 Research Article
 Open Access
 Published:
Sourceconfigured symmetrybroken hyperbolic polaritons
eLight volume 3, Article number: 14 (2023)
Abstract
Polaritons are quasiparticles that combine light with matter, enabling precise control of light at deep subwavelength scales. The excitation and propagation of polaritons are closely linked to the structural symmetries of the host materials, resulting in symmetrical polariton propagation in highsymmetry materials. However, in lowsymmetry crystals, symmetrybroken polaritons exist, exhibiting enhanced directionality of polariton propagation for nanoscale light manipulation and steering. Here, we theoretically propose and experimentally demonstrate the existence of symmetrybroken polaritons, with hyperbolic dispersion, in a highsymmetry crystal. We show that an optical diskantenna positioned on the crystal surface can act as an inplane polarized excitation source, enabling dynamic tailoring of the asymmetry of hyperbolic polariton propagation in the highsymmetry crystal over a broad frequency range. Additionally, we provide an intuitive analysis model that predicts the condition under which the asymmetric polaritonic behavior is maximized, which is corroborated by our simulations and experiments. Our results demonstrate that the directionality of polariton propagation can be conveniently configured, independent of the structure symmetry of crystals, providing a tuning knob for the polaritonic response and inplane anisotropy in nanophotonic applications.
1 Introduction
Exploiting ultraconfined and highly directional polaritons [1,2,3,4,5,6,7,8,9,10] at the nanoscale is essential for developing integrated nanophotonic devices, circuits and chips [11,12,13,14,15]. Highsymmetry crystals have been extensively studied for this purpose, with particular focus on hyperbolic polaritons (HPs), including outofplane hyperbolicity [16,17,18] in hexagonal crystals (e.g., boron nitride) [5, 6], and inplane lowloss HPs in trigonal (e.g., calcite) [7] and orthorhombic (e.g., αMoO_{3}) [8,9,10] crystals. However, the inplane HP propagation in highsymmetry optical crystals usually exhibits four mirrorsymmetric beams, which reduces the directionality and energy transporting efficiency. Recently, hyperbolic shear polaritons [19,20,21,22], characterized with mirrorsymmetrybroken hyperbolic wavefronts, have been discovered in lowsymmetry monoclinic crystals, which exhibit enhanced directional propagation despite suffering from large losses. The nontrivial asymmetries of these shear polaritons arise from the intrinsic nonHermitian permittivity tensor of the lowsymmetry crystals [19,20,21,22], which thus are not available in highsymmetry crystals.
Here, we investigate the impact of inplane linearly polarized sources [23,24,25,26,27,28,29] on generating symmetrybroken HPs with enhanced directional propagation in highsymmetry, lowloss systems. We theoretically and experimentally demonstrate that controlling the nearfield excitation source can configure the excitation and propagation of inplane HPs. This leads to the breaking of polariton mirror symmetry without the need of low crystalline symmetry. Our sourceconfigured approach enables the tuning of the polariton asymmetry propagation over a broad frequency range, thereby establishing in a new degree of freedom [27,28,29] for the dynamic and robust control of light guiding [30,31,32] and propagation on the nanoscale. Our results expand the possibilities for manipulating asymmetric polaritons and can be applied to reconfigurable polaritonic devices [33,34,35,36,37,38] for polarizationdependent nanophotonic circuits [37] or optical isolation [38].
2 Results and discussion
To distinguish mirrorasymmetric HP behaviors in low and highsymmetry crystals, we conducted numerical simulations of dipolelaunched polaritons as shown in Fig. 1. First, we compared the HPs launched by a vertical dipole (p_{x} = p_{y} = 0, p_{z} ≠ 0) on monoclinic (Fig. 1a) and orthorhombic (Fig. 1b) crystals, where offdiagonal terms ε_{xy} associated with shear phenomena were present and absent in their permittivity tensors, respectively.
Our results showed highly asymmetric hyperbolic shear polaritons in the lowsymmetry monoclinic crystal, as evidenced by the atypical tilted hyperbolic wavefronts with the mirror symmetry being broken along the realspace crystal axis (perpendicular black lines) (Fig. 1d). The corresponding Fourier transform also indicated a mirrorasymmetric wavevector distribution with respect to axes O1 and O2 (Fig. 1g). In contrast, when the permittivity tensor was diagonalized in the orthorhombic crystal, the wavefronts in real space became mirrorsymmetric with respect to the crystal axis (Fig. 1e), and the kspace dispersion also exhibited symmetric hyperbola with respect to both the O1 and O2 axes (Fig. 1h). Note that, the tilted crystal axis (or effective optic axes) were corrected for the orthorhombic crystal and were found to be along the principal orthogonal x and y (or k_{x} and k_{y}) directions.
