As a biaxial vdW material with a low crystalline structural symmetry, it has multiple RBs ranging from the middle-infrared to far-infrared spectral region20, 34, 35, as shown in Fig. 1a and b. Notable, there are additional spectral regimes where theses RBs overlap, resulting in fascinating polaritonic properties. Specifically, the overlapped spectral region between the RB Ⅰ (545 cm− 1~851 cm− 1) and RB Ⅱ (820 cm− 1~972 cm− 1) refers to two resonance peaks of optical phonons: the transverse optical phonon (TO) along [100] axis (x-axis) at \({\omega }_{\text{T}\text{O}}^{\text{x}}\) = 820 cm−1, and the longitudinal optical phonon (LO) along [001] axis (y-axis) at \({\omega }_{\text{L}\text{O}}^{\text{y}}\) = 851 cm− 1. Importantly, both the real part of \({\epsilon }_{\text{x}}\) and \({\epsilon }_{\text{y}}\) are negative in this peculiar spectral region (820 cm−1~851 cm− 1), while \({\epsilon }_{\text{x}}/{\epsilon }_{\text{y}}\) is calculated up to be several thousands, indicating a remarkably large optical anisotropy. Additionally, it has been recently reported that the theoretical discovery of canalization effects under resonant conditions6, 36. Therefore, it is anticipated that there is the possibility of PhPs canalization in a single-layer α-MoO3 flake. Nevertheless, such proximity near the TO along the x-axis (\({\omega }_{\text{T}\text{O}}^{\text{x}}\)) has finite influence on the propagation loss of canalized PhPs in the α-MoO3 flake, which we will prove both theoretically and experimentally. So far, the observation of canalized PhPs in a single-layer α-MoO3 flake under this intersected spectral region remains to be addressed, technically due to unavailable continuous-wave excitation light sources for the monochromatic scattering-type scanning near-field optical microscopy (s-SNOM) 20, 37, impeding the fundamental understanding and practical exploitation of polariton canalization modes in the single-layer α-MoO3 flake, a natural vdW crystal.
In our experiment, a 160-nm-thick α-MoO3 flake was transferred on the SiO2/Si substrate by an all-dry transfer method (see Methods). A global cartesian coordinate is defined that the x- and y-axis are along the [100] and [001] crystal directions of the α-MoO3 flake, respectively. To characterize the crystalline quality of the α-MoO3 flake. Raman spectrum was measured to show classical optical phonon resonance frequencies within different RBs of the α-MoO3 crystal (Fig. 1a), all indicating excellent crystalline conditions. As shown in Fig. 1b, the in-plane dielectric permittivities of the α-MoO3 crystal were calculated by multiple Lorentzian oscillator models (see Section 1 in Supplementary Information, SI). The gray shaded region in Fig. 1b shows the overlapped spectral region between the RB I and RB II in the α-MoO3 crystal (820 cm− 1~851 cm− 1), where the real parts of \({\epsilon }_{\text{x}}\) and \({\epsilon }_{\text{y}}\) are both negative. Particularly, the wave-vector ratio of PhPs along the y-axis (\({k}_{\text{y}}\)) and x-axis (\({k}_{\text{x}}\)) is quite large (see Section 2 in SI), suggesting the high axial ratio of the corresponding IFCs (thus being close to flat-band) and hence the possibility of highly directional propagation of PhPs, i.e. canalization modes.
To corroborate the above statement, numerical simulations were carried out based on the finite-element method (FEM). A z-oriented dipole was used as the extreme anisotropic PhPs launcher, and the distance between the dipole and the top surface of the α-MoO3 flake was 200 nm. We recorded the z-component of the electric fields (\({E}_{z}\)) at a distance 10 nm above the top surface of the α-MoO3 flake. Figure 1c and d show the simulation results with the frequency (\({\omega }_{\text{i}\text{n}\text{c}}\)) of the incident laser at 823 cm− 1 and 840 cm− 1, respectively. As shown in Fig. 1c, the energy flow of PhPs with \({\omega }_{\text{i}\text{n}\text{c}}\) at 823 cm− 1 is highly collimated along the x-axis, while extremely compressed in the y-axis. As the \({\omega }_{\text{i}\text{n}\text{c}}\) is increased to 840 cm− 1, the PhPs show hyperbolic-like propagation behaviors. To understand their dispersions features in momentum space, we performed the fast Fourier transform of the Re(\({E}_{z})\) fields and calculated the corresponding analytical dispersion relationships (see Section 2 in SI) to extract the corresponding IFCs, which are shown in Fig. 1g and h. Surprisingly, both IFCs with \({\omega }_{\text{i}\text{n}\text{c}}\) at 823 cm− 1 and 840 cm− 1\(\)show an extreme open hyperbolic-like curve, despite that \({\epsilon }_{\text{x}}<0\) and \({\epsilon }_{\text{y}}<0\) in this intersected spectral regime. This is different from the conditions for conventional hyperbolic IFCs that \({\epsilon }_{\text{x}}\) and \({\epsilon }_{\text{y}}\) should be in the opposite sign.
