# Radial bound states in the continuum for polarization-invariant nanophotonics | Panda Anku

### Design and implementation of the radial BIC concept

The symmetry-protected and radially distributed electromagnetic BIC states are accessed through a carefully designed ring structure incorporating symmetry-broken double rod unit cells, where individual resonators are rotated to satisfy radial alignment (Fig. 1c). At resonance, adjacent rods support opposing electric dipoles, which leads to a suppression of the total dipole moment and consequently of the radiative losses in the system. In contrast to previous 2D or 1D resonator arrangements, our approach leverages a semi-infinite ring geometry30,31, providing a polarization-invariant optical response and avoiding edge effects, which allows a higher number of resonator elements to participate in the qBIC state for increased resonance quality32. Owing to the unique circular arrangement, the coupling between elements over the entire ring is improved as well, resulting in much higher Q factors compared to linear resonator chains (Supplementary Fig. S4). As a result, the radial BIC platform provides the highest quality factor to footprint ratio amongst comparable BIC-based nanophotonic approaches18,19,20,32 (Fig. 1). Furthermore, even though neighboring resonators are set at a small angle with respect to each other, the radial arrangement of rods with identical lengths produces a non-radiating BIC state with negligible coupling to the far field (Fig. 2a).

One of the key parameters to control the optical characteristics of the radial BIC mode is the asymmetry parameter ΔL, i.e., the length difference between the two constituent rods of the unit cell. In the symmetric case (ΔL = 0), we observe a BIC-like state, which is decoupled from the radiation continuum and exhibits negligible electromagnetic near-field enhancement (Fig. 2a). For nonzero asymmetry (ΔL > 0), the BIC transforms into a radial quasi-BIC mode (termed radial BIC from now on), which can be accessed from the far field, demonstrating strong field enhancements and the characteristic field pattern of oppositely oriented electric dipole moments observed in asymmetric metasurfaces26 (Fig. 2b). Due to the engineered inter-resonator coupling within the ring, linearly polarized light efficiently couples to the full radial BIC mode extended along the ring and excites spectrally distinct resonances with high-Q factors (Supplementary Fig. S1). Our numerical calculations confirm the presence of highly enhanced near fields outside the resonator volume and highlight how the field magnitude can be tailored via the structural asymmetry (Fig. 2b). We engineer the radial BIC system to exhibit high-Q resonances at around 770 nm and find that the corresponding numerically simulated Q factors exceed 8000 toward the smallest asymmetries (Supplementary Fig. S2). Furthermore, we observe a clear correlation between the electric near fields (Fig. 2b, for field intensities, see Supplementary Fig. S6) and the values of the Q factors, where the lowest asymmetry is associated with the maximum values of both parameters.

Symmetry-broken ring structures with varying geometry were fabricated from a 120 nm thick amorphous silicon layer on a glass substrate using electron beam lithography and reactive ion etching (see “Methods” for details). All measurements shown in the manuscript are based on ring structures with 24 annular unit cells and a base rod length L0 = 335 nm with width w = 115 nm. Symmetry-breaking is introduced by shortening one rod within the unit cell as indicated by the corresponding asymmetry parameter ΔL (Fig. 2b). The choice of the asymmetries is based on tradeoff between highest Q factors (and associated strong near fields) and the resulting modulation of the radial BIC resonances in the optical experiments. Although higher field enhancements and ultrahigh Q factors above 8000 are numerically predicted for asymmetries approaching ΔL = 0 nm (see Supplementary Fig. S2), an optimum parameter range was found for asymmetry values around ΔL = 25 nm, taking into account nanofabrication accuracy and the noise characteristics of our spectroscopy setup.

Optical characterization of the structures is carried out with a commercial confocal microscopy setup under collimated white light illumination. We first examine the optical response of the radial BIC structures with different radii and find precise tuning of the resonance wavelength via the ring radius R (Fig. 2c). Importantly, our versatile design provides additional degrees of freedom for tailoring the resonance, such as the number of unit cells constituting the ring and the base length L0 of the all-dielectric rods (Supplementary Fig. S8).

