Visible to mid-infrared giant in-plane optical anisotropy in ternary van der Waals crystals

Birefringence is at the heart of photonic applications. Layered van der Waals materials inherently support considerable out-of-plane birefringence. However, funnelling light into their small nanoscale area parallel to its out-of-plane optical axis remains challenging. Thus far, the lack of large in-plane birefringence has been a major roadblock hindering their applications. Here, we introduce the presence of broadband, low-loss, giant birefringence in a biaxial van der Waals materials Ta2NiS5, spanning an ultrawide-band from visible to mid-infrared wavelengths of 0.3–16 μm. The in-plane birefringence Δn ≈ 2 and 0.5 in the visible and mid-infrared ranges is one of the highest among van der Waals materials known to date. Meanwhile, the real-space propagating waveguide modes in Ta2NiS5 show strong in-plane anisotropy with a long propagation length (>20 μm) in the mid-infrared range. Our work may promote next-generation broadband and ultracompact integrated photonics based on van der Waals materials.

By measuring the changes in the polarization state of light after reflection from the sample surface, the optical properties can be analyzed by spectroscopic ellipsometry.
The changes are represented by the ellipsometric parameters Psi (Ψ) and Delta (Δ), which are related to ratio of the pand s-polarization components as follows 13 : ( where rp and rs denote the Fresnel reflection coefficients of pand s-polarized light, respectively.
For the spectroscopic ellipsometry analysis, a four-phase model consisting of the Si/SiO2/Ta2NiS5/air ambient was established.The azimuthal angle-dependent spectroscopic ellipsometry spectra is shown in Supplementary Figure 3. Considering To unambiguously extract the dielectric functions of the Ta2NiS5 layer, we first employed the point-by-point method 14 , which is a mathematical inversion process that requires the pre-knowledge of the thickness of each layer.The ellipsometric equation of the four-phase structural model can be expressed as: The dielectric function of Si (εSi), SiO2 (εSiO2) are adopted from previous literature 15 .And the thickness of thermal SiO2 layer was pre tested by spectroscopic ellipsometry on the clean substrate and its value is 290 nm which is fixed in the following spectroscopic ellipsometry analysis.The thickness of the Ta2NiS5 nanosheet was determined by the atomic force microscope (AFM) measurement, which is consistent with the results (421.0±0.5 nm) of later Lorentz model fitting.Therefore, with two measured parameters ψ and Δ at each wavelength, the ε1 and ε2 of Ta2NiS5 can be directly calculated through point-by-point method.
For the validation of the Kramers-Kronig consistency of the mathematical inversion result, we also considered the Lorentz model: where E is the photon energy, and Ai, Ei, and Γi are the amplitude, center energy and damping coefficient of each oscillator.

Identify of the crystallographic features from spectroscopic ellipsometry data
For clarity of the crystallographic features from spectroscopic ellipsometry spectra,

