Polaritonic Fourier crystal

Abstract

Polaritonic crystals - periodic structures where the hybrid light-matter waves called polaritons can form Bloch states - promise a deeply subdiffractional nanolight manipulation and enhanced light-matter interaction. In particular, polaritons in van der Waals materials boast extreme field confinement and long lifetimes allowing for the exploitation of wave phenomena at the nanoscale. However, in conventionally patterned nanostructures, polaritons are prone to severe scattering loss at the sharp material edges, making it challenging to create functional polaritonic crystals. Here, we introduce a concept of a polaritonic Fourier crystal based on a harmonic modulation of the polariton momentum in a pristine polaritonic waveguide with minimal scattering. We employ hexagonal boron nitride (hBN) and near-field imaging to reveal a neat and well-defined band structure of phonon-polaritons in the Fourier crystal, stemming from the dominant excitation of the first-order Bloch mode. Furthermore, we show that the fundamental Bloch mode possesses a polaritonic bandgap even in the relatively lossy naturally abundant hBN. Thus, our work provides an alternative paradigm for polaritonic crystals essential for enhanced light-matter interaction, dispersion engineering, and nanolight guiding.

Fig. 1. Polaritonic Fourier crystal.

a Illustration of a 1D Fourier crystal based on a harmonically corrugated and gilded Fourier surface supporting a pristine polaritonic material. The inset shows the profile of a bi-harmonic Fourier surface used in the drawing. b Schematic of an hBN-based single-harmonic 1D Fourier crystal. When the hBN slab is placed on the gilded Fourier surface, the varying distance between hBN and gold induces the spatial modulation of the HPhP momentum in hBN. c Modes of different order experience a significantly different modulation depth in the Fourier crystal. Thus, the manifestation of the polaritonic bandgap is predicted only for the fundamental Bloch mode.

Fig. 2. Fabricated Fourier crystal with hBN.

a Optical microscope image of a large hBN flake on the gold-covered 1D Fourier surface with visible corrugation period of 485 nm. Nanotip schematically shows the near-field scan areas far from the hBN edges. b AFM height profile across the hBN edge, showing the well-preserved sinusoidal profile of the gold surface and flat hBN crystal. c Near-field amplitude (top) and phase (bottom) images measured by s-SNOM above the sample far from the hBN edges. Both near-field amplitude and phase show strong dependence on the excitation frequency, revealing different propagation regimes of polaritons in the Fourier crystal.

Fig. 3. Spectral map of the near-field distribution.

a Experimentally measured profiles of the near-field amplitude (left) and phase (right) in the Fourier crystal at different excitation frequencies. Every near-field scan is taken across the area of 4 × 2 μm², followed by integration to obtain a linear profile, which is then normalized. b Numerically calculated near-field amplitude (left) and phase (right) over the Fourier crystal with the period P = 485 nm and the surface modulation depth 70 nm; x = 0 corresponds to the maximal gap between the hBN and gold surface. Black dashed lines are guides for the eye indicating the corresponding near-field features and showing a blue shift of the calculated data (also highlighted by the red dashed line). c Numerically calculated band diagram of the HPhP modes in the Fourier crystal along the direction of surface corrugation. Dashed and dot-dashed lines indicate a correspondence between the near-field and photonic bands.

Fig. 4. Analysis of M0 and M1 modes.

a Numerically calculated band structure of HPhP in the Fourier crystal across the upper Reststrahlen band of hBN. The same model is used as in Fig. 3c. b Independently calculated band structures of M0 (left) and M1 (right) modes, where the Fourier crystal with hBN is modeled as a 2D sheet of spatially modulated conductivity according to the analytical dispersion of modes in hBN over a conductive surface. Band structures in (a and b) are calculated for the direction of surface corrugation. c Numerically calculated near-field amplitude profiles independently generated by M0 (left) and M1 (right) modes. Here, the hBN slab is modeled as a 2D sheet of uniform conductivity which supports only one desired eigenmode, placed over a metallic Fourier surface. d Same as in (c), but for the near-field phase. Both (c and d) demonstrate that experimentally and numerically obtained near-field patterns correspond to the M0 mode.

Fig. 5. HPhP probing near the hBN edge.

a Optical microscope image of the hBN edge in the Fourier crystal, with a coordinate system used for the near-field analysis: the edge is parallel to the x-axis, and the y-axis is at 32° angle to the lattice vector G. The hBN edge launches the polariton mode in y-direction (yellow wavy arrow) which produces interference fringes in the near-field images, while the s-SNOM nanotip excites the Bloch modes in the Fourier crystal (red wavy arrow). b Near-field images of the hBN edge shown in (a), measured at the excitation frequencies around the expected upper limit of the M0 bandgap in the direction of G. Dashed lines indicate the maxima of the underlying gold surface in contact with hBN. c Corresponding Fourier spectra of the near-field images in (b). Yellow arrows mark the spectral signature of the near-field interference due to the edge-launched HPhP; the white dashed circle indicates k = G. d Numerically calculated band structure of the HPhP propagating in y-direction (yellow wave in a). Yellow circles indicate the experimentally measured momentum of the edge-launched mode: 1.0, 1.25, and 1.65 μm^–1 at 1420, 1440, and 1460 cm^–1, respectively. Red tick marks indicate the corresponding experimental frequencies and the red dashed lines show the expected M0 bandgap in the direction of G from 1410 to 1470 cm^–1. Red dot-dashed lines show the analytically calculated M0 dispersion in G-direction (also shown in Fig. 4b).