Consistent with numerical simulations, we observe the highest resonance sharpness for the smallest values of the asymmetry ΔL with Q factors exceeding 500 (Fig. 2d, for fitting details and full spectra, see Supplementary Figs. S9 and  S10). To our knowledge, this is one of the highest Q factors measured for symmetry-broken qBIC resonances in the visible wavelength range. For all asymmetries, the Q factor increases with decreasing radii (Supplementary Fig. S3), which we attribute to the improved coupling between the resonators. This scaling behavior is consistent with the fact that the coupling vanishes for infinite radii with infinite spacing between the resonators.

Strikingly, the annular arrangement of the resonator elements renders the Q factor mostly invariant under rotations of the incident polarization direction of light33, maintaining a value above 200 throughout the angular range (Fig. 2e). We attribute the observed spectral polarization invariance to the inherent C24 symmetry of our ring design which congruently transforms the ring onto itself for a rotation by one unit cell (15°). Furthermore, we do not observe any polarization angle dependence of the spectral response for smaller rotation angles ((0le varphi le 15^circ,) see Supplementary Fig. S7). As a result, we conclude that the spectral response of the radial BIC is fully polarization-invariant for any arbitrary polarization angle. Crucially, this polarization-independent performance enables simplified experimental measurements and greatly eases practical device applications.

### Enhanced biomolecular sensitivity

We leverage the high-Q resonances and strong surface-confined near fields of the radial BIC platform to demonstrate biomolecular sensing for different ring geometries and asymmetries based on a biotin-streptavidin binding bioassay (Fig. 3a). As an initial step, we assess the refractometric sensing performance by varying the local refractive index around the ring structure (see “Methods” for details) by means of magnetron sputtering of conformal silicon dioxide (SiO2) thin films with increasing thickness. Pronounced resonance shifts for all asymmetries can be detected, depending on the SiO2 layer thicknesses (Fig. 3b). For the calculation of the bulk refractive index sensitivity SB (BRIS), the resonance shifts mediated by the thin films are converted by considering both the electric near-field decay length of the resonators and the film thickness as introduced previously34 (see “Methods”).

Based on these BRIS values, we find that the sensing figure of merit

$${{{{{rm{FOM}}}}}}={S}_{B}/{{{{{rm{FWHM}}}}}},$$

(1)

where FWHM is the resonance full width at half maximum, is clearly correlated with the asymmetry of the radial BIC structures, with the highest FOM of above 20 per refractive index unit (RIU) observed for the lowest asymmetry (Fig. 3c). Despite the much smaller footprint, the FOM values of our radial BIC geometry for biomolecular sensing are mostly comparable with large array-based approaches relying on symmetry-protected metasurfaces18. Nevertheless, we believe that the FOM can be substantially boosted by utilizing even smaller asymmetries or by further engineering the inter-rod coupling, e.g., by optimizing the resonator shape. Additional advantages of the radial BIC approach can be realized by a concentric arrangement of several rings for multiplexed operation without increasing the footprint, or by integrating multiple rings with identical resonance wavelengths to enhance light absorption efficiency per area.

For the implementation of a model multistep bioassay, we first functionalize our structures with (3-Aminopropyl) triethoxysilane (APTES), followed by the attachment of biotin molecules and the binding of different concentrations of streptavidin protein. After each step, we observe a clear redshift of the radial BIC resonance (Fig. 3d), induced by the higher number of molecules on the surface and the associated increase of the environmental refractive index. To compare the biomolecular sensing performance for all asymmetries, we normalize the resonance shift by the respective FWHM and plot it for each streptavidin concentration to obtain a map of biomolecular sensing performance (Fig. 3e).

Notably, a clear and streptavidin-dependent sensor response over a broad range of concentrations and resonance linewidths is observed. Depending on the sensitivity and spectral resolution of the spectroscopic equipment in experiments and thus the required resonance modulations as well as Q factors, we demonstrate a wide parameter space for picking the appropriate structural asymmetry. Especially, field-deployed applications with low spectral resolution may require lower Q resonances—whereas when pushing for highest sensitivity, high-Q resonances and consequently a high FOM is favored. Our versatile radial BIC platform allows to cover all such use cases in an ultrasmall footprint, pushing the limits of on-chip multiplexing applications for biosensing and opening up new avenues for the precise on-demand tailoring of FOM, Q factor, resonance position, and resonance modulation.

