Asymmetric metasurfaces with high- Q resonances governed by bound states in the continuum
Kirill Koshelev, Sergey Lepeshov, Mingkai Liu, Andrey Bogdanov, Yuri Kivshar
AAsymmetric metasurfaces with high- Q resonances governed by bound states in the continuum Kirill Koshelev , , Sergey Lepeshov , Mingkai Liu , Andrey Bogdanov , and Yuri Kivshar , Nonlinear Physics Centre, Australian National University, Canberra ACT 2601, Australia and ITMO University, St. Petersburg 197101, Russia
We reveal that metasurfaces created by seemingly different lattices of (dielectric or metallic) meta-atoms withbroken in-plane symmetry can support sharp high- Q resonances that originate from the physics of bound statesin the continuum. We prove rigorously a direct link between the bound states in the continuum and the Fanoresonances, and develop a general theory of such metasurfaces, suggesting the way for smart engineering ofresonances for many applications in nanophotonics and meta-optics. Metasurfaces have attracted a lot of attention in the recentyears due to novel ways for wavefront control, advanced lightfocusing, and ultra-thin optical elements [1]. Recently, meta-surfaces based on high-index resonant dielectric materials [2]have emerged as essential building blocks for various func-tional meta-optics devices [3] due to their low intrinsic loss,with unique capabilities for controlling the propagation andlocalization of light. A key concept underlying the specificfunctionalities of many metasurfaces is the use of constituentelements with spatially varying optical properties and opticalresponse characterized by high quality factors ( Q factors) ofthe resonances.Many interesting phenomena have been shown for meta-surfaces composed of arrays of meta-atoms with broken in-plane inversion symmetry (see Fig. 1), which all demonstratethe excitation of high- Q resonances for the normal incidenceof light. The examples are the demonstration of imaging-based molecular barcoding with pixelated dielectric metasur-faces [4] and manifestation of polarization-induced chiralityin metamaterials [5], which both are composed of asymmet-ric pairs of tilted bars [see Fig. 1(a)], observation of trappedmodes in arrays of dielectric nanodisks with asymmetricholes [6] [see Fig. 1(b)], sharp trapped-mode resonances inplasmonic and dielectric split-ring structures [7, 8] [see, e.g.,Fig. 1(c)], broken-symmetry Fano metasurfaces for enhancednonlinear effects [9, 10] [see Fig. 1(d)], tunable high- Q Fanoresonances in plasmonic metasurfaces [11] [see Fig. 1(e)],trapped light and metamaterial-induced transparency in arraysof square split-ring resonators [12, 13] presented in Fig. 1(f).Here, we demonstrate that all such seemingly different struc-tures can be unified by a general concept of bound states inthe continuum, and we prove rigorously their link to the Fanoresonances.Bound states in the continuum (BICs) originated fromquantum mechanics as a curiosity [14], but later they wererediscovered as an important physical concept of destructiveinterference [15] being then extended to other fields of wavephysics, including acoustics [16] and optics [17, 18]. A trueBIC is a mathematical object with an infinite Q factor andvanishing resonance width, it can exist only in ideal losslessinfinite structures or for extreme values of parameters [19–21]. In practice, BIC can be realized as a quasi-BIC in theform of a supercavity mode [22] when both Q factor and res-onance width become finite at the BIC conditions due to ab- d ca be f FIG. 1: Top: Schematic of light scattering by a metasurface. Bottom:Designs of unit cells of metasurfaces with a broken in-plane inversionsymmetry of constituting meta-atoms supporting sharp resonances,analysed in Refs. [4–13]. sorption, size