We further considered a dipole exciting HPs on the orthorhombic crystal, with the dipole orientation changed to the inplane direction (p_{x} ≠ 0, p_{y} ≠ 0, p_{z} = 0) as shown in Fig. 1c. Compared with the results shown in Fig. 1e, the HPs now mostly spread in the first or third quadrant with respect to the coordinate system of the orthorhombic plane, resulting in a clear broken mirror symmetry of wavefronts in the highsymmetry crystal (Fig. 1f). However, in this case the wavefronts were not rotated, which is different from hyperbolic shear polaritons. Anomalously, the propagation of HPs was weakly emitted and even forbidden along the dipolar orientation (brown double arrow). This behavior contrasted with the excitation of polaritons on the inplane isotropic polar crystals such as SiC (Additional file 1: Fig. S5). These observations inspired a new proposal of realizing symmetrybroken HP propagation. Instead of reducing the lattice symmetry or engineering the energy band of materials, symmetrybroken HPs can also be realized by configuring the polarization of the excitation source, as confirmed via the Fourier transform of the realspace electric field E_{z} in Fig. 1i.
To shed light on the mechanism of asymmetric hyperbolic polaritons caused by an excitation source, we used the theory of nearfield interference [28], which describes the angular spectrum of dipole sources while considering momentum conservation. Because we experimentally excited HPs with a transverse magnetic (TM) wave, we only considered a TMpolarized component wherein the dipole moment was defined per unit length as p = [p_{x}, p_{y}, p_{z}]. Furthermore, the dipole’s electric field was divided into spatialfrequency components k_{x} and k_{y} in an isotropic medium with permittivity ε_{1} and permeability μ_{1} via a dyadic Green function (cf. Additional file 1: Note S1):
where \({k}_{t}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\) and \({k}_{z}=\sqrt{{\varepsilon }_{1}{\mu}_{1}{k}_{0}^{2}{k}_{t}^{2}}\) are the inplane and outofplane wavevectors, respectively, and \({k}_{0}=\omega \sqrt{{\mu }_{0}{\varepsilon }_{0}}\) is the wavevector in free space.
The electric field intensity distribution, denoted as \({E}_{z}^{p}\), can be mapped onto the k_{x}k_{y} momentum space using Eq. (1), given the polarization of the dipole p in free space. Panel 1 in Fig. 2a shows the contour distribution of a linearlypolarized electric dipole with p = [cosφ, sinφ, 0], where φ represents the polarization angle of dipole polarization p relative to the yaxis in a Cartesian coordinate system. Note that the direction of minimum contour \({{E}_{z}^{p}}_{\mathrm{min}}\) (indicated by the yellow dashed line) is perpendicular to both realspace dipolar orientation (red arrow) and the direction of \({{E}_{z}^{p}}_{\mathrm{max}}\) (red solid line), leading to several intriguing consequences, as described below. Subsequently, we investigate the unique dispersion of anisotropic hyperbolic systems in Panel 2. In this scenario, the sourcefree Maxwell equations in the anisotropic system with an arbitrary dielectric tensor \(\widetilde{\varepsilon }\) must be solved, considering the polariton wave carrying a fixed frequency of wavevector (k_{x}, k_{y}, k_{z}). The hyperbolic dispersion can be derived from the determinant of secular equation as shown in Additional file 1: Note S2:
It should be note that the hyperbolic isofrequency contour (IFC) in Eq. (2) is solely determined by the dielectric constant \(\widetilde{\varepsilon }\), which reflects the crystal properties. Additionally, the dipolar orientation has no effect on the open angle θ of the IFC (θ denotes the angle between the asymptote of the hyperbolic IFC and k_{y} axis in momentum space). Finally, we explore the influence of the \({E}_{z}^{p}\) distribution on the hyperbolic IFC, as demonstrated in Panel 3. Coupling with the asymmetric \({E}_{z}^{p}\) induced by the horizontal dipole, certain propagation modes in the hyperbolic dispersion can be selectively excited (indicated by the red solid line in Panel 3) or not excited (indicated by the red dotted line in Panel 3). Consequently, our sourceconfigured method can support symmetrybroken HP propagation at any anisotropic material systems in theory (Additional file 1: Fig. S6).
