Abstract
High quality(Q) factor optical resonators are indispensable for many photonic devices. While very large Qfactors can be obtained theoretically in guidedmode settings, freespace implementations suffer from various limitations on the narrowest linewidth in real experiments. Here, we propose a simple strategy to enable ultrahighQ guidedmode resonances by introducing a patterned perturbation layer on top of a multilayerwaveguide system. We demonstrate that the associated Qfactors are inversely proportional to the perturbation squared while the resonant wavelength can be tuned through material or structural parameters. We experimentally demonstrate such highQ resonances at telecom wavelengths by patterning a lowindex layer on top of a 220 nm silicon on insulator substrate. The measurements show Qfactors up to 2.39 × 10^{5}, comparable to the largest Qfactor obtained by topological engineering, while the resonant wavelength is tuned by varying the lattice constant of the top perturbation layer. Our results hold great promise for exciting applications like sensors and filters.
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Introduction
HighQ photonic nanoresonators constitute a core element in highperformance optoelectronic and photonic devices in modern optical communications. While very large Qfactors have been obtained with microring resonators and microdisks via the optical fiber excitation through inplane nearfield coupling^{1}, it is very challenging to achieve an ultrahigh Qfactor in experiments via freespace excitation. Resonant waveguide gratings and photonic crystal slab are viewed as an ideal platform to achieve highQ guidedmode resonances (GMRs)^{2,3,4}, which can be accessed in free space. They have been widely used in in filtering^{5,6,7,8,9,10}, sensing^{11,12,13}, and wavefront shaping^{14,15}. Although the Qfactors of GMRs in theory can be very high, the largest Qfactor in experiments has been so far limited by fabrication imperfections (i.e., roughness, disorder and nonuniformity) causing increased radiation loss. The nanofabrication process usually involves dryetching to define these nanostructures, introducing additional surface roughness^{16}, imperfect vertical side walls^{17}, and longrange nonuniformity^{18}. To date, most of the reported Qfactors of GMRs in experiments fall in the range of from few hundreds to thousands, as summarized in Table 1. HighQ resonances up to 2340 have been realized with plasmonic metasurfaces^{19} by harnessing surface lattice resonances, but such plasmonic approaches are widely recognized to be limited by intrinsic material loss. Another candidate to achieve highQ resonances is the alldielectric metasurface exploiting electromagnetically induced transparency phenomena^{20}.
Recently, bound states in the continuum (BICs) have introduced a viable alternative to realize ultrahighQ resonances in a free space setting^{21,22,23,24,25}. The giant field enhancement typically associated with largeQ factors can benefit applications ranging from lasing^{26,27,28,29,30}, sensing^{31,32}, strong coupling^{33,34} to enhanced nonlinear harmonic generations^{35,36,37,38}. Most of the quasiBICs(QBICs)based highQ modes reported so far are achieved by patterning the highindex semiconductor thin film as a single nanoparticle^{39,40,41} or an array of nanoparticles on substrate^{21,22,26,29,37,42,43,44,45,46}. Table 1 summarizes the measured Qfactor of QBIC modes based on alldielectric metasurfaces. Typically, the Qfactors of QBICs are limited by scattering loss caused by unavoidable fabrication imperfections (i.e., roughness, disorder). Also, the presence of a substrate may also introduce the additional leaky channel due to outofplane asymmetry, further reducing the Qfactors of QBICs. Of course, ultrahighQ modes can still be constructed by carefully arranging the structure parameters. For example, topological BICs, arising from merging multiple BICs into a single one, are found to be less sensitive to fabrication imperfections^{47,48,49,50,51,52}. However, achieving such a high Qfactor requires delicate engineering of the structure parameters and may not always be compatible with other design goals. Moreover, the substrate must be removed to satisfy the requirement for environmental symmetry. In addition, generalizing this achievement to visible wavelengths is an outstanding challenge that may prove difficult due to materials and fabrication constraints. To promote widespread adoption in practical applications, it is highly desirable to develop a universal yet simple design strategy to realize ultrahigh Qresonances that minimize the impact of fabrication imperfections, mitigate the substrate effect, and do not need careful engineering based on complex topological concepts.
