold film, and R being the scattered amplitude of the (0, 0) order mode in the twisted PC. At ϕ = 0 (Figure 2b), ΔR displays a peak at 885.1 cm⁻¹. At the same frequency, the R p 10 amplitude (blue curve) shows a clear resonant peak in its spectrum, indicating the emergence of a PhP Bragg resonance in (±1, 0) diffraction order. Indeed, the latter largely exceeds the contribution from the other nearest diffraction orders R p 1 − 1 and R _p11, as follows from their spectra, shown by the red and yellow lines, respectively. Apart from the (±1, 0) order mode, other Fourier field harmonics can also display resonant peaks, such as at 881.4 cm⁻¹, where the dominant amplitudes come from the (±1, ±1) and ± 1 , ∓ 1 orders. In these pairs, the sign of the left number matches the sign of the right number (up for up, down for down).

Figure 2:

Twist-tunable Bragg resonances in a PC. (a, e, i) Schematics illustrating the top view of a twistable PC. The α-MoO₃ layer is aligned with the xy axis, whereas the gold HA (with the lattice vector basis, L ₁ and L ₂ ) undergo an anticlockwise twist of 0°, 15°, and 30°, respectively. (b, f, j) Spectra of different Fourier field harmonics amplitudes for normal incident light linearly polarized along the [100] axis. Black line represents ΔR = R _b − R, where R _b is the reflection of α-MoO₃ on top of bare gold layer, and R is amplitude of the (0, 0) field harmonic. The blue, yellow, and red lines represent R p 10 , R p 11 , and R p 1 − 1 (p-polarization components of the Fourier field harmonics). The shift of the resonance is indicated by a gray dashed line. (c, g, k) Electric field distributions for ϕ = 0°, 15°, and 30°, at the (±1, 0) resonance frequencies, 885.1 cm⁻¹, 882.5 cm⁻¹, and 875.7 cm⁻¹, respectively. The Bragg vector G ₁₀ , perpendicular to the wave fronts and aligned to L ₁ lattice vector, is indicated by the black arrow. (d, h, l) Color plot (representing ∑ R σ N 2 for σ ∈ s , p ) illustrates the IFC of the PC at the same twist angles and frequencies as in (c, g, k). The reciprocal lattice and the Bragg vector, G ₁₀ , are indicated by the blue points and the red arrow, respectively. IFC of the empty lattice for the diffraction order (0, 0) is represented by the green dashed line.

The emerging Bragg resonances can be explained by matching the IFC of the M1 PhP mode in the structure without holes with the reciprocal space vectors G n 1 n 2 = n 1 g 1 + n 2 g 2 , pointing from the origin toward the reciprocal space points (the grid of integers), marked by the blue dots in Figure 2d, h, and l. For instance, at 885.1 cm⁻¹, the IFC of the M1 PhP mode (indicated by the green dashed line in Figure 2d and also seen as the bright maxima of the color plot – see Supplementary Material, Section 2 for more details about color plot) meets the reciprocal vector G ₁₀ , thus fulfilling the Bragg resonance condition for the (±1, 0) order. When the resonance condition is met, the electric field pattern, E z x , y , represents a standing wave with the reciprocal vector G ₁₀ , as shown in Figure 2c.