To demonstrate the propagation of polariton with broken mirror symmetry, we intentionally selected two points, A and B (shown in Panel 3 of Fig. 2a), which were located close to the direction of open angles, to calculate the variation of \({\Delta E}_{z}\) (i.e. \(\left{E}_{z}^{\mathrm{A}}{E}_{z}^{\mathrm{B}}\right\)) based on polarization angle φ and open angle θ. As shown in Fig. 2b, for any θ, the difference \({\Delta E}_{z}\) only became zero when the inplane dipolar orientation was aligned with the crystal axis of highsymmetry crystals, satisfying the conditions φ = 0° or ± 90° in our sample (indicated by green dashed lines). This led to symmetrically hyperbolic polariton behaviors. Otherwise, different degrees of mirror symmetry breaking occurred (i.e. \({\Delta E}_{z}\ne 0\)), where mirror symmetry breaking refered to the absence of any mirror symmetry axis in either the realspace wavefronts or the kspace wavevector. Specifically, when the minimum contour \({{E}_{z}^{p}}_{\mathrm{min}}\) coincides with the direction of open angle, that is φ = θ ± 90°, the difference \({\Delta E}_{z}\) reached its maximum. We defined this maximum difference \({{\Delta E}_{z}}_{\mathrm{max}}\) for the “symmetrybroken” state, as illustrated by black dashed lines in Fig. 2b (More details in Additional file 1: Fig. S2).
Using our definition, we simulated the distributions of E_{z}, Re(E_{z}) and fast Fourier transform FFT[Re(E_{z})] in one of the symmetrybroken cases at φ = 90° − θ = 55° (where θ is 35° at 1450 cm^{−1} marked by the yellow triangle patch) in calcite [7]. As expected, both field distributions of E_{z} and Re(E_{z}) exhibited symmetrybroken propagation in Fig. 2c, d. The Fourier analysis in Fig. 2e coincided with our previous theoretical IFC in Panel 3, where the contour of \({{E}_{z}^{p}}_{\mathrm{min}}\) (yellow dashed line) overlapped with the direction of open angle θ. Interestingly, the polariton propagation along the direction of dipolar polarization p was lost in Fig. 2c, d, which cannot emerge in an inplane isotropic system. This unusual feature can be attributed to the hyperbolic anomalous dispersion (Additional file 1: Fig. S5), causing a mismatch between the directions of energy flow and wavevector [16, 17]. The demonstrated asymmetry contrast of polariton in Fig. 2c could be further enhanced theoretically. The maximum symmetrybroken behavior occurred at θ = 45° (i.e. \({{E}_{z}^{p}}_{\mathrm{min}}\) and \({{E}_{z}^{p}}_{\mathrm{max}}\) overlap with the directions of both θ and –θ, respectively) when φ was ± 45°, as marked by the white circle in Fig. 2b.
We initiated the study by simulating the most asymmetric HPs in calcite at 1470 cm^{−1}, with an open angle 45° denoted by θ. Considering the even symmetry of \(\left{\Delta E}_{z}\left(\varphi ,\theta \right)\right\) in Fig. 2b, we analyzed the change in the symmetry of HPs with varying polarization angle φ in the range of 0°–90°. Figure 2f provides a visual representation of the impact of dipolar orientation on the symmetry of polaritonic rays. As previously analyzed, the mirror asymmetry of polaritonic rays appears when the the dipolar orientation is not aligned with the directions of crystal axis in calcite. To determine the defined symmetrybroken state, the degree of polariton asymmetry must be quantified with respect to polarization angle φ. The amplitude profiles E_{z} along the horizontal dashed white lines in Fig. 2f were analyzed, and the distinct growth and extinction trends of electric field intensities in Peak I and Peak II were observed by analyzing the peak intensity of the symmetric conditions (φ = 0° or 90°) as shown in Fig. 2g. These asymmetric behaviors are derived from a nearfield interference effect of the dipole source. Furthermore, their difference (ΔS) in Fig. 2h shows a trend of first rising and then falling, reaching its maximum at φ = 45°, corresponding to the peak of \({E}_{z}^{\mathrm{I}}\) and the dip of \({E}_{z}^{\mathrm{II}}\), confirming the validity of our theoretical analysis related to the cause of mirror asymmetry. Hitherto, we demonstrated a feasible method for sourceconfigurable symmetrybroken polaritons from the perspective of theoretical calculation and simulation.
