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[岩長 祐伸](https://orcid.org/0000-0002-8930-6940)

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[Comprehensive Analysis of Optical Resonances and Sensing Performance in Metasurfaces of Silicon-Nanogap Unit](https://mdr.nims.go.jp/datasets/36b5d6e9-df1c-4ef7-8830-d63b871d7114)

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Comprehensive Analysis of Optical Resonances and Sensing Performance in Metasurfaces of Silicon Nanogap UnitCitation: Iwanaga, M. ComprehensiveAnalysis of Optical Resonances andSensing Performance in Metasurfacesof Silicon Nanogap Unit. Photonics2024, 11, 1053. https://doi.org/10.3390/photonics11111053Received: 14 October 2024Revised: 4 November 2024Accepted: 8 November 2024Published: 10 November 2024Copyright: © 2024 by the author.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).ArticleComprehensive Analysis of Optical Resonances and SensingPerformance in Metasurfaces of Silicon Nanogap UnitMasanobu IwanagaResearch Center for Electronic and Optical Materials, National Institute for Materials Science (NIMS), 1-1 Namiki,Tsukuba 305-0044, Japan; iwanaga.masanobu@nims.go.jpAbstract: Metasurfaces composed of silicon nanogap units have a variety of optical resonances,including bound states in the continuum (BIC). We show comprehensive numerical results onmetasurfaces of Si-nanogap units, analyze the optical resonances, and clarify optically prominentresonances as well as symmetry-forbidding resonances that are the BIC, based on the numericalanalyses of optical spectra and resonant electromagnetic field distributions. Introducing asymmetryin the unit cell, the BIC become optically allowed, being identified as magnetic dipole, electricquadrupole, and magnetic quadrupole resonances. Moreover, the optical resonances are examinedin terms of refractive index sensing performance. A pair of the resonances associated with electricfield localization at the nanogap was found to be sensitive to the refractive index in contact with themetasurfaces. Consequently, the gap mode resonances are shown to be suitable for a wide range ofrefractive index sensing over 1.0–2.0.Keywords: all-dielectric metasurface; silicon nanogap; BIC; optical sensing; refractive index1. IntroductionMetasurfaces have opened the way for diverse optical properties and applications.In addition to light-wave manipulations employing the complex subwavelength struc-tures [1–7], various types of resonances have been exploited to realize prominent effectsthat are helpful for practical applications such as metalenses [8–12], color splitting topixel [13–15], liquid-crystal-based active responses [16], and biosensing based on resonanceshift [17–19] and on fluorescence detection [20–23].Among many effects in all-dielectric metasurfaces, one of the popular resonances isquasi-bound states in the continuum (qBIC). Structural asymmetry allows us to access theresonances of qBIC that are forbidden in symmetric structures. Although this concept wasintroduced in an electronic system almost a hundred years ago [24], a similar effect to theelectronic system was shown in a nanophotonic system, which was a photonic crystal,about ten years ago [25], which stimulated nanophotonics studies on the qBIC.As applications of the qBIC in nanophotonics, refractive index sensing [26–28] andnonlinear effects [29–31] were explored to date. On the physical limit of refractive indexsensing using periodic structures, a formula for the resolution in the units of nm/refractiveindex unit (RIU) was recently shown [32] such that∆λ∆n= P (1)where λ, n, and P denote wavelength of light, refractive index of surrounding medium,and periodic length, respectively. Equation (1) indicates that the upper limit of refractiveindex sensing is determined by the periodic length P.Figure 1 illustrates structural features of metasurfaces addressed in this study. InFigure 1a,b, Si-nanogap units are symmetric and asymmetric in the unit cell on the xyplane, respectively. Keeping gap width g constant, we vary y-width, y1 and y2, whilePhotonics 2024, 11, 1053. https://doi.org/10.3390/photonics11111053 https://www.mdpi.com/journal/photonicshttps://doi.org/10.3390/photonics11111053https://doi.org/10.3390/photonics11111053https://creativecommons.org/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/photonicshttps://www.mdpi.comhttps://orcid.org/0000-0002-8930-6940https://doi.org/10.3390/photonics11111053https://www.mdpi.com/journal/photonicshttps://www.mdpi.com/article/10.3390/photonics11111053?type=check_update&version=2Photonics 2024, 11, 1053 2 of 15keeping the sum of y1 + y2 + g constant. In Figure 1b, a condition of y1 > y2 results in−y-direction offset of the center line in the nanogap (dotted line). We define the absolutevalue of the offset as α, which is an asymmetric parameter, and quantitatively characterizethe asymmetry in the unit cell, and can write it down such thatα = |y1 − y2|/2. (2)Introducing the asymmetry along only the y axis, the definition in Equation (2) is simpleand explicitly