effects [23], and other perturbations [24]. Nev-ertheless, the BIC-inspired mechanism of light localizationmakes possible to realize high- Q quasi-BIC in optical cavi-ties and photonic crystal slabs [18, 20, 22], coupled opticalwaveguides [25–27], and even isolated subwavelength dielec-tric particles [28].In this Letter, we reveal that sharp spectral resonances re-cently reported and even observed for many types of seem-ingly different plasmonic and dielectric metasurfaces origi-nate in fact from the powerful concept of BIC, and this find-ing allows us to predict and engineer high- Q resonances innanophotonics. We demonstrate that true BICs transform into a r X i v : . [ phy s i c s . op ti c s ] S e p T r an s m i ss i on
01 1563 1565 1567 1569 1571
Wavelength, nm
BIC θ= W a v e l eng t h , n m
10 20 30 4016001560152014801440 W a v e l eng t h , n m θ , deg
00 10 1 | E || H | T c bd Eigenmode analysis quasi-BIC
BIC quasi-BICBIC quasi-BIC
Transmission
HE k yzx a θ Si FIG. 2: (a) A square lattice of tilted silicon-bar pairs with a design of the unit cell. Parameters: the period is 1320 nm, bar semi-axes are 330nm and 110 nm, height is 200 nm, distance between bars is 660 nm. (b) Eigenmode spectra and transmission spectra with respect to pumpwavelength and angle θ . Error bars show the magnitude of the mode inverse radiation lifetime. (c) Evolution of the transmission spectra vs.angle θ . Spectra are relatively shifted by 1.5 units. (d) Distribution of the electric and magnetic fields for BICs and quasi-BICs. quasi-BICs when the in-plane inversion symmetry of a unitcell becomes broken, and we derive the universal formula forthe Q factor as a function of the asymmetry parameter. We de-velop a novel analytical approach to describe light scatteringby arrays of meta-atoms based on the explicit expansion of theGreen’s function of an open system into eigenmode contribu-tions, and demonstrate rigorously that reflection and transmis-sion coefficients are linked to the conventional Fano formula.We prove that the Fano parameter becomes ill-defined at theBIC condition which corresponds to the collapse of the Fanoresonance [29, 30]. Our findings pave the way towards a novelapproach to the engineering of resonances in meta-optics andnanophotonics.To gain a deeper insight into the physics of BICs in meta-surfaces with in-plane asymmetry, we focus on the design re-cently suggested for biosensing [4], namely, a square array oftilted silicon-bar pairs shown in Fig. 2(a). For calculations,we consider a homogenous background with permittivity .We calculate both the eigenmode and transmission spectra asshown in Fig. 2(b). An asymmetry parameter is introducedthrough the angle θ between the y -axis and the long axis ofthe bar. The ideal (lossless and infinite) structure supports atrue optical BIC at θ = 0 ◦ [31]. Such an ideal BIC is un-stable against perturbations that break the in-plane symmetry ( x, y ) → ( − x, − y ) , and it transforms into a quasi-BIC with afinite Q factor.The eigenmode and transmission spectra are shown in Fig. 2(b) as functions of θ , where T = | t | is the transmit-tance and t is the amplitude of the transmitted wave. We ob-serve that the BIC with infinite Q factor at θ = 0 ◦ transformsinto a high- Q quasi-BIC, whose radiation loss increases with θ . The detailed transmission spectra shown in Fig. 2(c) exhibita narrow dip that vanishes when the pair becomes symmetric,which confirms the results of the eigenmode analysis. Fig-ure 2(d) demonstrates similarity of the distributions of electricand magnetic fields in BICs and quasi-BICs within a unit cell.Analysis shows that the BIC carries a topological charge of 1(see Supplemental Material).We analyze the transmission spectra of such metasurfacesand prove rigorously