In this work, we introduce a strategy to implement ultrahighQ guidedmode resonances in an alldielectric metasurface without the need for topological engineering of the structure or introducing any symmetry breaking other than the discretization of translational symmetry. Instead of patterning the highindex layer to form a Mie resonator in a unit cell, we introduced an ultrathin photoresist layer as a perturbation layer on top of a multilayerwaveguide system so that guided modes are transformed into leaky modes that produce GMRs, which obviates the need for dryetching. High Qfactors of GMRs can be easily realized as they are inversely proportional to the perturbation parameter squared (Q ∝ α^{−2}). The perturbation can be reduced to a very small value because the fabrication of such a device only involves spincoating of resist and developing, enabling minimal perturbation by removing the roughness and disorder of the sample. The validity of such a design methodology is confirmed by patterning a lowindex photoresist thin film on a standard silicon on insulator (SOI) sample that serves as a simple waveguide system. The experimental results show that the Qfactor was as high as 2.39 × 10^{5}, comparable to the maximum Qfactor of topological BICs. In addition, the resonant wavelength was easily tuned by changing the period of the metawaveguide system. This design strategy is robust and powerful in obtaining ultrahighQ resonances at arbitrary operating wavelengths because it can be generalized to any other waveguide system made of any different high refractive index (n = 2∼5) as the core layer. Our results may find great potential in applications requiring very sharp spectral features, such as sensing and filtering.
Results
Guided modes in a multilayerwaveguide system
We start by investigating the leaky modes of a multilayer waveguide structure, consisting of a high refractive index n_{2} layer with finite thickness sandwiched between two semiinfinite layers with low refractive index being n_{1} and n_{3}, respectively. Without loss of generality, we assume that the electric field component is perpendicular to the plane of structure, which is E//y. Such an open system supports a series of FabryPerot resonant modes with lowQ factors (See Section 1 of supplementary materials and Fig. S1). If we apply a virtual periodic boundary condition with arbitrary periods, in addition to the lowQ leaky modes, such a simple geometry can also support guided modes (GMs) with an infinite Qfactor. Note that there are two ways of defining a periodic structure: a 1D grating or a 2D metasurface, as shown in Fig. 1a and Fig. S2. The band diagram for TE (E//y) and TM (H//y) modes can be obtained by folding the waveguide dispersion of GMs into the first Brillouin zone. The dispersion relationship of GMs has been derived in the text book^{53} and can be expressed as follows
Where \({\beta }_{1}=\sqrt{{k}^{2}{{n}_{1}^{2}k}_{0}^{2}}\), \({\beta }_{2}=\sqrt{{{n}_{2}^{2}k}_{0}^{2}{k}^{2}}\), \({\beta }_{3}=\sqrt{{k}^{2}{{n}_{3}^{2}k}_{0}^{2}}\), \(k\) is the propagating constants of guided mode, \({k}_{0}=\omega /c\), ω is the angular frequency and c is the speed of light in the vacuum.
It is necessary to mention that there is a cutoff frequency for different guided modes.
Where s = 1 for TE and \({s=n}_{2}^{2}\), c is the speed of light in the vacuum.
More detailed discussions on band structure calculation are given in section 2 of supplementary materials and Fig. S3.
Guided mode resonances and BICs
Here, we mainly focus on the GMs at Γpoint in the first Brillouin zone in momentum space. Alternatively, the eigenfrequencies of these GMs can be calculated quickly with the commercial software COMSOL Multiphysics. For the 1D grating system, these GMs are designated as TE_{ml} (TM_{ml})^{54,55}, where m and l are the numbers of antinodes of the electric (magnetic) field in the x and z directions, respectively. Fig. 1b, c shows two typical examples of GMs: modes TE_{21} and TE_{31}. These GMs can be transformed into GMRs by introducing an ultrathin lowindex metasurface on top of the highindex layer. For example, when the top perturbation layer is patterned as a grating, GMs TE_{21} and TE_{31} are converted into symmetryprotected BIC mode TE_{21} and GMR TE_{31} with a finite Qfactor, respectively. If the mirror symmetry of the top periodic structure is broken with respect to the zaxis, BIC mode TE_{21} is reduced to QBIC. Note that GMs like TE_{21} and TE_{31} at the Γ point share a similar feature with symmetry protected BICs. When the periodic perturbation is introduced in the multilayer waveguide system, such GMs with infinite Qfactors are successfully converted into either a BIC mode with an infinite Qfactor or a GMR with a finite Qfactor depending on the mode parity. If the perturbation approaches zero, the radiative decay rates of GMRs are reduced to zero. Rigorous derivation shows that Q values are inversely proportional to β^{2} (see section 3 of supplementary materials), where β is the perturbation parameter of the system. This is similar to the symmetryprotected BICs with Q ∝ α^{−2} (where α is the asymmetry parameter of unit cell). The GMs for the TM case are shown in Fig. S4. The following sections demonstrate that GMs provide much freedom and flexibility to design ultrahigh Qfactor resonance with a subwavelength metawaveguide system at an arbitrary operating wavelength.