expressed. Also, experimental tests in the future will be easy to conduct.Note that the asymmetric parameters for qBIC are usually normalized using a reasonablereference length to make the parameters dimensionless; for example, the offset can benormalized by the length of unit cell; however, to make the physical meaning of α clear, wesimply define the α in Equation (2). Thus, the present α means the offset of the gap centerfrom the center of unit cell; that is, α = 1 means that the offset is 1 nm.Figure 1(a)-(c) Figure_drawing_for_Photonics_2024.ai(c)xyzkin EinSiSiO2(a) (b)xySiSiSiSiαy1y2(d)SigFigure 1. Schematics of metasurface structure and optical configuration in this study. (a) xy-sectionview of unit cell of metasurface of Si-nanogap unit (purple). The gap g is indicated with a both-endarrow. The center line along the x axis is drawn with a broken line, which goes across the central pointof the unit cell. (b) Asymmetry introduced in the unit cell is indicated by a quantity α, which is offsetfrom the center line (broken line) to the middle line (dotted line) in the gap. (c) Three-dimensionalillustration of unit cell of metasurface of an asymmetric Si-nanogap pair. Optical configuration isshown together. The unit cells in (a–c) are set to have dimensions of 600 × 600 nm2 in the xy plane.Length of the Si nanoblock along the x axis is 400 nm and the sum of y1 + y2 + g is 400 nm. Height ofthe Si nanoblock is set to 200 nm. (d) Three-dimensional illustration of a metasurface of periodic arrayof asymmetric Si-nanogap units, which are assumed to be infinitely periodic in the computations.In Figure 1c, a 3D illustration of the unit cell of the metasurface (Figure 1b) is showntogether with spatial xyz coordinates and an optical configuration. Wavevector kin denotesthe wavevector of the incident plane wave, and vector Ein represents electric field (orpolarization) of the incidence. In this study, we mainly study normal incidence because theasymmetry enables us to access qBIC and allows us to explore various optical resonances.The metasurface is schematically illustrated in Figure 1d, which is assumed to be infinitelyperiodic in the computations. In practice, the metasurface is designed to be fabricatedthrough nanolithography on a Si-on-insulator (SOI) wafer.2. Materials and MethodsThe metasurfaces in this study are assumed to comprise dielectrics of Si and SiO2.SOI substrates are assumed to be used as base materials to fabricate the Si metasurfaces.In particular, the thickness of the top SOI layer, the middle buried oxide layer that isSiO2, and the base Si layer are set to 200 nm, 1000 nm, and 725 µm, respectively, inaccordance with SOI substrates available in reality. The permittivity of crystalline Si wastaken from the literature [33] and that of SiO2 was set to a representative value of 2.1316 inthe photon-energy range of present interest. Using these permittivities, realistic simulationswere conducted.Photonics 2024, 11, 1053 3 of 15Numerical implementation was conducted based on a method combining rigorouslycoupled-wave analysis (RCWA) [34], which solves Fourier-transformed Maxwell equationsfor infinitely periodic objects in the frequency domain, with the scattering-matrix (S-matrix)algorithm [35]. The S-matrix algorithm makes numerical calculations stable even for stackedlayer structures. The combined numerical method is a frequency-domain method andsuitable to precisely compute optical spectra such as reflectance and transmittance spectraof periodic structures; indeed, the method has been applied to reproduce experimentaloptical spectra in various periodically structured materials such as metamaterials andmetasurfaces [36–41]. The code executing the RCWA and S-matrix algorithm run onsupercomputers in a multiparallel implementation (MPI) manner. Optical spectra werecomputed in the MPI manner and therefore the net runtime was almost equal to totalruntime/(number of MPI), being substantially reduced. Also, the combined method enablesus to output electromagnetic (EM)-field distributions, which is helpful to understandfeatures of optical resonances in the metasurfaces. The RCWA is mathematically exact tothe Maxwell equation under infinite expansion of the Fourier series. In reality, the RCWAprovides approximate solutions by truncating the Fourier series at a finite order. We setthe truncation order at ±15 and more in this study and did not see changes in the opticalspectra. Note that, since we considered 2D periodic systems, the total truncation order was31 × 31 and more.3. Results3.1. Asymmetric Metasurfaces and qBIC ModesFigure 2a shows a series of reflectance spectra of metasurfaces of Si-nanogap units,illustrated Figure 1b–d. The reflectance is normalized and takes values in a range from 0 to1. The reflectance spectra are displayed with offset; accordingly, the zero lines for the offsetspectra are drawn with thin gray lines.The unit cell is set to be 600 × 600 nm2 in the xy plane, and the width of Si nanoblocksalong the x axis is set to 400 nm. The height of Si nanoblocks along the z axis is 200 nm. Thegap