that it can be described by the classicalFano formula, and the observed peak position and linewidthcorrespond exactly to the real and imaginary parts of the fre-quency of the eigenmode. While the analytical solution ofMaxwell’s equations does not exist, the description of thetransmission T with the Fano formula is still widely usedby introducing the Fano parameter phenomenologically [32].Here, we demonstrate the explicit correspondence between theFano lineshape of the transmission spectra and properties ofthe eigenmode spectra. It is worth mentioning that previouslythe Fano formula for BICs in periodic photonic structures wasdicussed or obtained only for special assumptions [17, 20, 33].To derive an analytical expression for light transmission,we expand the transmitted field amplitude into a sum of in-dependent terms where each term corresponds to an eigen- Asymmetry parameter α S Δ s α = sin θα = Δ s / S ba θ add remove Perturbation Δ LL L Δ L α = Δ L / L α = Δ L / L R ad i a t i v e Q f a c t o r FIG. 3: Effect of in-plane asymmetry on the radiative Q factor ofquasi-BICs. (a) Dependence of the Q factor on the asymmetry pa-rameter α for all designs of symmetry-broken meta-atoms shown inFigs. 1(a-f), in log-log scale. (b) Definitions of the asymmetry pa-rameter α for some of the structures. mode of the photonic structure. This becomes possible byapplying the recently developed procedure allowing for rigor-ous characterization of the Green’s function and, therefore, theeigenmode spectra of open optical resonators [34]. The eigen-modes of a metasurface are treated as self-standing resonatorexcitations with a complex spectrum describing both the res-onant frequencies ω and inverse lifetimes γ . Straightforwardbut rather cumbersome calculations (see Supplemental Mate-rial) reveal that the frequency dependence of the transmission T is described rigorously by the Fano formula, and the Fano parameters are expressed explicitly through the material andgeometrical parameters of the metasurface and dimensionlessfrequency Ω = 2( ω − ω ) /γ , T ( ω ) = T q ( q + Ω) + T bg ( ω ) . (1)Here q is the Fano asymmetry parameter, T and T bg describethe smooth background contribution of non-resonant modesto the resonant peak amplitude and the offset, respectively(see the explicit expressions in Supplemental Material [35]).Remarkably, the exact formula shows that the parameter q inEq. (1) for BICs supported by metasurfaces with unbroken in-plane symmetry becomes ill-defined, which corresponds to acollapse of the Fano resonance when any features in the trans-mission spectra disappear, and the resonant mode is trans-formed into a ” dark mode ” .Next, we describe analytically the behavior of the radia-tive Q factor of the quasi-BIC as a function of θ shown inFigs. 2(b-c). We consider the radiation losses as a perturba-tion which is a natural approximation valid when θ remainsrelatively small. Then, the inverse radiation lifetime γ rad canbe calculated as a sum of radiation losses into all open radia-tion channels. We focus on the quasi-BICs with the frequen-cies below the diffraction limit for which only open radiationchannels represent the zeroth-order diffraction. Then γ rad ofthe quasi-BIC takes the form (see Supplemental Material), γ rad c = | D x | + | D y | , (2a) D x,y = − k √ S (cid:20) p x,y ∓ c m y,x + ik Q xz,yz (cid:21) . (2b)Here k = ω /c , S is the unit cell area, p , m and ˆQ arethe electric dipole, magnetic dipole and electric quadrupolenormalized moments of a unit cell, which depend on θ . Theamplitudes D x,y govern the overlap coefficients between themode profile inside the photonic structure and the profile ofvertically propagating free-space modes of two orthogonal po-larizations, respectively. Thus, Eqs. (2a-b) show