For simplicity, we use a threelayer structure air/Si/SiO_{2} to design a highQ resonance at λ_{0} = 1550 nm. Without loss of generality, the thickness of the middle layer Si is chosen as 220 nm. The refractive index of Si and SiO_{2} are set as 3.47 and 1.46, respectively. The dispersion relationship of GMs in such a threelayer system can be obtained from Eq. (1) and is shown in Fig. 2. To support a GM at λ_{0} = 1550 nm (dashed black line in Fig. 2), the propagation constants should be\(\,k=5.6515*{{{{{\rm{\pi }}}}}}/{\lambda }_{0}\). Then, the virtual period can be obtained as p = 548 nm by applying \(k=2{{{{{\rm{\pi }}}}}}/p\). In principle, the resonant wavelength can be tuned to any other value by varying the period, which is discussed in the later section.
By introducing a top perturbation layer that is patterned as either a grating or a metasurface, the GM evolves into a BIC or GMR depending on the structure’s geometry and the mode’s parity. Their band structures are plotted in Figs. S5, 6. Here, it is worth noting that the refractive index of grating or metasurface can be of any value as long as the top layer serves as a perturbation layer. As an example, the refractive index of the top layer is set as 1.46, matching the index of SiO_{2}. We first consider a simple grating structure shown in Fig. 3a and study the effect of width and thickness on the Qfactor of GMR. From Fig. 3b, it can be found that the Qfactor of the GMR TE_{31} is proportional to t^{−2} when the width of the grating is fixed as 200 nm. Excellent agreement can be found between the calculated Qfactor (solid red line) and fitted Qfactor (dashed blue line) by applying the fitting equation Q=Ct^{−2}(C is the fitted constant). This is similar to the observation in symmetryprotected BICs, where Q ∝ α^{−2} (where α is the asymmetry parameter). Such a relationship is well explained by perturbation theory^{56} (see section 3 in supplementary materials). This is also confirmed by the reflection spectrum mapping as the thickness increases from 0 to 300 nm (See Fig. S7).
Similarly, the Qfactor shows a linear dependence on w^{−2} or (dw)^{−2} for a given thickness, as confirmed in Fig. 3c. Thus, it can be safely concluded that the effective perturbation area or volume plays the dominant role in governing the Qfactors of GMRs. Moreover, the resonant wavelength remains almost the same with the changing thickness or width because of the lowindex nature of the top perturbation layer (See Figs. S7, 8). As described in Fig. 1, except for the GMR TE_{31}, such a grating also supports a symmetryprotected BIC mode TE_{21} due to structural symmetry. Further breaking the symmetry can induce the transition from BIC to QBIC. For instance, by introducing an asymmetric air slit inside the rectangular nanowire, BIC mode TE_{21} is successfully converted into the QBIC. Fig. 3b shows the Qfactor of QBIC TE_{21} as a function of thickness. The case of TE_{31} is also plotted as a reference to make a comparison in Fig. 3d–f. Similar to the case of TE_{31} shown in Fig. 3b, the Qfactor of QBIC TE_{21} is proportional to t^{−2} when an asymmetric slit is introduced with g = 120 nm and xc = 10 nm. The left and right nanowires’ widths are 110 nm and 90 nm, respectively. Besides, we find that the Qfactor of QBIC mode TE_{21} is more than one order of magnitude higher than that of mode TE_{31}. This is understandable because the perturbation required for symmetrybreaking is much smaller for QBIC TE_{21} compared to GMR TE_{31}. Such a general conclusion also can be applied to the 3D metasurface. When the cuboid unit cell of the top perturbation layer is arranged as a square lattice, such a metasurface supports degenerate symmetryprotected BICs (TE_{211} and TE_{121}) and GMRs (TE_{311} and TE_{131}). The degeneracy could be easily lifted by choosing the different lattice constants along the x and yaxis (See Fig. S9). Further introducing inplane broken symmetry leads to the transition from BICs into QBICs for the TE_{211} mode. Fig. 3g–i shows the Qfactor of GMR TE_{311} and QBIC TE_{211} versus the thickness of top layer. Additionally, it is not surprising that Qfactor follows a similar trend as 1D grating, Q ∝ t^{−2}. Moreover, such a 2D metasurface supports BICs or GMRs for TM cases (See Fig. S10). Here, it is necessary to point out that the top layer can also be patterned as a rectangular lattice with any other shape such as photonic crystal slab because it just functions as a weak perturbation layer.