Next, we analyze the spectra of the PC when the α-MoO₃ layer is twisted by 15° and 30° with respect to the HA (schematics in Figure 2e and i), Figure 2f and j, respectively. In both cases, the frequency of the resonant peaks redshifts, to 882.5 cm⁻¹ for ϕ = 15° and to 875.7 cm⁻¹ for ϕ = 30°, respectively, thus amounting up to ∼7 % of the entire RB. This twist-induced shift is also seen in the reciprocal space representations shown in Figure 2h and l. Due to the twist of the lattice in the real space, the reciprocal space points undergo an anticlockwise rotation. Because of the anisotropy of α-MoO₃, the dispersion relation depends upon the orientation of the k-vector and thus the dispersion curves. Consequently, the frequency at which the IFC meets the reciprocal vector G ₁₀ varies, so that the Bragg resonance condition is fulfilled at 882.5 cm⁻¹ and 875.7 cm⁻¹ for 15° and 30°, respectively. At these frequencies, the field patterns shown in Figure 2g and k clearly display a Bloch standing wave whose fringes are oriented along the L ₁ direction. Consistently, the major contribution into the excited PhP near-field arises from the diffraction order ± 1,0 with the dominating field amplitude, R p 10 . Note that the Bragg resonance condition for other diffraction orders also shifts in frequency, as observed in the peak position of the field amplitudes R p 11 and R p 1 − 1 . Actually, for nonzero Φ, their spectra no longer coincide, due to the break of symmetry. For instance, for Φ = 15°, when the IFC meets 1,1 and − 1 , − 1 reciprocal space points, it does not meet those at 1 , − 1 and − 1,1 . Finally, remark that the frequency tunability of the Bragg resonance by rotating the lattice with respect to the optical axes of the crystal layer remains an important intrinsic feature of twisted PCs [16], [17].

The dependence of the Bragg resonances upon the rotation angle is also encoded into the band structure of the twisted PC, which is illustrated in Figure 3. The volumetric Wigner–Seitz cell of the twisted PC is schematically shown in Figure 3a–c, together with the high symmetry points of the first Brillouin Zone (BZ). Apart from the square lattice symmetry points Γ, X, and M, we have introduced an additional point, X′, as obviously due to the in-plane anisotropy of the α-MoO₃ layer, the PhP dispersion along 100 and the 010 axes is different. More specifically, the band structure along the path starting from Γ and finishing in X differs from that connecting Γ and X′. The color plot shown in Figure 3d illustrates the near-field intensity given by the summation of a large number of the field amplitudes, R σ n 1 n 2 , across the path Γ − >X − >M − >Γ − >X′ − >M (the arrows indicate the direction of the path, from the starting point to the end point) in the in-plane momentum plane. The bright maxima of the color plot approximately represent the “density of states” of PhP modes in the twisted PC. For a better interpretation of the band structure, we also plot in Figure 3d (gray dashed lines) the Φ-dependent empty grating dispersion branches, k n 1 n 2 ω = k M 1 ω + G n 1 n 2 (where k _{M 1} is the momentum of PhPs supported by α-MoO₃ on top of an unstructured metal film), for the diffraction orders −2 ≤ n ₁, n ₂ ≤ 2 and for Φ = 30°. The empty grating dispersion finds an excellent agreement with the maxima of the color plot. Note that, the band gaps of the PhPs in the twisted PC are visually indistinguishable in the color plot. We attribute it to the low refractive index contrast, and thus the small reflectivity of the PhPs, by the areas of the holes. On the other hand, the dispersion branches of the twisted PC can be shifted by the twist angle, as illustrated in Figure 3e–h, showing a zoomed-in area within the blue dashed rectangle in Figure 3d. For instance in the Γ point, while at Φ = 0° the two branches − k M 1 ω + G 10 and k M 1 ω + G − 10 meet at 885.1 cm⁻¹ (Figure 3e), by increasing Φ to 15°, 30°, and 45°, the crossing redshifts to 882.5 cm⁻¹, 875.7 cm⁻¹, and 863.9 cm⁻¹, respectively (in Figure 3f–h the shift is indicated by the gray dashed lines). To connect the tunability of the band structure with the Bragg resonance condition, in Figure 3i–l, we represent the IFCs at the crossing frequencies, 885.1 cm⁻¹, 882.5 cm⁻¹, 875.7 cm⁻¹, and 863.9 cm⁻¹, respectively. For the reference, the IFCs of the empty lattice PhPs with momenta ± k _{M 1}, ± k _{M 1} + G _±10, and ± k _{M 1} + G _0±1 are also rendered (green, yellow, and red dashed lines, respectively). For each Φ, we observe that the IFC meets the reciprocal space point ± 1,0 so that the Bragg resonance condition is fulfilled, implying the color plot maxima at this frequency at the Γ point observed in Figure 3e–h. Remark that, while our discussion has been centered on the zoomed-in region, these findings extend to the entire PhP band structure, so that all the bands undergo frequency shifts by twisting.