To understand this difference on the open hyperbolic-like IFCs, we calculated the polaritonic dispersion at RB I and RB II of α-MoO3 flake. Specifically, as \({\omega }_{\text{i}\text{n}\text{c}}\), increases, the real part of both \({k}_{\text{x}}\) and \({k}_{\text{y}}\) is increased accordingly. Due to the large real part value of \({k}_{\text{y}}\) (~λ0/30) at 823 cm− 1, the imaginary part of in-plane \(k\) (the loss component of PhPs) is great along y-axis, prohibiting its propagation along the above direction. By contrast, \({k}_{\text{x}}\) is relatively small with \({\omega }_{\text{i}\text{n}\text{c}}\) at 840 cm− 1, and the propagation loss of PhPs shows a decrease along x-axis since \({\omega }_{\text{i}\text{n}\text{c}}\) is far away from the TO phonon resonant frequency along x-axis (\({\omega }_{\text{T}\text{O}}^{\text{x}}\)=820 cm− 1). As a result, if we extend the range of the \({k}_{\text{x}}\) and \({k}_{\text{y}}\) to a wide momentum range, the corresponding IFCs derived from analytical dispersion relationships will be closed, as shown in the subfigures of Fig. 1g and h. However, within the accessible wave-vector range in the near-field experiment for exciting PhPs, the polaritonic dispersion still exhibits open hyperbolic-like curves, which is the major reason of hyperbolic-like PhPs in this spectral region.
Another nonnegligible discussion is on how the imaginary part of the permittivity along x-axis affects the propagation behavior of PhPs, since resonant material damping dominate at the TO phonon resonant frequency. It is widely recognized that materials become lossy near the resonance frequency of the Lorentzian model, but this does not necessarily hold true in our specific polaritonic systems. To clearly demonstrate this, we utilized the analytical dispersion relationships to calculate the in-plane wave-vector \({k}_{{\rho }}=\sqrt{{k}_{\text{x}}^{2}+{k}_{\text{y}}^{2}}\) and decay distance (1/e power decay away from the edge of bulk α-MoO3) for various frequencies and azimuth angles \(\theta =\text{atan}({k}_{\text{y}}/{k}_{\text{x}})\), as shown in Fig. 1e and f. The solid and dashed lines in Fig. 1e correspond to the real and imaginary parts of\({k}_{{\rho }}\), respectively. It is observed that despite the concerned frequency range near the resonance frequency at 820 cm−1, the imaginary component of \({k}_{{\rho }}\) of α-MoO3 remains low, leading to a considerable large propagation decay distance, particularly in the x-direction, which also can be found by the colored circles in the Fig. 1g and h. Furthermore, theoretical analysis indicates that as \(\left|{\epsilon }_{\text{x}}\right|\) approaches infinite, the imaginary component of \({k}_{\text{x}}\), which determines the propagation distance at the x-direction, will approach zero, regardless of whether real or imaginary part of \({\epsilon }_{\text{x}}\) (see Section 2 in SI for details).
To experimentally verify the striking hyperbolic-like and canalization modes with low-loss in a single-layer α-MoO3 flake, we performed the photo-induced force microscopy (PiFM, Molecular Vista) 38–41 for real-space nanoimaging PhPs. The PiFM instrument is based on the atomic force microscopy (AFM). It can simultaneously yield the nanoscale resolved topography, and gradient of the near-field optical force generated at the tip-sample region (Fig. 2a). During the near-field experiment, an oscillating Au-coated tip was illuminated by a pulsed p-polarized infrared laser (quantum cascade laser source from 758 cm− 1~1850 cm− 1, Block Engineering). The tip acts as an infrared antenna to launch PhPs. Then, PhPs propagate outwards, and are reflected by the sample edge towards the tip, giving rise to the PiFM interference fringes with a spacing distance of \({\lambda }_{p}/2\) (where \({\lambda }_{p}\) is the wavelength of PhPs). Figure 2b and c present typical PiFM images of the α-MoO3 flake at \({\omega }_{\text{i}\text{n}\text{c}}\) = 780 cm−1 and 880 cm− 1, both of which are inside hyperbolic IFCs regions. However, the propagation directions of these PhPs modes are different at those frequencies, since the fringes are parallel to the x direction with \({\omega }_{\text{i}\text{n}\text{c}}\) at 780 cm− 1 (Fig. 2b), and to the y direction at \({\omega }_{\text{i}\text{n}\text{c}}\)= 880 cm−1 (Fig. 2c). This is because of different open directions of the PhPs hyperbolic dispersion at different frequencies. In particular, it indicated that the topological transition of PhPs from open hyperbolic curves along the y-direction to that along x-direction was realized by simply sweeping the \({\omega }_{\text{i}\text{n}\text{c}}\) over the overlapped spectral region between the RB I and RB II. For a quantitative analysis of the anisotropic PhPs propagation, we plotted the PhP dispersions, \(\omega \left({k}_{\text{i}}\right) (i =x, y)\), from the measured PiFM images of the same α-MoO3 flake (See Fig. S5, and Fig. S6 in SI). The period of fringes in PiFM images were extracted to calculate \({k}_{\text{i}} (i=x, y)\) of PhPs. The experimental results agree well with the theoretical predictions shown as a 2D pseudo colored plot of the complex reflectivity in Fig. 2d (See Section 2 in SI).