that for atrue BIC mode symmetry mismatch leads to zero D x,y andvanishing radiation losses [40]. In other words, the electricfield components E x , E y of a BIC are odd with respect to theinversion of coordinates ( x, y ) → ( − x, − y ) so γ rad = 0 .For quasi-BICs, we perform straightforward transformationsof Eqs. (2a-b) (see Supplemental Material) to show that theradiative quality factor Q rad = ω /γ rad depends on θ as Q rad ( θ ) = Q [ α ( θ )] − , (3)where α = sin θ and Q is a constant independent on θ .In general, Eq. (3) remains valid for metasurfaces placed ona substrate as long as the quasi-BIC frequency is below thediffraction limit of the substrate [24].Next, we demonstrate that the quadratic dependence de-fined by Eq. (3) represents a universal behavior of the Q factorof a quasi-BIC mode as a function of the asymmetry parame- Q Campione, 2016Vabishchevich, 2018 Tittl, 2018 Lim, 2018Singh, 2011 A sy mm e t r y pa r a m e t e r α Wavelength, μ m DIELECTRIC METAL
Forouzmand, 2017
Scaling factor FIG. 4: Map of operating wavelengths and quality factors for siliconmetasurfaces with tilted-bar pairs by varying the orientation of thebars ( α = sin θ ) and by scaling the metasurface. The radiative partof the total Q factor is evaluated using Eq. (3), the non-radiative partis taken equal . All calculations are confirmed by direct numeri-cal simulations with realistic dispersion of silicon (see SupplementalMaterial). Geometric scaling factor is shown in the upper horizontalaxis. Colored rectangles correspond to the structures considered inthe previous studies, see Fig. 1. ter for all dielectric and plasmonic metasurfaces with broken-symmetry meta-atoms. We introduce the generalised asym-metry parameter α , which has distinct definitions for differentstructures but takes values between 0 and 1. We derive Eq. (3)using the second-order perturbation theory for open electro-magnetic systems and confirm the result by independent cal-culations of the eigenmode and reflectance spectra for all de-signs presented in Figs. 1(b-f) (see Supplemental Material).Since plasmonic metasurfaces possess significant absorption,we extract the bare radiative Q factor by evaluating the inverseradiative lifetime γ rad being a difference between the total in-verse lifetime γ and non-radiative damping rate evaluated at α = 0 .Figure 3(a) shows a direct comparison of the values of theradiative Q factor of quasi-BICs as functions of the asym-metry parameter α for dielectric and plasmonic metasurfaceswith various broken-symmetry meta-atoms in the unit cell.All curves are shifted relatively in the vertical direction tooriginate from the same point. As can be noticed fromFig. 3(a), for small values of α the behavior of Q rad for allmetasurfaces is clearly inverse quadratic. Importantly, formost of the structures the law α − is valid beyond applica-bility of the perturbation theory. Figure 3(b) introduces thedefinition of the asymmetry parameter α for different meta-surface designs.The quadratic scalability of the Q factor of quasi-BICs combined with linear scalability of Maxwell’s equations sug-gests a straightforward way of smart engineering of photonicstructures with the properties on demand. As an example, wefocus on a design of tilted silicon-bar pairs and suggest a verysimple algorithm for a design of metasurfaces with a widerange of Q factors and operating wavelengths ranging fromfrom visible to THz. First, since dispersion of the refractiveindex for silicon is relatively weak, we can tune the operat-ing wavelength from 0.5 µ m to 300 µ m by a linear geometricscaling of the structure in all dimensions. Second, we can con-trol the mode radiative Q factor in a wide range of parametersby changing the asymmetry parameter α = sin θ according