Guided mode resonances engineering
Note that many GMs can be easily constructed with such a simple metawaveguide system. We again consider the 1D grating structure shown in Fig. 1a with material parameters and geometry parameters given in Fig. 3a. Except for TE_{21} and TE_{31} mode, such a threelayer structure supports other highorder GMs. Fig. 4a shows the field distributions of six GMs: TE_{21} and TE_{31}, TE_{22} and TE_{32}, TE_{41} and TE_{51}. Their resonant wavelengths are presented in Fig. 4b. Note that all of these modes are dominated by the zeroorder diffraction. Of course, even higherorder GMs like TE_{42} and TE_{52} can be found. However, their Qfactor may decrease significantly due to the enhanced radiation from unwanted diffraction after the perturbation is introduced. Also, GMs may be designed at any predefined wavelength with different structure parameters and material parameters. For example, if the thickness of the Si layer is set as 220 nm or 340 nm, we can always tune the virtual period so that the resonant wavelengths of GM TE_{21} or TE_{31} could cover a broadband wavelength range from 1000 nm to 1800 nm, as displayed in Fig. 4c.
Similarly, from Fig. 4d, we found that the resonant wavelength can be tuned by varying the thickness of Si layer while the period is fixed as 450 nm (or 550 nm). One can also engineer the GM’s wavelength by choosing materials with different refractive indices. For example, to design a GM at 1550 nm, the period should be varied from 1000 nm to 460 nm when the refractive index n_{2} increases from 2 to 4. The corresponding result is shown in Fig. 4e. To verify the robustness of such a design strategy in realizing ultrahighQ resonances, we calculate the Qfactor of GMR TE_{31} in a grating system for the middle layer with various refractive index n_{2}. Indeed, all of the Qfactors decrease significantly with the increasing thickness of the top grating layer, as illustrated in Fig. 4f. Nevertheless, it is interesting that the Qfactor for highindex n_{2} = 3.5 is almost one order higher than the lowindex n_{2} = 2.0. This can be intuitively understood by the reduced index contrast between the substrate (or superstrate) and the sandwiched highindex layer, which leads to more energy radiation into the substrate (or superstrate) instead of confining inside the highindex layer. The robustness of such a design strategy could enable us to design ultrahighQ GMRs at ultraviolet and visible wavelength ranges by choosing appropriate core layers with highindex, including Si_{3}N_{4}, GaN, TiO_{2}. In the supplementary materials, we demonstrate two examples of designing highQ resonators in the visible wavelength by using GaN and Si_{3}N_{4} as core layer, as shown in Figs. S11, 12.
Experimental demonstrations of highQ GMRs
After gaining a solid understanding of designing ultrahighQ resonances from GMs, we move to the experimental demonstration of such a highQ resonance. Fig. 5 shows the experimental setup of a custommade crossedpolarization measurement system. Crossed polarization measurement has been widely used to measure the highQ resonances^{50,57}. We fabricate a series of photonic crystal slabs made of 330 nm thick photoresist (ZEP520) on an SOI substrate (220 nmSi/2 µmSiO_{2}/Si) using electronbeam lithography followed by developing process. Here, the photoresist layer instead of the SiO_{2} layer is patterned as a perturbation layer. On the one hand, the photoresist has a low refractive index of around 1.4. On the other hand, no other postprocessing (i.e., dry etching) is applied to the other layers so that higher sample quality is guaranteed.