To visualize the effect of source polarization on the symmetricbroken HPs in the nearfield imaging experiments, we fabricated a gold disk with a diameter of 1.6 μm and a thickness of approximately 50 nm as an optical nanoantenna on the surface of a bulk crystal calcite (see Methods). The nanoimaging concept is illustrated in Fig. 3a. Upon illumination with a TMpolarized infrared beam of electric field E_{in}, the gold disk focused the incident field into a nanoscale spot at its extremities. This nanoscale hotspot served as a local point source with an inplane dipolar moment for inducing highly confined HPs on the calcite surface. The induced polaritonic field E_{p} was coherently superimposed with the illumination field E_{in} during propagation. Eventually, the resultant field E_{in} + E_{p} and its phase spectrum were collected by the tip of scatteringtype scanning nearfield optical microscopy (sSNOM). Further discussion can be found in Additional file 1: Fig. S7. Figure 3b clearly shows typical dipolar distributions at the disk extremities and a nearly 180° phase jump at the disk center. Moreover, the orientation of the antenna was parallel to the direction of phase jump, as illustrated by polarization angle φ (black double arrow) with respect to yaxis. This orientation can be dynamically tailored by rotating the polarization direction of illumination field with a simple farfield control, which involves rotating the sample. The nearfield signal were demodulated at the second harmonic without specification.
The nearfield amplitude images of HPs were measured with illuminating frequency 1470 cm^{−1} where θ = 45° for calcite, at four different polarization angles φ (see also experimental data in Additional file 1: Fig. S8), as shown in Fig. 3c. These experimental images were corroborated with the numerical simulations in Additional file 1: Fig. S9, showing an excellent agreement in the amplitude images. Figure 3c exhibits two types of asymmetric HP behaviors: the universal asymmetric HP patterns on both sides of the antenna under the oblique incidence (with an angle 30° to the calcite surface) and their fringe spacing constantly changes with the variation of polarization angle φ, which is a typical characteristic of interference with the incidence field. The offset of intrinsic hyperbolic dispersion induced by the oblique incidence is responsible for the change of interference fringes E_{in} + E_{p}, as demonstrated in the Fourier analysis of Additional file 1: Fig. S9c. However, HP propagations still maintain a certain degree of mirror symmetry with respect to xaxis at φ = 90° (or yaxis at φ = 0°).
Figure 3c also shows another nonsymmetric behavior witnessed at φ = 70° or 45° when the antenna polarization was no aligned with the direction of realspace crystal axis. To avoid any illuminationcaused effects, the background signals were filtered out (see Methods), and the nearfield signals of polaritons E_{p} were extracted, as shown in Fig. 3d. All of these signals exhibit raylike polariton propagation along the normal direction of open angle θ without interference fringes. Additionally, numerical simulations under the normal incidence were performed in Additional file 1: Fig. S10b to remove the factor of oblique incidence completely. We highlight that backgroundfree polariton signals in Fig. 3d exhibited remarkable consistency with the simulations of normal incidence. This indicates that mirror symmetry breaking is not related to the interference with farfield illumination but rather to the dipolar moment induced by the polarization of the gold disk antenna, as discussed previously.
Figure 3e illustrated the electric field intensity line profiles extracted from both the experiment (Fig. 3d) and simulation (Additional file 1: Fig. S11b). They exhibited excellent consistency in terms of amplitude trends, displaying constructive or destructive interference behaviors similar to those observed in Fig. 2g. Hence, the theoretical trend of asymmetry degree in HPs can be obtained via fitting simulation results of ΔS at different φ. As shown in Fig. 3f, the theoretical trend of ΔS was consistent with experimental points (marked by star symbols) and shares very similar features with the trend of \({E}_{z}^{\mathrm{I}}{E}_{z}^{\mathrm{II}}\) in Fig. 2h. However, it exhibited an asymmetrical distribution on both sides of the maximum at φ = 45° owing to the existence of offdiagonal elements ε_{yz} in the trigonal calcite crystal (Additional file 1: Fig. S10). Notably, when the symmetrybroken state occurred at φ = 45° as θ equals 45° (see peak in Fig. 3f), the dipolar orientation of the gold disk antenna, where the scattering field intensity is strongest (i.e. the direction of \({{E}_{z}^{p}}_{\mathrm{max}}\) in Fig. 2a), coincided exactly with the direction of the open angle of hyperbolic IFCs. At this point, the ray propagated along the antenna polarization, and by our definition, was almost erased. On the other hand, the peak intensity of the other one reaches its maximum for all φ and is almost twice as much as the symmetric case in Fig. 3e (but it is not applicable to all θ, Additional file 1: Fig. S11c). This feature confirms that our sourceconfigured method not only breaks the mirror symmetry of HPs but also results in the hyperbolic energy flow being efficiently coupled into rays in one direction rather than being divided into four equal branches, provided that the antenna orientation was appropriately tuned with respect to the direction of open angle θ.
