Probing subwavelength in-plane anisotropy with antenna-assisted infrared nano-spectroscopy

Infrared nano-spectroscopy based on scattering-type scanning near-field optical microscopy (s-SNOM) is commonly employed to probe the vibrational fingerprints of materials at the nanometer length scale. However, due to the elongated and axisymmetric tip shank, s-SNOM is less sensitive to the in-plane sample anisotropy in general. In this article, we report an easy-to-implement method to probe the in-plane dielectric responses of materials with the assistance of a metallic disk micro-antenna. As a proof-of-concept demonstration, we investigate here the in-plane phonon responses of two prototypical samples, i.e. in (100) sapphire and x-cut lithium niobate (LiNbO3). In particular, the sapphire in-plane vibrations between 350 cm−1 to 800 cm−1 that correspond to LO phonon modes along the crystal b- and c-axis are determined with a spatial resolution of < λ/10, without needing any fitting parameters. In LiNbO3, we identify the in-plane orientation of its optical axis via the phonon modes, demonstrating that our method can be applied without prior knowledge of the crystal orientation. Our method can be elegantly adapted to retrieve the in-plane anisotropic response of a broad range of materials, i.e. subwavelength microcrystals, van-der-Waals materials, or topological insulators.

, where 1 and 2 are the indices of refraction of environment and sample. In our case 1 = 1 represents the air and 2 represents the sapphire. 1 and 2 are the incident and transmission angles, which follow Snell's law: 1 sin( 1 ) = 2 sin ( 2 ). In the ⊥ case, 2 = √ . In the // case, 2 ≈ √ eff and eff = √ .
To investigate the -dependence of Equivalently, the difference observed in sap ( , ) for different values is mainly from the far-field effect. Consequently, we postulate that if the experiment were done on a sapphire microcrystal, sap ( , ) would not vary with significantly.

Supplementary Note 2: Effective disk polarizability and behavior
As shown in the main text, the effective polarizability of the gold disk is approximately . Here we investigate the behavior of 1 = | 1 | around the epsilon-near-zero (ENZ) region closely. The amplitude of 1 can be expressed as where 1 = ( in−plane ) and 2 = ( in−plane ). While the phase is given by ).
In Supplementary Fig. 1(a), we show the | 1 | as a function of 1 and 2 . The global maximum occurs when 1 = 1 and 2 = 0. That is, the peaks in the bright dark spectra show up right next to the ENZ point as shown in Fig. 3(b), around 1 = 1. When 1 = 0, the amplitude of 1 is constantly 1, which means the "bright" and "dark" sides changes place as 1 switches sign. In Supplementary Fig. 1(b) the phase of 1 is plotted. It's clear that non-zero phase occurs only around the ENZ region and peak exactly at 1 = 2 = 0, which explains why the phase spectrum bright − dark in Fig The result for = 1, 2, 3, 4 is shown in Supplementary Fig. 2(b). Compared to the full 3D case 3 , the field patterns are the same except for some minor details. Thus, we deem the difference between 2D and 3D simulations is of secondary importance.
It has been demonstrated that the electric field near the tip apex is a quantitative gauge for the near-field interaction 3 . In the previous study, the volume-averaged field is considered. For simplicity, we show that the electric field taken at a single point (a few nm above the sample surface right under the tip apex) is already sufficient to capture the experimental observations. We verify this statement, together with the validity of our 2D simulations, by simulating the near-field spectrum of a SiO 2 sample. As shown in Next, we demonstrate through simulations that our proposed technique can be applied to measure the in-plane dielectric anisotropy of micro-crystals smaller than the wavelength. Assume the scenario where a 3 µm long antenna is fabricated on a sapphire micro-crystal of size 10 µm and thickness of 1 µm. The sapphire micro-crystal is placed on a substrate with permittivity = 2 ( Supplementary Fig. 3(a)). In Supplementary Fig.  3(b) we show the simulation results for two crystal orientations E⊥c and E//c. Compared to results shown in Fig. 4(e) and (f), although the peaks are broadened and the contrast is reduced here, the main spectral features due to the in-plane anisotropic phonon response remain observable. This broadening is likely due to the influence of the substrate and the finite size of the micro-crystal. We also would like to comment on the effect of the Au disk antenna size. In the 1/ analysis above we demonstrate that the contrast of bright to dark flips at = 0 in the sample. In FEM simulations we monitor In Supplementary Fig. 4 (a) and 4(b), we show the sapphire film thickness-dependence of the bright dark signal contrast. For both crystal orientations, the peaks at LO phonon frequencies are damped when the film thickness decreases.The spectra for // cases show relatively larger fluctuations than the E⊥c cases and the peak height at the LO phonon resonance is more sensitive to the decrease of film thickness. It can be attributed to the effect of phonon polariton waveguide mode for E//c cases 8,9 . Supplementary Fig.  4(c) shows the simulated electric field distribution of the hyperbolic phonon polariton mode based on finite element modeling (FEM) simulation. Supplementary Fig. 4(d) shows the imaginary part of p-polarized Fresnel coefficient r p , indicating the dispersion relation of the phonon polariton in the sapphire-substrate system. The propagating waveguide mode interferes with the plasmonic response of the gold disk and our method is not appropriate for this situation. However, in this case, the optical anisotropy can be quantitatively extracted from the ordinary and extraordinary waveguide modes in thin film, which was realized in the previous study 10 . Our method, together with the waveguide modes analysis, establishes a general methodology for the optical anisotropy measurement on samples of great variety. Next, we investigate the signal dependence on gold disk diameter. The overall value of the bright dark spectra increases monotonically with increasing disk diameter, indicates the positive correlation between the disk diameter and the effective polarizability of the system. The enhancement of plasmonic resonance can be utilized to magnify the resonance from the sample. In Supplementary Fig. 5 (a) and (b) we plot the simulated disk-diameter dependent bright dark spectra on bulk sapphire and on 300 nm sapphire film.

Supplementary
The amplitude of the peak and signal contrast for 4.5 μm disk diameter is magnified compared with the one for 0.9 μm disk diameter.