Fig. 6a shows the schematic drawing of the whole device, and Fig. 6b shows the scanning electron microscopy image of one typical fabricated sample. The lattice constants along the x and yaxis are identical in this device. We first investigate the effect of hole size on the Qfactor by fixing the lattice constant as 545 nm but varying the hole radius. Eigenmode calculations indicate that such a structure can support degenerate GMRs TE_{311} and TE_{131} at around 1550 nm. It also supports symmetry protected BICs TE_{211} and TE_{121}. However, they cannot be excited at normal incidence unless the inplane mirror symmetry is broken with respect to the yz plane.
The relevant results are put in Fig. 6d–f. It can be observed from Fig. 6d that shrinking the size of the hole on the top perturbation layer leads to the narrowing of resonance linewidth and slight redshift of resonance. We retrieved the Qfactors and resonant wavelengths of these highQ resonances by fitting them to a Fanoprofile. Fig. 4e shows one example of Fano fitting for R = 72 nm. The measured Qfactor for structures with different hole sizes is presented in Fig. 6f, where the highest Qfactor is up to 2.39 × 10^{5} at R = 72 nm. Such a high Qfactor is comparable to the recordhigh Qfactors in a topological metasurface^{50}. Nevertheless, our approach involves only patterning the top resist layer without the need for carefully engineering the topological features. Also, the calculated Qfactor is plotted as a reference and is found to show an excellent agreement with the experiment measurement. In fact, we also fabricate the sample with an even smaller radius R = 35 nm. However, it becomes very challenging to get an even higher Qfactor in experiment because the peak cannot be distinguished from the noise or background. Further optimization of the measurement system is needed. Such an ultrahighQ factor can be attributed to two factors: small hole radius and shallow hole with partial unetched resist film underneath. After taking both factors into account, the calculated Q is an order of million for such a size (See Fig. S13).
Finally, to demonstrate the tunability of resonant wavelength, we fabricate five samples with the period varying from 535 nm to 555 nm while their hole radii are fixed as 80 nm. The scattering spectra are plotted in Fig. 6g. Indeed, the resonant peaks show a linear dependence on the period (Fig. 6h), matching very well to the theoretical calculation. In addition, all of the measured Qfactor are ranged between 1.0 × 10^{5} and 2.0 × 10^{5} (Fig. 6i). Note that TE_{311} and TE_{131} are degenerate for the identical lattice constants along the x and yaxis. This degeneracy can be lifted by employing a rectangular lattice with different p_{x} and p_{y}. Thus, highQ resonance could be implemented at a different wavelength by switching the polarization (See Fig. S14). In other words, even a single structure can support two highQ modes at different wavelengths for x and ypolarization. An even higherQ factor can be obtained by exciting QBICs via breaking the unitcell’s mirror symmetry, as demonstrated in Fig. 3i. In addition to reducing the hole size, such an ultrahighQ mode may be enabled by using an ultrathin material as a perturbation layer. The recently developed twodimensional (2D) materials, especially 2D transition metal dichalcogenides (TMDCs)^{58,59} and hexagonal boron nitride (hBN)^{60}, are ideal candidates for realizing atomically thin film below 10 nm. Moreover, such a 2D TMDC multilayer with thickness below 10 nm can be easily patterned by induced coupled plasma (ICP) etching. Although the refractive index of TMDC is a bit high (n = 3∼4.5 in the nearinfrared), their atomically thin nature makes them an excellent candidate as a perturbation layer. We expect a highquality thin film like TMDCs and hBN multilayers, and mature fabrication process could help to realize an ultrahighQ factor larger than 10^{6} or even more.
Note: The results presented in this work are in bold. EITelectromagnetic induced transparency; SLR surface lattice resonance; GMRguided mode resonance; BICbound state in the continuum; NPnanoparticle; PCSphotonic crystal slab; SOIsilicon on insulator.