Finally, we investigated the difference in symmetrybroken HP excitation and propagation at different open angles θ by adjusting the illumination frequency. Our previous analysis (Fig. 2) enabled us to determine the response (φ = 90° − θ) to ideal symmetry breaking for an arbitrary given θ. Conversely, we could also determine θ when φ was known. Note that φ was derived from the theoretical IFC of calcite (Additional file 1: Fig. S4) in advance, whereas θ was measured after the experiment. Experimental visualization of the symmetrybroken HPs were shown in Fig. 4a, b, where three different polarization angles φ were calculated as followed: 50° at 1460 cm^{−1}, 55° at 1450 cm^{−1}, and 60° at 1440 cm^{−1}. Remarkably, the degree of mirrorsymmetry breaking of HPs in these three cases is largely maintained while ramping the illumination frequency ω over a large range. Furthermore, the energy flow of HPs did not travel in the direction of antenna polarization but was blocked under the symmetrybroken conditions (Fig. 4b). Relevant principles have been covered in Fig. 2c–e. The corresponding Fourier spectra (Fig. 4c) also confirm the mirror symmetry breaking in the wavevector distribution, providing an advantage in the dynamical broadband regulation with our source configuration methods. This advantage arises from the fact that the open angle θ can be finely tuned almost throughout the Reststrahlen band in calcite. We intentionally combined three FFT images into a colormap in Fig. 4d to visibly demonstrate the evolving symmetrybroken behaviors with the change in θ. Apart from the decisive effect of \({{E}_{z}^{p}}_{\mathrm{min}}\) on the mirror symmetry breaking, we also highlight the dominant role of \({{E}_{z}^{p}}_{\mathrm{max}}\) in the excitation efficiency of HPs (details in Additional file 1: Fig. S11). Note that the intensity of asymmetric polaritonic rays in Fig. 4b gradually weakenes as the open angle θ increases from 1460 cm^{−1} to 1440 cm^{−1}, in accordance with the decline of \({{\Delta E}_{z}}_{\mathrm{max}}\) when θ varies from 39° to 27° in Fig. 2b. This weakening occurred due to the different degrees of deviation between the dipolar orientation of the antenna (i.e., \({{E}_{z}^{p}}_{\mathrm{max}}\)) and the direction of the open angle under the symmetrybroken state. The relationship between θ and φ under symmetrybroken conditions is shown in Fig. 4e, which exhibits a good linear correlation within reasonable experimental error when the infrared frequency ω ranges within 1420–1480 cm^{−1}, satisfying the theoretical relation θ = 90° − φ as shown in Fig. 2b. These observations support the suitability of our theoretical study for the antennaexcited HPs in practice.
In summary, we have introduced and experimentally demonstrated a sourceconfigured method to break the mirror symmetry of HP propagation in highsymmetry crystals, which enables the manipulation of polariton asymmetry in a simple and robust manner. By controlling the nearfield excitation source, we can achieve a broad range of symmetrybroken behaviors in almost all HP responses across a wide frequency spectrum. Our simulated and experimental results provide a comprehensive understanding of asymmetrically polaritonic phenomena and contribute to the development of fully controllable engineering of light steering at the nanoscale. Furthermore, this approach can be extended to other polaritonic and nanophotonic systems, creating exciting opportunities for robust, dynamical, and ultrafast nanolight routing based on polarization. Our study shows that symmetrybroken polaritons in a highsymmetry crystal offer higher directionality of symmetrybroken polaritons, enabling energy coupling in specific directions, as well as the advantages of highsymmetry crystal polaritons with low loss and easy device processing. Additionally, symmetrybroken polaritons in a highsymmetry crystal can be combined with methods such as graphene heterojunction, twist engineering, and chemical doping, which offer potential applications in areas such as nanoimaging, nearfield radiation management, photonic circuits for nanoscale light propagation, and quantum physics.
Availability of data and materials
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Abbreviations
 AFM:

Atomic force microscope
 FFT:

Fast Fourier transform
 HP:

Hyperbolic polariton
 IFC:

Isofrequency contour
 sSNOM:

Scatteringtype scanning nearfield optical microscopy
 TM:

Transverse magnetic
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Acknowledgements
We also thank the Analytical and Testing Center of HUST for help with the measurements.
