Unidirectional ray polaritons in twisted asymmetric stacks

Abstract

The vast repository of van der Waals (vdW) materials supporting polaritons offers numerous possibilities to tailor electromagnetic waves at the nanoscale. The development of twistoptics-the modulation of the optical properties by twisting stacks of vdW materials-enables directional propagation of phonon polaritons (PhPs) along a single spatial direction, known as canalization. Here we demonstrate a complementary type of directional propagation of polaritons by reporting the visualization of unidirectional ray polaritons (URPs). They arise naturally in twisted hyperbolic stacks with very different thicknesses of their constituents, demonstrated for homostructures of $\alpha$ -MoO₃ and heterostructures of $\alpha$ -MoO₃ and $\beta$ -Ga₂O₃. Importantly, their ray-like propagation, characterized by large momenta and constant phase, is tunable by both the twist angle and the illumination frequency. Apart from their fundamental importance, our findings introduce twisted asymmetric stacks as efficient platforms for nanoscale directional polariton propagation, opening the door for applications in nanoimaging, (bio)-sensing, or polaritonic thermal management.

Fig. 1. Unidirectional ray polaritons in twisted asymmetric stacks.

Schematic of the two systems under study: a a 80 nm-thin $\alpha$-MoO₃ layer placed over a 3 µm-thick $\alpha$-MoO₃ layer (homostructure) and b a 100 nm-thin $\alpha$-MoO₃ layer on top of a 5 µm-thick $\beta$-Ga₂O₃ layer (heterostructure). Parameters ${\theta }_1$ and ${\theta }_2$ represent the twist angle of the thick bottom layer with regard to the thin top layer for both systems, respectively. Real-space electric fields $\mathrm{Re}\left({\mathrm{E}}_z\right)$ launched by a point dipole source at the thin $\alpha$-MoO₃ surface for the homostructure (c–e) and the heterostructure (f–h) at illumination frequencies $\omega =920\mathrm{c}{\mathrm{m}}^{-1}$ and 734 $\mathbf{c}{\mathrm{m}}^{-1}$, respectively. The twist angles are ${\theta }_1=0^{\mathrm{o}}$ (c), ${15}^{\mathrm{o}}$ (d), ${30}^{\mathrm{o}}$ (e) and ${\theta }_2={150}^{\mathrm{o}}$ (f), ${120}^{\mathrm{o}}$ (g), ${90}^{\mathrm{o}}$ (h). The in-plane direction of ray-like propagation is marked with dashed gray lines at an angle defined by ${\phi }_1,{\phi }_1^{\prime }$, and ${\phi }_1^{″}$ (c–e) and ${\phi }_2$ (f–h) with respect to the [100] crystal direction of the top $\alpha$-MoO₃ layer. The Fourier Transforms (2D-FFTs) of the simulated real-space images are shown in the insets of c–h. The same scale has been used for every simulated real-space map.

Fig. 2. Observation of unidirectional ray polaritons in twisted asymmetric homostructures.

Near-field amplitude image in a twisted structure made of a $80$-nm thin $\alpha$-MoO₃ layer over a 3-μm thick α-MoO₃ layer with twist angles $\theta ={30}^{\mathrm{o}}$ (a–c) and ${60}^{\mathrm{o}}$ (d) at an illuminating frequency of $\omega =880\mathrm{c}{\mathrm{m}}^{-1}$, $900\mathrm{c}{\mathrm{m}}^{-1}$, $920\mathrm{c}{\mathrm{m}}^{-1}$ and $900\mathrm{c}{\mathrm{m}}^{-1}$ for a–d, respectively. A 200-nm diameter hole allows efficient launching of the phonon polaritons, whose wavefronts and direction of propagation are visualized by s-SNOM (scattering-type scanning near-field optical microscopy). For a–c the in-plane direction of propagation is marked with gray lines at an angle defined by $\phi$ with respect to the [100] crystal direction of the top $\alpha$-MoO₃ layer, see Fig. 1a. The experimental isofrequency curves (Fast Fourier Transform (2D-FFT) of the near-field image) are shown in the insets, verifying polariton unidirectional propagation in the direction defined by $\phi$. e–h Simulated near-field amplitude images of the system in a (e), b (f), c (g) and d (h). The 2D-FFTs of the simulated images are shown in the top insets of e–h.