g between the Si nanoblocks is set to 50 nm, and the sum of y1 + y2 is set to 350 nm;consequently, the unit of Si nanoblocks has a dimension of 400 × 400 nm2 in the xy plane,being set at the center of the unit cell.Introducing the asymmetry represented by α, we can allow for the qBIC modes toappear in optical spectra, such as reflectance spectra in Figure 2a. The qBIC modes areindicated with MD, MQ, and EQ that denote magnetic dipole, magnetic quadrupole, andelectric quadrupole, respectively, and are indicated with arrows. They appear for α ≥ 1though the signals at α = 1 are quite small. The refractive index n in the incident layerand gaps between the Si nanoblocks is set to 1.0. The Si-nanogap unit cell is anisotropicfor x and y axes; accordingly, the reflectance spectra are also anisotropic. The x- and y-polarized reflectance spectra are shown from bottom to top with offset, using black-to-blueand red-to-brown colors, respectively. The parameter α varies from 0 to 25. The series ofreflectance spectra exhibit definite anisotropy for the incident polarizations and variousresonant responses in the shapes of the spectra. In the photon energy range over 1.41 eV,diffraction-associated resonant modes appear. We here focus on the resonances below 1.4 eV,which are diffraction-free and belong to a subwavelength range at the normal incidence.We briefly mention a general aspect in optical systems (e.g., Figure 1c) that containperiodic structures and plain waves that are incoming and outgoing. Even when theperiodic systems have inversion symmetry in the xy plane, oblique incidence to the xyplane yields asymmetry in the optical responses; consequently, qBIC modes can be observedunder oblique incidence. In this study, we mainly study the normal incidence and make theplain waves symmetric in the xy plane; therefore, at the normal incidence, the qBIC modesare observed only for the metasurface of asymmetric units (i.e., α > 0) that do not haveinversion symmetry in the xy plane. Note that, under the oblique incidence, the opticalsystems are asymmetric for the xy plane, and that the origin of asymmetry is generallyindistinguishable between the structural units and optical configurations.Photonics 2024, 11, 1053 4 of 15Figure 2(a) Y240819e-f, Y240820a,c,d. Offset 0, 1, 3, 10, 25 nm.(b) (c)(b)612 4 6102 4 6100α100246100024MD612 4 6102 4 6100α100246100024EQ(c)(a)MDEQMQFigure 2. Optical resonances emerged in the Si-nanogap unit metasurfaces. (a) Computed reflectancespectra at two incident polarizations, Ein ∥ x and Ein ∥ y for various asymmetric parameters α,defined in Figure 1, and a fixed gap g of 50 nm. The reflectance spectra are shown with offset,changing color from black to blue for Ein ∥ x and from red to brown for Ein ∥ y, in accordance withα = 0–25, respectively. EQ, MD, and MQ denote electric quadrupole, magnetic dipole, and magneticquadrupole, respectively, indicated by arrows. (b,c) Asymmetric parameter α and estimated quality(Q) factors of the MD (closed blue circles) and EQ modes (red diamonds), respectively. The Q factoris defined as Equation (3). Black lines denote fitted lines using a power function (see the text).The reflectance spectra in Figure 2a have several peaks with broad bands at both x andy polarizations. The peaks coexist in the photon-energy range with interference comingfrom the multilayer structure of the SOI substrate; the interference is easy to be verifiedat a low energy range between 1.1 and 1.2 eV at x and y polarizations, respectively. Thesecoexisting properties make the spectra complicated.In addition, the spectra in Figure 2a include qBIC resonances with narrow line widths;the EQ modes are located at 1.2717 and 1.2809 eV for x polarization, the MD modes at1.236 and 1.280 eV for y polarization, and the MQ modes at 1.403 eV for y polarization,as indicated by arrows. For smaller α, the resonant signals become smaller and finallydisappear at α = 0 for both polarizations, suggesting that these signals originate from qBIC.These resonances are quantified using quality (Q) factor, defined such thatQ =ω0∆ω(3)where ω0 and ∆ω denote resonant frequency and full width at the half maximum (FWHM)of the resonance in the frequency domain, respectively. In Figure 2b,c, the Q factors ofMD (blue closed circle) and EQ modes (red diamond) are plotted for the parameter α,respectively. Although two MD and EQ lines appear, we chose the lower-energy resonance.The reason why the two MD modes appear is described with examining resonant EM-fielddistributions below. As the parameter α becomes smaller, the Q factors increase and exceedPhotonics 2024, 11, 1053 5 of 151000 at α = 1. The Q factor was fitted using a power function, Q = Aαn + B, where Aand B are constants, being found that n = −1.22 and −0.70 for the MD and EQ modes,respectively. The fitted power functions are shown with black lines. If radiative lossis simply assumed to evaluate the Q factors, a relation of Q ∝ α−2 was suggested [42].However, nonradiative losses are able to occur, due to light absorption by the constituentmaterials and surface