Figure 3:

Tunability of the band structure of the twisted PC. (a) Schematic of the twisted PC for Φ = 30°. (b) A zoom-in view of the Wigner–Seitz cell of the PC. The thickness of α-MoO₃ layer and the gold film at d = 70 nm, and d _m = 30 nm, respectively. (c) A sketch of the reciprocal space directions, indicated by blue dashed lines, with the main points of the BZ labeled as Γ, X, M, and X′, respectively. The BZ is combined with a schematic of the x–y projection of the Wigner–Seitz cell. (d) Color plot (representing ∑ R σ N 2 for σ ∈ s , p ) indicates the band structure of the twisted PC for Φ = 30°. The empty lattice band structure is traced by the gray lines, for the diffraction orders n 1 , 0 , n 1 , ± 1 , and n 1 , ± 2 , with n ₁ restricted to −2 ≤ n ₁ ≤ 2. (e–h) Zoom-ins of the band structure for Φ of 0°, 15°, 30°, and 45°, respectively. The shift of the bands at the Γ point is indicated by the gray dashed lines and the arrow. The blue asterisk symbols indicate the extracted momentum from experiments (see details in Supplementary Material, Section 3). (i–l) Color plots indicating the IFCs of the twisted PC for Φ of 0°, 15°, 30°, and 45°, at the frequencies of 885.1 cm⁻¹, 882.5 cm⁻¹, 875.7 cm⁻¹, and 863.9 cm⁻¹, respectively. The reciprocal lattice is shown by the blue dots, and the IFCs of the empty lattice for the diffraction orders 0,0 , ± 1,0 , and 0 , ± 1 are represented by the green, yellow, and red dashed lines, respectively.

To corroborate our theoretical analysis, we conducted experimental near-field measurements on twisted PCs, which consist on a single α-MoO₃ layer placed on top of a gold film with a series of HA rotated at various angles (see Section 3). To visualize the PhPs Bloch modes, we employed near-field nanoimaging via scattering-type scanning near-field optical microscopy (s-SNOM) [23], see schematics in Figure 4a. The near-field excitation offers a significant advantage over the far-field approach, as (i) the latter is suppressed by the screening effect from the metal substrate and (ii) the s-SNOM tip allows one to access to large PhP momenta beyond the light cone. Both the sample and the s-SNOM tip are illuminated with a p-polarized mid-IR light at an incident frequency of 875.7 cm⁻¹, i.e., within the RB frequency range. By recording the scattered field signal (s ₃) (see Section 3), we produced the near-field images visualizing PhPs excited in the twisted PC at an angle Φ = 30° (color plot in Figure 4b). In the image, we observe a series of parallel fringes, marked by black solid lines, with a separation distance between them matching the periodicity of the HA. To interpret the observed near-field pattern, we conducted full-wave simulations employing a vertical point dipole source, mimicking the s-SNOM tip [23]. The simulated snapshot of Re E z shown in Figure 4c reveals an oscillating field distribution (the wavelength is indicated by the black lines) within a narrow sector around the 100 crystallographic direction. This distribution is characteristic of recently reported canalized PhPs [9], [12]. The canalization regime of PhPs in our twisted PCs can be explained by the plane shape of the IFC of the M1 PhPs in an α-MoO₃ layer on top of a metal substrate, similar to α-MoO₃ on SiC [28]. Remarkably, such a directional pattern is produced by placing the dipole at any point above the twisted PC. Nevertheless, in the near-field image constructed by scanning the dipole in the x–y plane (Figure 4d, see Section 3, near-field full-wave simulations section), one can clearly recognize a standing wave pattern, in good agreement with the near-field measurements (Figure 4b). Such periodic pattern clearly indicates a largely collective nature of the excited electromagnetic fields, inherent to PCs. Moreover, in both simulated and experimental near-field images, the direction perpendicular to the fringes (highlighted by the solid black lines) forms a 30° angle with the 100 crystallographic axis, thus being aligned with the lattice vector. This observation provides a hint that the ± 1,0 Bloch mode is excited due to the ± 1,0 Bragg resonance discussed above. To confirm our speculation, we performed full-wave simulations considering a normally incident plane wave illumination, thus completely excluding any effects induced by a localized source. The resulting distribution of Re E z is illustrated in Figure 4e. The normally incident wave “acquires” the momentum G _±10 provided by the lattice, so that the two excited contrapropagating PhPs plane waves form the ± 1,0 Bloch mode, which can be recognized in Figure 4e. Its field distribution agrees well in shape, wavelength, and fringe orientation with the near-field images in Figure 4b and d. Furthermore, by representing the Fourier transform (FT) of the experimental near-field image as a color plot in Figure 4f, we can observe two bright maxima labeled as 1,0 and − 1,0 that perfectly match the lattice vectors, G _±10. Overall, our simulations are consistent with our assumption that PhP Bloch mode is observed in the nanoimaging experiments.