Fig. 3. Analysis of the thickness disparity in twisted homostructures.

a–d Schemes of four systems made of $\alpha$-MoO₃ layers. The top thin layer in a–c has a thickness of $d_{top}=80$ nm while the bottom has $d_{bot}=80$ nm (a), $d_{bot}=500$ nm (b), $d_{bot}=3$ μm (c). Scheme d corresponds to a single thick layer of $\alpha$-MoO₃ (d= 3 μm). In all cases, the twist angle is $\theta ={30}^{\mathrm{o}}$ and the illumination frequency is $\omega =900\mathrm{c}{\mathrm{m}}^{-1}$. e–h Numerical simulations showing the near-field $\mathrm{Re}\left({\mathrm{E}}_z\right)$ generated by a point dipole located above the four structures represented in a–d, respectively. i–l Isofrequency curves (IFCs) obtained by performing the Fourier Transforms (2D-FFTs) of the near-field images in e–h, respectively. White dashed curves in i–k correspond to the analytic IFC of the bilayer case shown in a. White and orange lines in i–l are the asymptotes of the IFCs of phonon polaritons in the bottom and top layers, respectively.

Fig. 4. Observation of unidirectional ray polaritons in twisted asymmetric heterostructures.

Near-field amplitude image in a twisted heterostructure formed by a thin α-MoO₃ layer with thicknesses $d_{top}=200$ nm (a, c) and $400$ nm (b) on top of a $500$-μm thick (010) β-Ga₂O₃ substrate with twist angles $\theta ={45}^{\mathrm{o}}$ (a), $0^{\mathrm{o}}$ (b), and ${100}^{\mathrm{o}}$ (c) at an illuminating frequency of $\omega =734\mathrm{c}{\mathrm{m}}^{-1}$. A $1$-$\mathrm{\mu }\mathrm{m}$ diameter hole allows for the effective launching of the PhPs, whose propagation is visualized by s-SNOM (scattering-type scanning near-field optical microscopy). The experimental isofrequency curves, obtained by performing the Fourier Transforms (2D-FFT) of the near-field image, are shown in the insets. d–f Simulated near-field amplitude images of the system in a (d), b (e), and c (f). The 2D-FFTs of the simulated images are again shown in the insets. Both experimental and simulated images are aligned with the crystallographic axes of the top $\alpha$-MoO₃ layer.

Fig. 5. Analysis of the ray asymmetry in twisted asymmetric heterostructures.

a–e Simulated near-field amplitude images of a system made of a $100$-nm thin $\alpha$-MoO₃ layer over an isotropic substrate with permittivity ${\epsilon }_{sub}=-1$ (a), and over a 5-μm thick β-Ga₂O₃ substrate with twist angles $\theta ={28}^{\mathrm{o}}$ (b), ${57}^{\mathrm{o}}$ (c), ${86}^{\mathrm{o}}$ (d) and ${147}^{\mathrm{o}}$ (e). The illumination frequency is $\omega =734\mathrm{c}{\mathrm{m}}^{-1}$. f–j Simulated isofrequency curves of a–e obtained by performing the Fourier Transforms (2D-FFTs) of the simulated near-field images a–e, respectively. k–o. In-plane real permittivity values of $\alpha$-MoO₃ (red curve), an isotropic material with permittivity ${\epsilon }_{sub}=-1$ (green curve) and $\beta$-Ga₂O₃ (blue curve). The black straight lines represent the two angular directions of the $\alpha$-MoO₃ asymptotes which are defined by a zero-permittivity $\alpha$-MoO₃ value (red spots). The permittivity value of $-1$ for $\beta$-Ga₂O₃ is marked by a green spot. When the green and red spots are aligned, unidirectional ray-like propagation occurs along the corresponding asymptote. The black (green) dashed curve corresponds to the value $\epsilon =0$ ($\epsilon =-1$) and is present for visual guidance.