toEq. (3). The total Q factor of a quasi-BIC is limited by ab-sorption, which can be estimated as Re( ε ) / Im( ε ) . For siliconin the range from near-IR to THz, it is more than .Using this approach, we calculate the dependence of to-tal Q factor of the quasi-BIC vs. operating wavelength andasymmetry parameter for a square lattice of tilted silicon-barpairs, and summarize the results in Fig. 4. For all calculationsdone for this map, we make only one numerical simulationwith fixed material parameters to obtain the value of Q re-quired for use of Eq. (3), then exploit the advantages of thegeometrical scaling combined with rotation of bars. We ob-serve that the values of available Q factors can be achieved ina broad range from up to for each wavelength domain,by using the same material and design. The colored rectanglesshow the range of Q factors and operating wavelengths re-ported in the previous studies of metasurfaces with symmetry-broken meta-atoms [4, 8–11, 13]. We verify the applicabil-ity of the proposed analytical approach by three-dimensionalelectromagnetic simulations with the silicon dispersion by us-ing the finite-element method in COMSOL, and the results arein Supplemental Material. Both approaches agree well, thusjustifying the validity of our analytical scaling method.We believe our approach based on the BIC concept candescribe many other cases of symmetry-broken metasurfacesand also photonic crystral slabs, studied earlier with differ-ent applications in mind [41–45]. Also, our approach can behelpful to get a deeper physical insight into many other prob-lems in optics, including dark bound states in dielectric in-clusions coupled to the external waves by small non-resonantmetallic antennas [46] and electromagnetically induced trans-parency [47]. We argue that almost any problem involving theso-called ” dark states ” can find its rigorous formulation withthe powerful theory of BIC resonances.In conclusion, we have demonstrated that high- Q reso-nances recently observed in metasurfaces composed of dis-similar meta-atoms with broken in-plane inversion symmetryare associated with the concept of bound states in the contin-uum. We have proven rigorously a direct link between pe-culiarities of the transmission spectra, Fano resonances, andthe existence of high- Q quasi-BIC resonances. We have ex-plained analytically the variation of the Q factor with a changeof the unit-cell asymmetry that paves the way towards smartengineering of sharp resonances in meta-optics for nanolasers,light-emitting metasurfaces, optical sensors, and ultrafast ac-tive metadevices.The authors acknowledge a financial support from the Aus-tralian Research Council, the AvH Foundation and the Rus-sian Science Foundation (17-12-01581), and useful discus-sions with H. Altug, H. Atwater, F. Capasso, A. Krasnok, S.Kruk, Th. Pertsch, M. Rybin, R. Singh, V. Tuz and N. Zhe-ludev. [1] N. Yu and F. Capasso, Flat optics with designer metasurfaces,Nat. Mater. , 139 (2014).[2] A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y.S. Kivshar, and B. Lukyanchuk, Optically resonant dielectricnanostructures, Science , aag2472 (2016).[3] S. Kruk and Y. Kivshar, Functional meta-optics and nanopho-tonics governed by Mie resonances, ACS Photonics , 2638(2017).[4] A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.Y. Choi, D.N. Neshev,Y.S. Kivshar, and H. Altug, Imaging-based molecular barcod-ing with pixelated dielectric metasurfaces, Science , 1105(2018).[5] M. Liu, D. A. Powell, R. Guo, I. V. Shadrivov and Y. S.Kivshar, Polarization-induced chirality in metamaterials via op-tomechanical interaction, Adv. Opt. Mater. , 1600760 (2017).[6] V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K.L. Domina, S.Xu, H. Wang, and H.