Discussion
In summary, we theoretically proposed and experimentally demonstrated highQ guided resonances in a metawaveguide system. The designing strategy builds upon adding a perturbation layer (i.e., photoresist) on top of a conventional waveguide system, which converts GMs into the BICs or GMRs. The Qfactors of GMRs strongly depend on the perturbation of a metawaveguide system, suggesting a broadly adaptable way of realizing ultrahighQ resonances without resorting to topological concept, as demonstrated in the recent literature [46]. The ultrahigh Qfactor of GMRs can be realized because the fabrication of proposed structure only involves resistcoating and developing, but no deep etching process, successfully eliminating the common source of roughness and disorder of the sample and enabling a controlled tiny perturbation in realistic devices. We experimentally demonstrated the feasibility of this design strategy in achieving highQ resonances at around 1.55 µm by fabricating a thin photoresist layer photonic crystal slab on top of 220 nm Si/2 μm SiO_{2}/Si (standard SOI sample). The measured Qfactor reaches up to 2.39 × 10^{5}, comparable to the value of merging BICs by topological charge engineering. Also, the resonant wavelength can be tuned by varying the lattice constant of the top resist layer. Furthermore, by choosing different lattice constants along the x and yaxis, we realized polarizationdependent GMRs. Our findings open a new avenue to design optoelectronic and photonic devices in which ultrasharp spectral features may improve their performance, such as biosensors and filters.
Methods
Numerical simulation
The eigenmodes of metawaveguide are calculated by commercial software Comsol Multiphysics based on finiteelementmethod (FEM). The reflection and transmission spectra of metastructures are calculated by rigorouscoupled wave analysis (RCWA).
Fabrication
Our devices were fabricated on a silicononinsulator (SOI) wafer with a 220 nm silicon layer and 2 μmthick buried layer. The SOI was firstly coated with the 330 nmthick resist layer (ZEP520). Then, the device patterns were defined on the resist by electronbeam lithography. After that, the devices were carefully developed and fixed by dimethyl benzene and isoPropyl alcohol, respectively.
Optical Characterization
The incident light source was a tunable telecommunication laser (santec TSL550), the wavelength of which could be tuned from 1480 nm to 1630 nm. Light first passed through the polarizer, lens and beam splitter, and then focused on the rear focal plane of the objective lens (10X), to make sure the incident light on the sample was close to normal incidence. The reflected signal was collected by a photodiode (PDA10DTEC), which could be switched to a CCD to locate the sample. A lockin amplifier with the help of chopper (SR540) was connected to the photodiode to pick out the signal from the background noise. A pair of orthogonal polarizers were placed at the front and the middle part of the light path, to introduce crossedpolarization which could efficiently decrease the impact from background reflection and promote the relative intensity of the signal.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The code used in this work is available upon request from corresponding author.
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Acknowledgements
L.H. and A.E. Miroshnichenko were supported by the Australian Research Council Discovery Project (DP200101353) and the UNSW Scientia Fellowship program. L.H. was also supported by Shanghai Science and Technology Committee (22PJ1402900). C.Z. was supported by the National Natural Science Foundation of China (Grants No. 12004084 and 12164008), Guizhou Provincial Science and Technology Projects (ZK[2021]030). R.J., G.L., and X.C. were supported by National Natural Science Foundation of China (62222514, 62204249, 61991440); Youth Innovation Promotion Association of Chinese Academy of Sciences (Y2021070); Strategic Priority Research Program of Chinese Academy of Sciences (XDB43010200); Shanghai RisingStar Program (20QA1410400); Shanghai Science and Technology Committee (23ZR1482000, 20JC1416000 and 22JC1402900), Natural Science Foundation of Zhejiang Province (LR22F050004), and Shanghai Municipal Science and Technology Major Project (2019SHZDZX01). A.O. and A.A. were supported by the Air Force Office of Scientific Research and the Simons Foundation. This work was partially carried out at the Center for Micro and Nanoscale Research and Fabrication in University of Science and Technology of China.
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L.H. and A.E.M. conceived the idea. L.H. performed the theoretical calculation and numerical simulation. C.Z. fabricated the sample and helped with numerical simulations. R.J. and G.L. carried out the morphology characterization, built up the optical system and implemented the measurement. A.O., L.X., and F.D. helped with the numerical simulation. X.C., W.L., A.A., and A.E.M. supervised the project. L.H. and A.E.M. prepared the manuscript with input from all authors.
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Huang, L., Jin, R., Zhou, C. et al. UltrahighQ guided mode resonances in an Alldielectric metasurface. Nat Commun 14, 3433 (2023). https://doi.org/10.1038/s41467023392275
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DOI: https://doi.org/10.1038/s41467023392275
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