roughness that could give rise to in nanofabrication processes, andsubstantially reduce the large Q factors over 1000 [43]. In the present numerical evaluations,the absorption loss is explicitly incorporated based on the permittivity of crystalline Si inthe literature [33]. Furthermore, the high-Q modes are not isolated from the broad-bandinterference mode in the present metasurfaces; therefore, the EM-field confinement isconsidered to be reduced, though it is difficult to quantitatively evaluate the net amount ofdissipation stemming from the interference mode.Representative resonant EM-field distributions of qBIC are shown in Figure 3, wherethe asymmetric parameter α is chosen to be 10. EQ-mode EM-field distributions are shownin Figure 3a–c, where |E|, Re(Ez), and |H| components are displayed, respectively. Theboundaries of Si nanoblocks are shown with dotted white lines in Figure 3c. From the EMfields, several features are observed, as follows:1. EM-field intensity is significantly enhanced on the resonance. For the incident inten-sities of |Ein|2 = 1 and |Hin|2 = 1, the maxima of |E|2 and |H|2 reach 420 and 841in Figure 3a,c, respectively. Note that the nanogap does not contribute to the largeEM-field enhancement in this mode.2. The field pattern of Re(Ez) in Figure 3b indicates that the mode has quadratic oscil-lation on and around the Si-nanogap unit. Therefore, this mode is attributed to anEQ mode.3. Anisotropy is introduced in the unit structure along the y axis and results in theanisotropy of the EM-field distributions, as observed in Figure 3a–c.4. The resonant EM fields predominantly consist of near-field distributions in and aroundthe Si-nanogap unit. As a result, the imaginary parts of EM fields contribute to theenhanced intensities. This is verified from the fact that the maximum of |Re(Ez)| inFigure 3b is approximately two-fold smaller than that of |E| in Figure 3a.For y polarization, the qBIC modes are magnetic modes. Magnetic dipole resonance isvisualized in Figure 3d–f, where |E|, |H|, and a snapshot of Re(Hz) are shown, respectively.Due to the nanogap between the two Si nanoblocks, strong localization of electric fields inthe nanogap emerges, as shown in Figure 3d; accordingly, the magnetic fields are weaker atthe center of the unit cell, as seen in Figure 3e; however, they exhibit a modified magnetic-dipole field distribution that is mainly localized in the Si domain, as shown in Figure 3f.Thus, the EM-field distributions retain a feature of magnetic-dipole resonance.We here remark multipole approximations for the qBIC modes, which were reportedfrequently. First of all, it is to be noted that there is no analytical solution for the presentmetasurfaces that have complicated unit structures, in comparison with ideal spheres.Therefore, any trial to conduct multipole expansion for the metasurfaces will result innumerically approximated analyses. The issue is that it is difficult to evaluate the precisionof the approximation in a quantitative manner. As shown in Figure 3d–i, the MD modes aresignificantly modified due to the nanogap and another mode of the gap mode coexists.As aresult, if we should conduct the multipole expansion, the approximated multipolar valueswill be difficult to justify. Second, we already conducted and reported a multipole analysisfor the Si-nanogap unit using a finite-element method [28]. A similar result is expected andis not repeated here. Instead, the information on the EM-field distributions is shown inFigure 3.Photonics 2024, 11, 1053 6 of 15Figure 3(a-c) Y240821m_e1(d-f) Y240821n_e1(g-i) Y240821n_e4(a) 20.50(b) 9.5−9.5(c) 29.00(d) 57.30(e) 28.70(g) 19.80(h) 26.40xy(f)(i)4.2−4.211.0−11.0|E||E||E||H||H||H|HzHzEzFigure 3. Resonant electromagnetic (EM)-field distributions on the qBIC resonances. (a) xy-sectionview of |E| distribution at 1.272 eV. The xy axes are shown together and the xy sections were takenat the half height of the Si-nanogap pair. These settings are in common with the other panels inthis figure. (b) Snapshot of Re(Ez) component, corresponding to (a), presents a signature of electricquadrupole (EQ) mode. (c) |H| distribution, corresponding to (a). The incident polarization wasset to be Ein ∥ x in (a–c). (d–f) |E|, |H|, and Re(Hz) distributions at 1.236 eV, respectively. Themagnetic-field distribution shows a signature of magnetic dipole (MD) mode. (g–i) |E|, |H|, andRe(Hz) distributions at 1.402 eV, respectively. The magnetic-field distribution shows a signature ofmagnetic quadrupole (MQ) mode. The incident polarization was set to be Ein ∥ y in (d–i). The colorbars indicate values of resonantly enhanced EM fields, when the absolute values of incident fieldswere set to unity, that is, |Ein| = 1 and |Hin| = 1.In Figure 3d, the electric field is strongly localized in the nanogap and takes themaximum intensity at the center of x positions. In other words, the electric field is a singlenode in the nanogap along the x axis. We note that another MD mode in Figure 2a is locatedat a higher photon energy than this MD mode and has two maxima