Supplementary Figure 5a and 5b displays the azimuthal angle-dependent Psi and
Delta spectra with azimuthal angle from 0 degree (c-axis) to 90 (a-axis).Significant differences in the spectroscopic ellipsometry data between the two crystallographic orientations can be observed.Thereby, spectroscopic ellipsometry offers an unambiguous and non-destructive method to identify the crystallographic features of the Ta2NiS5. is the dielectric tensor of Ta2NiS5 sample, s  is the dielectric constant of substrate, d is the thickness of sample flake.
To fitting the experiment reflection spectrum, we used a genetic algorithm (GA) for the evaluation of parameters of Supplementary Equation (4).As a result, the inplane dielectric model parameters of Ta2NiS5 are shown below: Supplementary Table 2. In-plane dielectric model parameters of Ta2NiS5.
Then, the anisotropic in-plane permittivity of Ta2NiS5 can be described with the ) where   is the high frequency dielectric constant, the second term represents the free carriers contribution, Supplementary Figure 6 shows the experimental FTIR reflection spectrum of another Ta2NiS5 flake and the calculated reflection spectrum using above model parameters.The good agreement further verified the applicability the model parameters we have used.It is notable an uptick in the k (Figure 1e in main text) for aand c-axes in the wavelength longer than 10 microns occurs, which is possibly induced by the free carrier absorption rather than phonon.Since the absorption peaks of optical phonons are usually much sharper than the electronic absorption bands, this is not like the situation in our observation of the very broadband absorption in the 10-16 μm.Besides, the reported infrared spectra of Ta2NiS5 shows a phonon energy range below 50 meV (i.e., > 24.8 micron) 18 , which is much longer than our observed range.The free carrier is another possible origin account for this absorption.In our research, Drude-Lorentz model was applied to describe the optical response in the MIR region, and the fitted plasma frequencies from the Drude part are 1155.75and 1545.28 cm -1 for cand a-axis, respectively, which are close to the uptick of the MIR region.This gives us a clue to venture a guess that the absorption originates from free carriers.In the analysis of dielectric waveguides, we ignore the contribution of this interference, because the momentum of optical waveguides is small thus the reflectivity is low.
Moreover, additional intrinsic loss during the round-trip propagation (from the tip to the edge and back to the tip) further weakens the reflected waveguide propagating.
Therefore, we do not consider this mechanism in the analysis of dielectric waveguide fringes formation.Besides, in addition to the waveguide mode excited by the SNOM tip, the edge of the sample will also launch the waveguide mode under laser irradiation (P3).Because the size of the focused laser is very small, edge excitation is possible only when tip is very close to the sample edge.Meanwhile, the optical paths of edge excitation-tip scattering and tip excitation-edge scattering are same, so edge-launched modes generate the same fringes as the tip-launched modes.Considering the above factors, we also do not consider the influence of edge excitation in the analysis of interference mechanism.
According to Supplementary Figure 9b, based on the geometrical optics theory and momentum conservation relationship between incident light and dielectric waveguide modes, the relationship between near-field experiments observed sample surface wavevector qobs and the genuine wavevector qwm is derived in reference 19 (13)   where k0 is the free-space wavevector, α is the angle of the incident laser beam relative to the sample surface, β is the sample edge orientation angle.Here in 0 cos sin cos   Raman Shift (cm -1 ) A

10 Supplementary Figure 1 .e
Characterizations of the Ta2NiS5 crystals.a Optical microscope images of the exfoliated Ta2NiS5 flakes.b Crystal structures of Ta2NiS5.The red dash region represents the unit cell of Ta2NiS5.c Energy dispersive spectroscopy (EDS) results of an exfoliated Ta2NiS5 flake on CaF2 substrate.Insets show morphology and the component of the flake.d Raman spectra of Ta2NiS5 flakes.The angle-resolved polarized Raman spectra of Ta2NiS5.f The peak intensity ( 3 Ag mode) as a function of polarization angle.
and the horizontal plane of the sample.The sample is illuminated by a collimated and polarized beam containing pand s-components.

Supplementary Figure 3 .
the surface of our sample observed under optical microscopy, the roughness layer was not introduced during the analysis.Measured ellipsometry spectra with different azimuthal angle.The azimuthal angle-dependent Psi (a) and Delta (b) spectra with azimuthal angle from 0-350 degree by step of 10 degrees.

4 .Supplementary Figure 4 .
The extracted dielectric functions of the point-by-point method and Lorentz model are displayed in Supplementary Figure The fitting results of the two methods at different azimuth angles show very good agreement.The above results indicate that the spectroscopic ellipsometry is a precise and non-destructive method to identity the optical features of anisotropic vdW materials.Dielectric functions of Ta2NiS5 in visible range.The real a and imaginary b part of dielectric functions extracted by the point-by-point method (unfilled symbols) and Lorentz model (solid lines).