Funding
We acknowledge the support from the National Natural Science Foundation of China (Grant No. 62075070 and 52172162), National Key Research and Development Program of China (Grant No. 2021YFA1201500), Hubei Provincial Natural Science Foundation of China (Grant No. 2022CFA053) and the Innovation Fund of WNLO. Z.D. acknowledges support from the Natural Science Foundation of Guangdong Province (2022A1515012145), Shenzhen Science and Technology Program (JCYJ20220530162403007), and Key Research and Development Plan of Hubei Province. W.M. acknowledges the support from the Fundamental Research Funds for the Central Universities, HUST (Grant No. 2022JYCXJJ009).
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Contributions
ZD, XY, XZ, and PL conceived the study. CH, TS fabricated the samples. CH, ZY performed the theory analysis coordinated by WM and PL, CH, ZY, and TS performed the sSNOM measurements with the help of WM, CH, TS, and WM performed the simulations. ZD, XY, XZ, and PL coordinated and supervised the work. CH, ZD, XY, and PL wrote the manuscript with input from all coauthors. All authors read and approved the final manuscript.
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Supplementary Information
Additional file 1: Note S1.
Field calculation from an electric dipole. Note S2. Dispersions in high and lowsymmetry crystals. Note S3. More details of numerical simulations. Additional figures.
Methods
Methods
1.1 Permittivity of the calcite crystal
We employed calcite, a common trigonal crystal, with a diagonalizable permittivity tensor that is represented by \(\widetilde{\varepsilon }\) = diag [ε_{⊥}, ε_{}, ε_{⊥}], where ε_{} and ε_{⊥} correspond to the principal components parallel and perpendicular to the optical axis, respectively. These components exhibit opposite signs in the midinfrared Reststrahlen band, spanning from 1410 to 1550 cm^{−1}, ε_{}> 0 and ε_{⊥} < 0. To fit its permittivity, we used Lorentz oscillator models with two Lorentz oscillators for ε_{⊥} and one oscillator for ε_{}, according to the equation:
where \({\omega }_{\mathrm{TO},i}\) and \({\omega }_{\mathrm{LO},i}\) are the ith TO and LO phonon frequencies for different Lorentz oscillators, respectively. The parameters Г_{i} and ε_{∞,i} represent the damping constant and the highfrequency permittivity. Detailed information regarding all of the parameters is available in reference [7].
1.2 Numerical simulations
To simulate the nearfield distributions of hyperbolic polariton along the surface of calcite, we employed COMSOL, a finiteelement simulation software. We used the scattered field in the electromagnetic wave frequency domain module to directly launch the surfaceconfined HPs without any background. For further details regarding each simulation figure, please refer to Additional file 1: Note S3.
1.3 Experimental background subtraction
In order to remove the effect of background, we first selected a clean background area away from the interference fringes (a circular area with a radius of 1 μm in our paper). We then calculated the mean of the real part and imaginary part of total field signals within the truncated region, respectively. By performing a complexvalued subtraction, the backgroundfree nearfield amplitude E_{p} can be obtained using the equation:
where A_{2}(x, y) and Ψ_{2}(x, y) are the amplitude and phase in the demodulated experimental signals measured at the arbitrary position, respectively. And A_{2,bg}(x, y) and Ψ_{2,bg}(x, y) are the mean value of the amplitude and phase in the selected background areas.
1.4 sSNOM measurements
For our nearfield imaging experiments, we employed a commercial sSNOM system from Neaspec GmbH that consists of an atomic force microscope (AFM). The Ptcoated AFM tip was operated in tapping mode with an oscillation amplitude of ~ 50 nm at a cantilever resonance frequency Ω ≈ 270 kHz. The system was continuously tuned using a continuouswave quantum cascade laser ranging from 1310 to 1470 cm^{−1}.
The backscattered field signal was collected with a pseudoheterodyne interferometer, and the interferometric detector signal was demodulated at a higher harmonic nΩ (n ≥ 2) to suppress farfield background contribution in the tipscattered field, thereby yielding nearfield images of amplitude s_{n} and phase φ_{n} with a high spatial resolution of ~ 20 nm.
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Hu, C., Sun, T., Zeng, Y. et al. Sourceconfigured symmetrybroken hyperbolic polaritons. eLight 3, 14 (2023). https://doi.org/10.1186/s43593023000471
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DOI: https://doi.org/10.1186/s43593023000471
Keywords
 Hyperbolic polaritons
 Nearfield excitation
 Crystal symmetry
 Inplane polarization
 Calcite