on the electric-fieldintensity in the nanogap; that is, it has two nodes in the nanogap. Thus, the two MD modescorrespond to the first and second electric-field nodes in the nanogap. We mention that thethird node is not induced because the length of the Si nanoblocks along the x axis is notenough. A similar type of resonances appears later (Section 3.2).Magnetic quadrupole resonance is visualized in Figure 3g–i that shows |E|, |H|, and asnapshot of Re(Hz), respectively. The electric and magnetic field distributions show fourminimum and maximum points in the Si domain, respectively, which indicates quadrupole.This feature is confirmed from the Re(Hz) distribution in Figure 3i.Summing up Figure 3 briefly, the qBIC modes are found to be EQ, MD, and MQ modes.They have resonantly enhanced EM fields, that reach hundred-to-thousand-fold intensity,compared to that of incidence.3.2. Resonant Shift for Refractive IndexTwo series of numerically calculated reflectance spectra of metasurfaces of asymmetricSi-nanogap units are shown in Figure 4a, where the asymmetric parameter α was set to10 in the unit cell. We note that the spectra for n = 1.0 are identical to those of α = 10 inFigure 2a and that y1 = 185 nm, y2 = 165 nm, and g = 50 nm.Photonics 2024, 11, 1053 7 of 15Figure 4(a) Y240821a, Y240922h-l (α=10)(b) Analyzed data in IGOR file, BIC_Q-factor_Si-pair_MSFs_2024.8-10.pxpOriginal data: Y240922h-l, Y241029a-e(c) Depicted on this pptx file.(a)(c)yzSiO2Sinnn2.01.91.81.71.61.51.41.31.21.11.0Refractive Index n1300120011001000900 1st, 285.9 nm/RIU 2nd, 231.1 nm/RIU(b)Figure 4. (a) Reflectance spectra dependent on refractive index n in the medium contacting themetasurface. Black and red curves show x- and y-polarized spectra, respectively, which are displayedwith offset. Red arrows indicate a pair of resonances responsive to the index n over a wide range ofn = 1.0–2.0. (b) Resonance shift of the pair of resonances. The first and second ones are shown withclosed and open red circles, respectively. The slopes, fitted using a linear function and shown withdashed black lines, approximating performance as optical sensors in the units of nm/RIU. See moredetails in the text. (c) Schematic of a yz-section-view metasurface and medium of refractive index n.Dotted lines section the unit cell along the y axis.Refractometric responses of the two resonances for n = 1.5, indicated by red arrows inFigure 4a, are plotted for the refractive index n over a range from 1.0 to 2.0 in Figure 4b,where the lower and higher energy peaks are shown with closed and open red circles,respectively. The wide range of the refractive index enables us to detect diverse materialsfrom gas to liquid. Note that the vertical axis in Figure 4b is in the units of nm, which isinversely proportional to eV. The peaks were fitted using a linear function; dashed blacklines represent the fitted line, reproducing the peak profiles over n = 1.0–1.5 in a goodapproximation; the first and second peaks plotted for the wavelength show refractometricresponses of 285.9 and 231.1 nm/RIU, respectively. One of the advantages in the range ofn = 1.0–1.5 is the approximately linear responsivity to the wide range.Generally, refractometric responses can be fitted using quadratic functions [32] becausedispersion relations between wavenumber and photon energy are mostly quadratic in 2Dperiodic systems. Indeed, the two peak profiles in Figure 4b can be traced for the wholerange of n = 1.0–2.0 using quadratic functions.Thus, quantitative tracing of the two peaksis possible. Other various nanostructures including metasurfaces have been proposedPhotonics 2024, 11, 1053 8 of 15as refractive-index sensors [26–28,32,43]; however, almost all of them do not guaranteesuch a wide-range responsivity and do have a narrow range of target, and are limited to anarrower range of targets than the present metasurface.The medium of refractive index n is assumed to exist in the incident layer and gapsbetween the Si nanoblocks, as drawn in Figure 4c. Practically, liquid or gas media areassumed to be the target of refractive-index sensing. In addition, large molecules suchas biomolecules can be detected as resonant shift, similarly to the conventional surfaceplasmon resonance method [44].Figure 5 shows EM-field distributions at the two reflectance peaks at y polarization,indicated by the red arrows in Figure 4a. The peaks correspond to resonances responsive tothe surrounding refractive index n. In Figure 5, the index n was set to 1.5.Figure 5(a-c) Y241011a(d-f) Y241011b(a)|E|xy(b)|E|(c)12.00|H|18.50yzxy11.8015.00(d)|E|(e)|E|(f)|H|Figure 5. Resonant EM-field distributions on the two resonances responsive to surrounding refractiveindex, which is set to n = 1.5. (a,b) xy- and yz-section views of |E| distributions, respectively. The yzsection cuts across the center of Si nanoblocks in the x direction. (c) |H| distribution in xy-sectionview. The photon energy is 1.124 eV in (a–c). (d–f) |E| and |H| distributions at 1.167 eV, displayed in asimilar manner to (a–c). The incident polarization was set to