By scanning \({\omega }_{\text{i}\text{n}\text{c}}\) from 830 cm− 1 to 870 cm− 1, the wavefront evolution of PhPs can be visualized in the PiFM images. Figure 3a clearly display hyperbolic PhPs at \({\omega }_{\text{i}\text{n}\text{c}}\) = 870 cm−1. As the exciting frequency sweeping from 870 cm− 1 to 830 cm− 1, the measured PiFM images appear to be the open hyperbolic propagation of the wavefront of PhPs around a nanostructure (marked by a gray semicircle icon). It is actually ascribed to the two-lobed curve of IFCs with an increased axial ratio, which is in a good agreement with the theoretical predictions in Fig. 1g and h. Interestingly, at \({\omega }_{\text{i}\text{n}\text{c}}\) \(\)= 830 cm− 1, the PhPs propagate towards the x-axis, which is highly directional and diffraction-free, manifesting a feature of the PhPs canalization mode. Such experimentally observed evolution of the wavefront of PhPs agrees well with the numerical simulations in Fig. 3b. Moreover, we extracted the corresponding IFCs in momentum space by performing fast Fourier transform (FFT) of the experimental PiFM images, as plotted in Fig. 3c. This shows the evolution of the topology of IFCs from the open hyperbola to closed shapes via decreasing \({\omega }_{\text{i}\text{n}\text{c}}\) below 851 cm− 1 (\({\omega }_{\text{L}\text{O}}^{\text{y}}\)), which is remarkable in a single-layer α-MoO3 flake. To further improve the performance of the canalized PhPs effect, ultra-low-loss PhPs in the α-MoO3 crystal could be realized through the isotope enrichment method42.
To further demonstrate the diffraction-less propagation nature of PhPs canalization, PiFM measurement was performed around nanohole arrays in the α-MoO3 flake. In Fig. 4a, it shows the AFM image of nanohole arrays in the α-MoO3 flake. The diameter and periodicity of nanohole arrays are 300 nm and 600 nm, respectively. At \({\omega }_{\text{i}\text{n}\text{c}}\) = 828 cm−1, the measured PiFM image presents the highly directional polaritonic flow along the x-direction as shown in Fig. 4b. It reveals that the canalized PhPs propagated unidirectionally and was sub-diffractive confined along the y-direction, which is perpendicular to the propagation direction. In Fig. 4c, it indicates that the numerical simulation of Re(\({E}_{z}\)) at the top surface of the nanohole array in the α-MoO3 flake is in good accordance with the experimental data (Fig. 4b), featuring the high-performance canalized PhPs modes. To quantitatively analyze the canalized propagation property of PhPs, we extracted profiles of the AFM topography and PiFM image by drawing cutting lines, denoted as white dashed lines in Fig. 4a and b, respectively. These two parallel lines are along the y-direction with a separation distance of about 700 nm. In Fig. 4d, the upper and bottom row represent topography data (denoted as the black line) and PiFM amplitudes (marked as the red line), respectively. Notably, the line profile of topography data shows a gentle boundary (Fig. 4d, upper) owing to the topography convolution induced by the nanoscale geometry between the conic AFM tip and the cross-section shape of nanoholes. Nevertheless, the PiFM line profile could exactly reproduce the trend in topography data even at a separation distance of 700 nm. Therefore, it is reasonable to conclude that the canalization angle of PhPs is near zero, and the full-width-at-half-maximum (FWHM) of PhPs canalization modes is estimated around 150 nm, which is ~ λ0/80 (where \({\lambda }_{0}\) is the free-space wavelength). Moreover, the FWHM of PhPs canalization modes may be further reduced if the thickness of the α-MoO3 flake is thinner43, 44.