### Enhanced SHG in an evanescently coupled monolayer of MoSe2

To further demonstrate the versatility of our design, we employed the radial BIC platform for enhanced light–matter interaction between the all-dielectric resonator system and a two-dimensional excitonic material20,35,36. Specifically, we demonstrate and localize enhanced second-harmonic generation (SHG) in a non-centrosymmetric monolayer crystal of the transition metal dichalcogenide (TMD) MoSe2, facilitated by the highly enhanced electromagnetic near fields that are associated with the radial BIC resonances. For this purpose, we deterministically transferred a monolayer of MoSe2 on top of the radial BIC platform (Fig. 4a). Crucially, this experiment is enabled by the low footprint of the radial BIC platform, which allows full spatial overlap between the micron-sized MoSe2 layer and the ultracompact ring geometry. Atomic force microscopy (AFM) imaging confirms the homogeneous coverage of the resonators (Fig. 4b) and indicates that the monolayer is in good contact with the silicon structures. In Fig. 4c, we show the quadratic dependence of the integrated SHG signal on the input power for the monolayer together with its polarization-resolved signal (inset, see “Methods”), confirming the origin of the signal from the nonlinear harmonic light generation in the atomically thin crystal37.

To demonstrate spectrally selective SHG, we excite the coupled system of a MoSe2 monolayer on top of a tailored radial BIC structure at three different excitation wavelengths with the use of a tunable femtosecond pulsed laser (Fig. 4d, e, see “Methods” for details). For this purpose, we choose a ring with ΔL = 50 nm to balance the high near-field enhancements with a sufficient resonance modulation. Although we expect even higher near-field and thus SHG enhancements for smaller asymmetries (see Supplementary Fig. S6), these resonances are often strongly damped by the TMD monolayer in experiments. For our SHG measurements, we excite the system at 744 nm in resonance with the radial BIC mode, at 800 nm close to the MoSe2 exciton peak, and at 900 nm, spectrally distinct from both the exciton and the radial BIC resonance (Fig. 4e). At each excitation wavelength, we raster the focused laser beam across the sample, direct the light to an avalanche single photon detector, and plot the obtained SHG signal intensity for each point. In Fig. 4d, we show the maps of the SHG signal on the ring normalized to the bare MoSe2 monolayer flake to guarantee consistency between the measurements and to eliminate wavelength-dependent processes such as detector and SHG efficiency as well as transmission of the signal in the optical beam path (see Supplementary Fig. S11b for raw data without background).

For radial BIC-resonant excitation, we observe a fivefold SHG intensity enhancement confined to the ring structure, illustrating the highly localized generation of the second-harmonic signal. In contrast, for off-resonant excitation, we observe a suppression of the SHG (Fig. 4d) within the same region. The suppression of the SHG signal when excited spectrally apart from the radial BIC resonance can be attributed to strain introduced in the monolayer TMD by the ring structure38, which is outperformed by the highly enhanced near fields of the resonant excitation. The wavelength dependence is a clear indication that the SHG enhancement is driven by the locally enhanced electromagnetic near fields of the radial BIC resonances. We further examined the SHG signal originating from the bare silicon radial BIC structures and observed only a negligible contribution to the overall SHG intensity (see Supplementary Fig. S11a).

We further conducted control experiments where we suppressed the radial BIC resonance by investigating a symmetric structure (ΔL = 0 nm, Supplementary Fig. S12). In this case, we observe no enhancement of the SHG signal when the monolayer is on top of symmetric rings and probed at the wavelength associated with the radial BIC resonance at 744 nm, further confirming that the observed SHG enhancement arises from the near-field coupling of the MoSe2 monolayer with the radial BIC mode. We also perform linearly polarized excitation and show the corresponding polarization-dependent SHG maps in Supplementary Fig. S13. Crucially, the SHG generation experiments are highly facilitated by the compact footprint and polarization invariance of our ring geometry since the constraints on the spatial alignment and orientation of the TMD monolayer are released.