Ein ∥ y in common. These xy sections aretaken at the half-height of the Si-nanogap pair. The yz sections are through the center of the unit cell.The first resonance at 1.124 eV is visualized in Figure 5a–c. The |E| component ispresented in xy- and yz-section views, respectively. The color bar indicates the absolutevalues in common. A feature observed in the yz-section view is that the enhanced electricfields localize at the upper side of the nanogap. Since the xy section was taken at thehalf height of the Si nanoblocks, the enhanced electric fields are not seen; in contrast, themagnetic fields are enhanced on the xy section.The second resonance at 1.167 eV is visualized in Figure 5d–f. The way of presentationis similar to that in Figure 5a–c. Obviously, the electric fields are enhanced at the nanogap.The magnetic fields are mostly localized in the Si nanoblocks.From the enhanced electric-field distributions, the refractometric response in Figure 4ais most likely to originate from the locally enhanced electric fields at the nanogap in Figure 5,which are considered to enhance responses susceptible to the refractive index. Indeed, asshown in Figure A1, in a metasurface of symmetric Si nanoblock unit without any nanogap,resonances corresponding to the two resonances in the asymmetric metasurface (Figure 4)are not observed, and the refractometric response becomes small and is not found for awide range of n = 1.0–1.5.3.3. Nanogap Modes with Hugely Enhanced Electric FieldsReflectance spectra of an asymmetric metasurface with α = 5 and g = 10 nm areshown in Figure 6a. The sum of y1 + y2 + g is set to 400 nm. The unit cell is the samePhotonics 2024, 11, 1053 9 of 15dimension as that for Figure 2. The setting of incident wave is in common with thatfor Figure 3.Figure 6(a) Y240907a. V6a, n=1.0.(b-d) Y240920a(e-g) Y240920b(b) 66.70(c) 23.7011.4(d)(e)xy(f) (g)73.4030.406.7−6.7−11.4|E| |H| Hz|E| |H| Hz(a)Figure 6. Optical resonances in an asymmetric metasurface of narrow gap g = 10 nm. (a) Reflectancespectra at x and y polarizations for refractive index n = 1.0, shown with black and red curves,respectively. (b–d) xy-section views of |E|, |H|, and snapshot of Re(Hz) at 1.178 eV, respectively.(e–g) xy-section views of |E|, |H|, and snapshot of Re(Hz) at 1.233 eV, respectively. The incidentpolarization was set to Ein ∥ y in common. The xy sections were taken at the half-height of theSi-nanogap pair.The reflectance spectrum in Figure 6a presents definite anisotropy for the incident xand y polarizations, shown with black and red curves, respectively. High-Q modes withnarrow line widths appear at 1.178 and 1.233 eV at the y polarization, suggesting thatthey are qBIC modes. We examine the features in terms of the EM-field distributions inthe following.In Figure 6b–d, the resonant EM fields, |E|, |H|, and Re(Hz), are visualized, respec-tively, at the condition of the left arrow in Figure 4a, that is, 1.178 eV and y polarization.The EM fields are shown in a xy-section view of the unit cell; the section was taken atthe half height of the Si-nanogap unit. The refractive index in the medium contacting themetasurface is set to 1.5. Evidently, strong localization of the electric field takes place at theSi nanogap, reaching the maximum of |E| = 66.7 (or intensity |E|2 = 4448.9). In particular,the electric field localizes around the center of the nanogap. The magnetic field mainlydistributes in the Si nanoblocks (Figure 6b) and the major component is found to be Hz(Figure 6d). The magnetic-field distribution indicates that this resonance is basically an MDmode. Introducing the nanogap, the electric field is able to localize inside the nanogap.In Figure 6e–g, the resonant EM fields correspond to the condition of the right arrowin Figure 4a, that is, 1.233 eV and y polarization. The most prominent feature is seen atthe electric-field distribution that has two maxima points regarding |E| in the nanogap.The maxima mean that the intensity |E|2 exceeds 5387, which is a huge enhancement,compared to the incidence of |Ein|2 = 1. The magnetic field is qualitatively similar to thatin Figure 6b,c. Thus, the two resonances in Figure 6 are basically MD modes associatedwith the first and second localized electric-field localizations in the nanogap.This asymmetric metasurface of 10 nm gap can be compared to symmetric metasur-faces of Si-nanogap unit (Figure A2), which has a 10 nm gap. In the symmetric case, there isPhotonics 2024, 11, 1053 10 of 15no qBIC mode, in contrast to the asymmetric case in Figure 6. Further details are describedin Appendix A.Regarding refractometric responses, the interaction volume with the surroundingmedium becomes small in this asymmetric metasurface. Consequently, the responsivenessis similar to that in Figure A2, being quantitatively smaller than that in Figure 4a.4. DiscussionExcept for the qBIC modes, the metasurfaces have prominent optical resonances inthe reflectance spectra. When the parameter α = 10, it was found that the two gap modesemerge and are susceptible linearly to a wide range of refractive