-B. Sun, High-quality trapped modes inall-dielectric metamaterials, Opt. Express , 2905 (2018).[7] V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, andN. I. Zheludev, Sharp trapped-mode resonances in planar meta-materials with a broken structural symmetry, Phys. Rev. Lett. , 147401 (2007).[8] A. Forouzmand and H. Mosallaei, All-dielectric C -shapednanoantennas for light manipulation: Tailoring both magneticand electric resonances to the desire, Adv. Opt. Mater. ,1700147 (2017).[9] S. Campione, S. Liu, L. I. Basilio, L. K. Warne, W. L. Langston,T. S. Luk, J. R. Wendt, J. L. Reno, G. A. Keeler, I. Brener,and M. B. Sinclair, Broken symmetry dielectric resonators forhigh quality factor Fano metasurfaces, ACS Photonics , 2362(2016).[10] P. P. Vabishchevich, S. Liu, M. B. Sinclair, G. A. Keeler, G. M.Peake, and I. Brener, Enhanced second-harmonic generation us-ing broken symmetry III-V semiconductor Fano metasurfaces,ACS Photonics , 1685 (2018).[11] W.X. Lim and R. Singh, Universal behaviour of high-Q Fanoresonances in metamaterials: Terahertz to near-infrared regime,Nano Convergence , 5 (2018).[12] V.V. Khardikov, E.O. Iarko, and S.L. Prosvirnin, Trapping oflight by metal arrays, J. Opt. , 045102 (2010).[13] R. Singh, I.A. Al-Naib, Y. Yang, D. Roy Chowdhury, W. Cao,C. Rockstuhl, T. Ozaki, R. Morandotti, and W. Zhang, Observ-ing metamaterial induced transparency in individual Fano res-onators with broken symmetry, Appl. Phys. Lett. , 201107(2011).[14] J. Von Neuman and E. Wigner, ¨Uber merkw¨urdige diskreteEigenwerte, Phys. Z. , 467 (1929).[15] H. Friedrich and D. Wintgen, Interfering resonances and boundstates in the continuum, Phys. Rev. A , 3231 (1985).[16] R. Parker, Resonance effects in wake shedding from parallelplates: some experimental observations, J. Sound Vibr. , 62 (1966).[17] D.C. Marinica, A.G. Borisov, and S.V. Shabanov, Bound statesin the continuum in photonics, Phys. Rev. Lett. , 183902(2008).[18] E.N. Bulgakov and A.F. Sadreev, Bound states in the continuumin photonic waveguides inspired by defects, Phys. Rev. B ,075105 (2008).[19] R.F. Ndangali and S.V. Shabanov, Electromagnetic bound statesin the radiation continuum for periodic double arrays of sub-wavelength dielectric cylinders, J. Math. Phys. , 102901(2010).[20] C.W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S.G. Johnson, J.D.Joannopoulos, and M. Soljaˆci´c, Observation of trapped lightwithin the radiation continuum, Nature , 188 (2013).[21] F. Monticone and A. Al´u, Embedded photonic eigenvalues in3D nanostructures, Phys. Rev. Lett. , 213903 (2014).[22] M. Rybin and Y. Kivshar, Supercavity lasing, Nature , 164(2017).[23] M. A. Belyakov, M. A. Balezin, Z. F. Sadrieva, P. V. Kapi-tanova, E. A. Nenasheva, A. F. Sadreev, and A. A. Bogdanov,Experimental observation of symmetry protected bound statein the continuum in a chain of dielectric disks, arXiv preprint,arXiv:1806.01932 (2018).[24] Z.F. Sadrieva, I.S. Sinev, K.L. Koshelev, A. Samusev, I.V. Iorsh,O. Takayama, R. Malureanu, A.A. Bogdanov, and A.V. Lavri-nenko, Transition from optical bound states in the continuum toleaky resonances: Role of substrate and roughness, ACS Pho-tonics , 723 (2017).[25] Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Sza-meit, and M. Segev, Experimental observation of optical boundstates in the continuum, Phys. Rev. Lett. , 183901 (2011).[26] M.I. Molina, A.E. Miroshnichenko, and Y.S. Kivshar, Surfacebound states in the continuum, Phys. Rev. Lett. , 070401(2012).[27] G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S.Longhi, Observation of surface states with algebraic localiza-tion, Phys. Rev. Lett. , 220403 (2013).[28] M.V. Rybin, K.L. Koshelev, Z.F. Sadrieva, K.B. Samusev, A.A.Bogdanov, M.F. Limonov, Y.S. Kivshar, High- Q supercavitymodes in subwavelength dielectric resonators, Phys. Rev. Lett. , 243901 (2017).