Figure 4:

Near-field imaging of the PhP modes in the twisted PC and their analysis. (a) Schematics of the PC and the s-SNOM. The upper α-MoO₃ layer is twisted by Φ = 30° with respect to the HA in a thin gold layer. The structure is suspended in air, thus there is no substrate under the HA. An s-SNOM tip scans the PC under the presence of an incident field E _in, while the scattered field E _out is recorded. (b) A near-field s-SNOM image taken at the third harmonic at ω = 875.5 cm⁻¹. Black lines indicate the fringes of the PhP Bloch wave. (c) The simulated R e E z x , y generated by a vertical point dipole on top of a finite-size twisted PC. The wavelength of the PhP along the x-direction is indicated by the black lines. (d) Simulation of the near-field image shown in (b). The L ₁ vector of the reciprocal lattice and the 100 crystallographic axis from α-MoO₃ are indicated by the red arrows, whereas the position of the holes are marked by the black dashed circles. Black solid lines indicate the fringes. (e) Simulated R e E z x , y generated by a normally incident plane wave polarized along 100 direction. Black lines indicate the fringes. (f) Color plot depicting the FTs of the near-field image shown in (b). The reciprocal space vectors G ₁₀ and G ₋₁₀ are indicated by the gray arrows, forming an angle of 30° with respect to the k _x -axis.

Our PhP nanoimaging experiments can be extrapolated to other twist angles, Φ, as illustrated in Figure 5a–d. From panels a–d, α-MoO₃ layer maintains its crystallographic axes 100 and 001 aligned with the x and y axes, respectively, whereas the HA is twisted anticlockwise. In Figure 5a–d, the near-field images are shown at frequencies 884.3 cm⁻¹, 882.2 cm⁻¹, 875.7 cm⁻¹, and 864.1 cm⁻¹, corresponding to ± 1,0 Bragg resonance condition for Φ = 0°, 15°, 30°, and 45°, respectively (near-field images of PhP Bloch modes in other diffraction orders n 1 , n 2 apart from ± 1,0 are shown in Supplementary Material, Section 4). Note that the case of Φ = 30° in Figure 5e and f is intentionally copied from Figure 4b, d, and f for a better visualization of Bragg resonance rotation as the twisting angle increase. For each twisting angle, a standing wave with fringes perpendicular to the L ₁ lattice vector is observed, where the fringe positions are indicated by black solid lines. These measurements are in good agreement with the full-wave simulations conducted via the scanning dipole method, shown in Figure 5e–h. To analyze the PhP wavelength and the orientation of the fringes, we performed FTs of the near-field data, represented as color plots in the insets of Figure 5a–d. The positions of the FT maxima confirm that the reciprocal lattice vector, G ₁₀, is twisted by an angle Φ with respect to the x axis. Specifically, for Φ = 45° in the near-field image (Figure 5d and h), in addition to the standing wave with fringes oriented along the lattice vector L ₁, we observe another standing wave oriented along L ₂ lattice vector (the fringes are indicated by gray lines). The appearance of the second standing wave arises due to the lattice symmetry. Indeed, for smaller twist angles, the Bragg resonance condition was fulfilled for two reciprocal lattice vectors, G _±10, since the IFC of the α-MoO₃ simultaneously intersects with only two reciprocal space points, namely ± 1,0 . However, for Φ = 45°, the crystallographic axes of α-MoO₃ are positioned such that the intersection of the IFC with four reciprocal space points, ± 1,0 and 0 , ± 1 , becomes possible. This effect becomes even clearer in the Fourier transform of the measured data shown in the inset of Figure 5d, where four bright maxima match with reciprocal lattice vectors G ₁₀ and G ₀₁. Using our near-field images, it is also possible to reconstruct the momentum of the PhP Bloch modes for each angle Φ by measuring the separation distance between the fringes, λ _exp (details provided in Supplementary Material, Section 3). Due to the limited efficiency of exciting Bragg mode solely via the lattice, a circular ring surrounding the PC was introduced to boost the near-field, thereby enhancing the generation of polaritonic Bragg modes in the HA region. The s-SNOM tip detects the scattered near-field and evanescent waves generated by the interaction between the ring and the HA, allowing us to map the resulting polaritonic modes. We then translate the measured wavelength into momentum space (G _exp = 2π/λ _exp), which falls into the second BZ. To properly represent this momentum in the band structure, we fold this momentum from the second BZ to the first BZ by subtracting g ₁ − G _exp. This folded momentum is depicted by blue asterisk symbols in Figure 3e–h, in good agreement with the theoretical predictions. Overall, in Figure 5, we have experimentally demonstrated tunability of Bloch modes by the twist angle.