Supplementary Figure 5 . 22 (r
Identify of the crystallographic features from spectroscopic ellipsometry.The azimuthal angle-dependent a Psi and b Delta spectra with azimuthal angle in the range of 0-90 degree.The complex in-plane dielectric of Ta2NiS5 in MIR range was extracted by fitting the experimental reflection spectrum with the calculated ones.The reflection of a flake sample with an unpolarized incident light can be calculated by are the Fresnel's reflectivity coefficients of the flake sample for p-and s-polarized light respectively,12 , ps r and23 , ps r are the Fresnel's reflectivity coefficients of the first interface between air and sample flake and the second interface between sample flake and substrate for pand spolarized light respectively, , ps t is the Fresnel's transmissivity coefficient of the first interface for p/s-polarized light.For more details, where  is the angle of incident light, ,, a b c


are the plasma frequency and free carriers damping rate respectively, and the last term represents the Lorentzian oscillator contribution, are the strength, oscillation frequency and damping rate of the i-th Lorentzian oscillator respectively.

Supplementary Note 5 :Supplementary Figure 8 .Supplementary Note 6 :
Theoretical calculation of the optical properties of Ta2NiS5We performed density functional theory (DFT) calculations to further study the anisotropic optical properties of Ta2NiS5.As shown in Supplementary Figure8, the calculated optical properties including dielectric function, refractive and extinction match well with our experimental results.The shift of peaks relative to the experimental results is likely due to the inadequacy of Perdew-Burke-Ernzerhof (PBE) functional in describing the band gap and optical properties of semiconductors.Experimental and calculated optical properties of the Ta2NiS5.a, b Experimental complex dielectric function along different crystal axis in the visible wavelength.c, d Experimental refractive and extinction in the visible and mid-infrared (MIR) wavelength along different crystal axis.e, f Density functional theory (DFT) calculated complex dielectric function along different crystal axis in the visible wavelength.g, h DFT calculated refractive and extinction in the visible and MIR wavelength along different crystal axis.Discussion of fringe formation mechanismsAs discussed in the main text, the real space fringes on the sample surface observed in our s-SNOM measurements are formed by the interference between the photons scattered by the tip (path P1) and the edge scattering photons (path P2).In addition to mechanism discussed above and in the main text, Supplementary Figure9adepicted other possible interference mechanisms.In path P4, the tip-launched waveguide modes could be reflected backward to the tip after reaching the sample edge.The interference between path P1 and P4 plays an important role in the formation of surface plasmon polaritons (SPPs) and phonon polariton (PhPs) (eg.graphene SPPs and α-MoO3 PhPs).

Figure 16 Supplementary Figure 16 .
Figure16 shows the calculated propagation length of anisotropic waveguide mode in Ta2NiS5 flake on CaF2 substrate.

Supplementary Figure 17 .Supplementary Figure 18 .Supplementary Figure 19 .
Near-field characteristics of Ta2NiS5 flakes with various thicknesses on SiO2 substrate.a Near-field images of Ta2NiS5 flake with various thicknesses.The excitation wavelength λ = 633 nm.The Ta2NiS5 flakes are placed aaxis on the left and fringes are extracted along c-axis.Scale bar: 1 μm.b, c Real-space fringe profiles and the corresponding Fourier transform (FT) profiles of a. d, e Experimental dispersion data points and theoretical dispersion relations of the TM-and TE-polarized waveguide modes of a. Near-field characteristics of Ta2NiS5 flakes with various thicknesses on SiO2 substrate.a Near-field images of Ta2NiS5 flake with various thicknesses.The excitation wavelength λ = 4.545 μm.The Ta2NiS5 flake are placed with β = 225° and fringes are extracted along c-axis.Scale bar: 3 μm.b, c Real-space fringe profiles and the corresponding Fourier transform (FT) profiles of a. d, e Experimental and theoretical dispersion relations of the TM-and TE-polarized waveguide modes of a. Near-field characteristics of Ta2NiS5 flakes with various thicknesses on CaF2 substrate.a Near-field images of Ta2NiS5 flake with various thicknesses on CaF2 substrate.The excitation wavelength λ = 4.545μm.The Ta2NiS5 flake are placed with β = 225° and fringes are extracted along c-axis.Scale bar: 3 μm.b c Real-space fringe profiles and the corresponding Fourier transform (FT) profiles of a. d, e Experimental and theoretical dispersion relations of the TM-and TE-polarized waveguide modes of a.

Table 1 .
Comparison of the in-plane structural anisotropy in different categories of van der Waals (vdW) materials.