[29] L. Fonda, Bound states embedded in the continuum and the for-mal theory of scattering, Ann. Phys. , 123 (1963).[30] C.S. Kim, A.M. Satanin, Y.S. Joe, and R.M. Cosby, Resonanttunneling in a quantum waveguide: Effect of a finite-size attrac-tive impurity, Phys. Rev. B , 10962 (1999).[31] B. Zhen, C.W. Hsu, L. Lu, A.D. Stone, and M. Soljaˆci´c, Topo-logical nature of optical bound states in the continuum, Phys.Rev. Lett. , 257401 (2014).[32] M.F. Limonov, M.V. Rybin, A.N. Poddubny, and Y.S. Kivshar,Fano resonances in photonics, Nat. Photonics , 543 (2017).[33] C. Blanchard, J. P. Hugonin, C. Sauvan. Fano resonances inphotonic crystal slabs near optical bound states in the contin-uum. Phys. Rev. B , 155303 (2016).[34] T. Weiss, M. Mesch, M. Schferling, H. Giessen, W. Lang-bein, E. A. Muljarov. From dark to bright: First-order perturba-tion theory with analytical mode normalization for plasmonicnanoantenna arrays applied to refractive index sensing. Phys.Rev. Lett. , 237401 (2016).[35] See Supplemental Material [link] for details on the rigorousderivation of the classical Fano formula for the transmissioncoefficient of a metasurface, the derivation of Eqs. (2)-(3), theresults on eigenmode and reflection spectra dependence on theasymmetry parameter α for designs shown in Figs.1(b-f), the comparison of the analytical approach shown in Fig. 4 with nu-merical simulations using the realistic dispersion of silicon andthe calculation of the BIC topological charge, which includesRefs. [36–39].[36] F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers,Quasinormal-mode expansion of the scattering matrix, Phys.Rev. X , 021035 (2017).[37] A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Lukyanchuk,and B. N. Chichkov, Optical response features of Si-nanoparticle arrays, Phys. Rev. B , 045404 (2010).[38] J. S. T. Gongora, G. Favraud, and A. Fratalocchi, Funda-mental and high-order anapoles in all-dielectric metamateri-als via Fano-Feshbach modes competition, Nanotechnology ,104001 (2017).[39] A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov,Optical theorem and multipole scattering of light by arbitrarilyshaped nanoparticles. Phys. Rev. B , 205434 (2016).[40] J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos,M. Soljaˆci´c, and O. Shapira, Observation and differentiationof unique high- Q optical resonances near zero wave vectorin macroscopic photonic crystal slabs, Phys. Rev. Lett. ,067401 (2012).[41] E Pshenay-Severin, A. Chipouline, J. Petschulat, U. H¨ubner, A.T¨unnermann, and T. Pertsch, Optical properties of metamateri- als based on asymmetric double-wire structures, Opt. Express , 6269 (2011).[42] C. Wu, N. Arju, G. Kelp, J.A. Fan, J. Dominguez, E. Gonzales,E. Tutuc, I. Brener, and G. Shvets, Spectrally selective chiralsilicon metasurfaces based on infrared Fano resonances, Nat.Commun. , 3892 (2014).[43] A. Jain, P. Moitra, T. Koschny, J. Valentine, and C.M. Souk-oulis, Electric and magnetic response in dielectric dark statesfor low loss subwavelength optical meta atoms, Adv. Opt.Mater. , 1431 (2015).[44] Y. Zhang, W. Liu, Z. Li, Z. Li, H. Chang, S. Chen, and J. Tian,High-quality-factor multiple Fano resonances for refractive in-dex sensing, Opt. Lett. , 1842 (2018).[45] V. Liu, M. Povinelli, S. Fan, Resonance-enhanced optical forcesbetween coupled photonic crystal slabs, Opt. Express , 21897(2009).[46] A. Jain, P. Tassin, T. Koschny, and C.M. Soukoulis, Largequality factor in sheet metamaterials made from dark dielectricmeta-atoms, Phys. Rev. Lett. , 117403 (2014).[47] J. Hu, T. Lang, Z. Hong, C. Shen, and G. Shi, Comparisonof Electromagnetically-induced transparency performance inmetallic and all-dielectric metamaterials, J. Lightwave Technol.36