index of n = 1.0–1.5(Figure 4a) and quadratically to a further wide range of n = 1.0–2.0 (Figure 4b); in the linearrange, the responses were evaluated to be 286 and 231 nm/RIU. The two peaks of reflectanceenable us to clearly identify the refractometric responses in the multi-peak reflectancespectra. Such a wide-range response has hardly been ensured for other metasurfaces, whichare often limited to sensing for narrow refractive-index ranges [17,26–28,45–49]. Table 1selectively lists the reported quantities based on Si-based nanostructures. Other platformssuch as plasmonic structures were already addressed in other references [32,46–50].Table 1. Comparison of Si metasurfaces and lattice in terms of refractive-index sensing. MSF, Exp, andR denotes metasurface, experiment, and reflectance, respectively. C band means a telecommunicationband of 1530–1580 nm. Detected n ranges without any indication resulted from simulations.Structure Feature Detected n Response ReferenceSi MSF 2D array of 50 nm height pellets. 1.33–1.35, Exp Linear [17]Shift at 840–850 nm. 227 nm/RIUSi MSF qBIC at C band. Asymmetric 1.0–1.1 Linear [26]Si bars. 440 nm/RIUSi MSF 2D array of 450 nm height pairs. 1.3–1.7 Linear [27]Shift at 1330–1600 nm.612 nm/RIUSi MSF qBIC MD mode at C band. 1.0–1.4 Linear [28]30 nm nanogap. 258 nm/RIUSi MSF qBIC MD mode at C band. 1.33–1.36, Exp Linear [28]30 nm nanogap. 317 nm/RIUSi lattice 1D line and space. 1.0–1.5 (wide) Quadratic [32]Large R change at 900–1000 nmSi lattice 2D square nanoblocks. 1.0–1.5 (wide) Quadratic [32]Large R change at 745–790 nmSi MSF qBIC MD mode at C band. 1.33–1.36, Exp Linear [43]Long-short bar pair. 231 nm/RIUSi MSF 2D ring and bar unit. Shift at 1.40–1.44, Exp Linear [45]1340–1360 nm. 289 nm/RIUSi MSF 2D cylinders with thin spokes. 1.000–1.005 Linear [46]Shift at 1381.99–1382.82 nm.166 nm/RIUSi lattice 1D grating of pedestal unit. 1.33–1.47 Linear [47]Shift at 1480–1555 nm.536 nm/RIUSi MSF qBIC at C band. 2D bar-pair unit. 1.33–1.48 Linear [48]Shift at 1340–1360 nm.501 nm/RIUSi MSF qBIC. Asymmetric Dimer. 1.4–1.6, Exp Linear [49]Shift at 710–760 nm. 140 nm/RIUPhotonics 2024, 11, 1053 11 of 15Table 1. Cont.Structure Feature Detected n Response ReferenceSi MSF Asymmetric Si-nanogap unit. 1.0–1.5 (wide) Linear This studyShift at 950–1120 nm.286, 231 nm/RIUSi MSF Asymmetric Si-nanogap unit. 1.0–2.0 (wide) Quadratic This studyShift at 950–1270 nm.Traceable double R peaksAs listed in Table 1, refractive index sensing was frequently studied for narrow rangesof the refractive index. In recent papers [26–28,43,48,49], qBIC modes were explored forrefractive index sensing. Although the motives were to find highly sensitive refractiveindex sensors, the responses in the units of nm/RIU are competing with other resonantnanostructures. Indeed, the physical limit of refractive index sensors comprising periodicstructures is given by Equation (1), indicating that the limit is proportional to the periodiclength and that larger periodic lengths tend to exhibit larger sensitivity in nm/RIU. Theperformance of the qBIC-based sensors [26–28,43,48,49] is 20–70%, compared to the physicallimit. Thus, there is a room to seek for better performance. Indeed, a 1D grating structureof a large periodic length of 820 nm showed an improved value of the sensitivity [47],compared with the qBIC-based sensors. The present metasurface sensors have a featurethat they are responsive to a wide range of refractive indices, showing a practical potential.Other refractive index sensors using plasmonic lattices, optical fibers, and THz absorberswere discussed in a previous publication [32], where 569.1 nm/RIU was shown using a 1Dplasmonic lattice and means 94.9% realization of the physical limit.Regarding qBIC resonances, it is an experimental issue to obtain high-Q qBIC modesbecause nanofabricated metasurfaces have inevitable scattering loss that originates fromroughness at the outermost surface of the nanostructures, even when Si is etched through aBOSCH process that allows for high-contrast, deep reactive ion etching of Si [43]; in fact,we refer to the fact that the experimental limitation for high-Q modes exists at Q ≈ 3000in the Si-nanogap metasurfaces. In the structures described in this study, Q factors over1000 will be a practical goal when experimentally pursuing the high-Q qBIC modes in thepresent Si-nanogap metasurfaces.5. ConclusionsMetasurfaces of Si-nanogap units were numerically examined in this study. Intro-ducing asymmetry in the unit cell, qBIC modes became detectable. As the asymmetricparameter α became larger, the qBIC signals in the reflectance spectra became evident. Thisis a feature of qBIC. Examining the resonant EM-field distributions, the EQ mode wasobserved at x polarization, and the MD and MQ modes were observed at y polarizations(Figure 2a). The MD modes were modified into two modes in accordance with the first andsecond electric-field nodes in the nanogap (Figure 6). Adjusting the gap g and the α, a hugeelectric-field-intensity enhancement exceeding 5000 was found. This is an outstanding fea-ture in this all-dielectric metasurface. Considering application for refractive index sensing,the present metasurfaces are responsive to a very wide range of refractive indices from 1.0to 2.0 (Figure 4). The linear approximation for sensitivity holds for n = 1.0–1.5, whereasthe whole range of n = 1.0–2.0 is reproduced using quadratic functions. The changes inreflectance spectra are traceable and calibrated by measuring the pair of two reflectancepeaks. Such a feature responding to the wide n range, to the best of our knowledge, hasnot been shown in other systems so far.Funding: This study was partially supported by JSPS KAKENHI Grant Number JP24K01389.Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.Photonics 2024, 11, 1053 12 of 15Data Availability Statement: Data in this study are accessible upon reasonable request to the author.Acknowledgments: The author thanks Keisuke Watanabe for discussion on Si-nanogap structures.Numerical implementations were conducted on supercomputers SX-AOBA in Cyberscience Center,Tohoku University, Japan.Conflicts of Interest: The author declares no conflicts of interest.AbbreviationsThe following abbreviations are used in this manuscript:qBIC quasi-bound state in the continuumSOI Si-on-insulatorRCWA rigrously coupled-wave analysisS matrix scattering matrixMPI multiparallel implementationEM electromagneticEQ electric quadrupoleMD magnetic dipoleMQ magnetic quadrupoleFWHM full-wave half maximumQ factor quality factorAppendix A. Reflectance Spectra of Symmetric MetasurfacesHere, we show two illustrative results using metasurfaces of symmetric units. InFigure A1, the unit has a single square Si nanoblock. In Figure A2, the unit has a symmetricSi pair. The refractometric responses are shown for comparison with Figure 4a.Figure A1a shows the unit cell in a xy-section view, which is set to a dimension of600 × 600 nm2. The square Si nanoblock is 400 × 400 × 200 nm3 in the unit cell located onthe SiO2 and Si base substrate, similarly to the unit cell in Figure 1.Figure A1(a) Figure_drawing_for_Photonics_2024.ai(b) 13th Sym Test. Y240902h, Y240922c-g. (a)xySi(b)Figure A1. Reflectance spectra of metasurface of a single symmetric Si-nanoblock unit. The spectradepend on refractive index n. (a) Unit cell in xy-section view. (b) Reflectance spectra dependent onrefractive index n in the contacting medium. They are displayed with offset.Reflectance spectra for different refractive index n in the medium contacting themetasurface are shown in Figure A1b. The spectra exhibit small responsiveness to therefractive index, compared with the asymmetric metasurface in Figure 4a. This resultindicates that the unit of a simple Si nanoblock is unsuitable for refractive index sensing.Metasurface of symmetric Si-nanogap unit is tested in a configuration of refractive-index sensing. The unit structure has two Si nanoblocks of 400 × 200 × 200 nm3, and thePhotonics 2024, 11, 1053 13 of 15gap is set to 10 nm, as illustrated in Figure A2a. The Si-nanoblock pair is placed at thecenter of unit cell of 600 × 610 nm2 in the xy plane and is assumed to be on the SiO2 layer,similarly to Figure 1. The refractive index n changes from 1.0 to 1.5 in the medium incontact with the metasurface in Figure A2b.Figure A2(a) Figure_drawing_for_Photonics_2024.ai(b) 17th Sym Test. Y240918a, Y240922x1-x5. (a)xySiSi(b)Figure A2. Reflectance spectra of metasurface of symmetric Si-nanogap unit. The spectra depend onrefractive index n. (a) Unit cell in xy-section view. (b) Reflectance spectra dependent on refractiveindex n in the contacting medium. Solid and dashed curves indicate x- and y-polarized spectra,respectively. They are displayed with offset.Response to refractive index n is evaluated for the y-polarization resonance thatappears at 1.170 eV for n = 1.0, as shown with the red dashed curve in Figure A2. It turnedout that the sensitivity is 193.6 nm/RIU, which is approximately 10% smaller than that ofthe asymmetric metasurface in Figure 4. This reduction probably comes from less enhancedEM fields in this symmetric Si-pair metasurface than that in the asymmetric metasurface(Figure 6); in particular, the maximum of |E| component in the nanogap of this symmetricmetasurface is 10.9, which is approximately 16.3% of the maximum value in Figure 6a.References1. 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MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.http://dx.doi.org/10.1103/PhysRevLett.121.193903http://www.ncbi.nlm.nih.gov/pubmed/30468599http://dx.doi.org/10.1063/5.0158793http://dx.doi.org/10.1016/S0925-4005(98)00322-0http://dx.doi.org/10.1038/ncomms6753http://dx.doi.org/10.3390/opt2030018http://dx.doi.org/10.3390/nano12101748http://dx.doi.org/10.1021/acs.nanolett.3c03089http://dx.doi.org/10.1364/OE.514787http://dx.doi.org/10.1021/acsnano.2c03310 Introduction Materials and Methods Results Asymmetric Metasurfaces and qBIC Modes Resonant Shift for Refractive Index Nanogap Modes with Hugely Enhanced Electric Fields Discussion Conclusions Reflectance Spectra of Symmetric Metasurfaces References