Figure 5:

Near-field imaging of the twist-tunable PC for different twist angles. (a, c, e, g) s-SNOM near-field images for twist angles of Φ = 0°, 15°, 30°, and 45° at the frequency of 884.3 cm⁻¹, 882.2 cm⁻¹, 875.7 cm⁻¹, and 864.1 cm⁻¹, respectively. Color plots representing the FTs of the near-field images for each Φ are included as insets, where the reciprocal lattice vectors, G ₁₀ and G ₀₁, are indicated by the gray arrows. (b, d, f, h) Simulated near-field images for a finite-size twisted PC, for the Φ and ω as in (a, c, e, g). The black solid lines indicate the fringes of the Bloch PhP wave in ± 1,0 diffraction orders, while in (g, h) the gray lines indicate those in 0 , ± 1 diffraction orders.

To summarize, we have introduced and experimentally realized a tunable PC based on a heterostructure consisting of a twistable α-MoO₃ crystal layer on top of a gold HA. In stark contrast to previously reported twisted PCs, our design preserves the pristine α-MoO₃ layer and allows it to be rotated without the need to modify the lattice beneath it. While by the theoretical analysis we have demonstrated the twist-tuneability of the emerging Bragg resonances in our PC, our experiments clearly demonstrate the tuneability of the PC band structure and the polaritonic Bloch modes. Remarkably, collective lattice effects largely dominate the natural polaritonic canalization effect. The latter is due to the intrinsically flat isofrequency curve of polaritons in a crystal layer above a metal substrate. Importantly, although demonstrated for α-MoO₃, our general concept is valid for any similar heterostructure based on in-plane anisotropic crystal layers supporting polaritons. It is important to note that, while twist-tunable systems have already shown great promise for real-time optoelectronic applications, several challenges remain before they can be fully implemented. Advances in microelectromechanical systems [29] are expected to provide precise and rapid control of twist angles at the nanoscale, enabling high-speed, real-time tuning. Additionally, mechanical robustness, particularly under repeated twisting cycles, is also critical, as these systems must maintain their structural integrity and performance over time. Despite these challenges, ongoing efforts to integrate twist-tunable systems into scalable platforms compatible with existing optoelectronic technologies are yielding promising results. Our findings expand the vision of twistoptics in a wide perspective and particularly hold promise for the development of actively rotatable two-dimensional polaritonic elements.

3 Methods