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Abstract
Metasurfaces supporting surface lattice resonances (SLRs) with narrow linewidths and high quality factors have become an exciting platform for diverse applications. Here we numerically show, for the first time, that narrowband out-of-plane Mie electric dipole SLRs (ED-SLRs) can be excited together with the in-plane ED-SLRs and magnetic-dipole SLRs in periodic silicon disks under oblique incidence with TM polarization. Simulation results show that the out-of-plane ED-SLR can have four times larger quality factors than the in-plane one under the same excitation conditions, and can have distinct near-field distributions and dispersion relationships compared with the plasmonic counterpart in periodic metallic nanodisks. We further show that the out-of-plane ED-SLR can define a symmetry-protected bound state in the continuum (BIC) at normal incidence, which transits into a quasi-BIC when the excitation field symmetry is slightly broken by the small incidence angle. We expect this work will advance the engineering of Mie SLRs for applications in metasurface-based nanolasers, nonlinear optics, and optical sensing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Plasmonic surface lattice resonances (SLRs) in periodic metal nanostructures are associated to the coherent interference between the localized surface plasmon resonances (LSPRs) of each nanoparticle and the in-plane diffracted light [1]. Characterized by narrow linewidths, high quality factors, and strong near field enhancements extended over large volumes, SLRs have attracted increasing interest in manipulating light–matter interactions at the nanoscale [1–3]. Over the years, diverse exciting applications have been realized based on SLRs. These include metasurface-based nanolasers with directional beam emission [4], millimeter-scale spatial coherence [5] and ultralow-threshold [6], nonlinear optics with enhanced conversion efficiency [7–9], and optical sensing with high figure of merit [10].
In early experiments, the measured quality factors of SLRs are only $Q\sim 40-60$ [11,12], which are several times of those of LSPRs ($\sim 10-20$). After years’ effort, the quality factors of SLRs have achieved considerable progress [13]. For example, Li et al. [14] demonstrated high quality factors of 500 in the mid-infrared regime with periodic ITO nanorods on a gold mirror. Le-Van et al. [15] reported experimental quality factors higher than 330 using silver nanoparticles in the visible range. Deng et al. [16] demonstrated high quality factor of 430 in the near-infrared regime by using annealed metallic nanoparticles. Bin-Alam et al. [17] experimentally demonstrated ultra-high quality factor of 2340 in the near-infrared regime with gold metasurfaces.
Compared with the above in-plane plasmonic SLRs, which are excited under normal incidence, out-of-plane SLRs launched under oblique incidence are believed to have higher quality factors due to stronger inter-particle coupling [18]. For example, in 2011 Zhou and Odom [19] observed subradiant out-of-plane SLRs with quality factor of 150, which was the record in the visible range at that time. Additionally, it has been shown that out-of-plane SLRs can maintain high quality factors over a large spectral range, within which the resonance wavelength can be tuned by varying the incident angle [20,21]. Quite recently, some of the authors made use of gold nanohemisphere array and numerically obtained exceptionally high quality factor of 794 in the visible regime [22]. We further proposed out-of-plane quadrupolar SLR and numerically achieved high quality factor reaching 1036 based on horizontal metal-insulator-metal gratings [23].
In order to suppress the absorption loss suffered by plasmonic resonances, recently the scope of SLRs has gradually extended from plasmonic to all-dielectric [3]. For periodic all-dielectric nanoparticles, the SLRs emerge from the enhanced radiative coupling of localized Mie resonances in the individual nanoparticles [24]. In other words, localized Mie resonances in the Mie SLRs are the analogues of LSPRs in plasmonic SLRs. Although the Mie SLRs in all-dielectric metasurfaces have been intensively investigated quite recently, from the fundamental physics [25–29] to the potential applications [30–33], very few effort has been put on improving the quality factor. Wang et al. [32] showed that the electric dipole SLR (ED-SLR) in a silicon nanoparticle array has relatively high quality factor of 120 in the visible regime. Murai et al. [33] experimentally achieved high quality factor of 298 at the wavelength of 617 nm for the ED-SLR, and numerically showed that the magnetic dipole SLR (MD-SLR) in periodic silicon nanoparticles has ultra-high quality factor of $\sim 8970$ at 628 nm. Zhang et al. [34] experimentally demonstrated high quality factor of $\sim 500$ for the MD-SLR in the visible regime by using monocrystalline silicon nanoantenna arrays. However, these high quality factors are achieved by in-plane Mie SLRs that are excited under normal incidence, even though Mie SLRs under oblique incidence have also been investigated [32–38]. Questions arise such as whether out-of-plane Mie ED-SLRs can be excited, do these ED-SLRs have larger quality factors than the in-plane ones, just like their plasmonic counterparts, and are their near-field maps and dispersion relationship similar to those of out-of-plane plasmonic ED-SLRs.
In this work, we numerically report, for the first time, the excitation of out-of-plane Mie ED-SLR in periodic silicon disks under oblique incidence with TM polarization. Simulation results will show that this resonance can coexist with the in-plane Mie ED-SLR, MD-SLR, and magnetic quadrupole SLR (MQ-SLR), and has four times larger quality factors than the in-plane ED-SLR under the same excitation conditions. We find that the near-field distributions and the dispersion relationship of the out-of-plane Mie ED-SLR can be distinct from those of the plasmonic one in periodic metallic nanodisks. Strikingly, we further find that the out-of-plane ED-SLR and the in-plane MQ-SLR can define two symmetry-protected bound states in the continuum (BICs) at normal incidence, which transit into quasi-BICs for small incidence angles. The effects of the silicon disk’s size and the scenario of TE polarization will also be discussed.
2. Design and simulation setup
Figure 1(a) illustrates the silicon metasurface composed of two-dimensional periodic disks of square lattice with periodicity $\Lambda =700$ nm. The silicon disks with diameter $d=500$ nm and height $h=200$ nm are embedded in homogeneous dielectric environment of $n_0=1.45$, and are illuminated by linearly polarized plane wave light with incidence angle $\theta$ and unitary amplitude ($|E_0|=1$). Note that here $\theta$ denotes the incident angle in the free space.
Fig. 1. (a) Schematics of the two-dimensional periodic silicon disks under oblique incidence with TM or TE polarization. The disks have diameter $d$, height $h$, and lattice periods $\Lambda$ in $x$ and $y$ directions. (b) Simulated reflectance and transmittance spectra of the silicon disk array under oblique incidence of $\theta =15^\circ$ with TM polarization. The vertical dashed line indicates the $(-1,0)$ RA wavelength. (c)–(f) Near-field electric field distributions $|E|^2$ (color for intensity and arrows for directions) and (g)–(j) Poynting vector maps at the four resonance wavelengths indicated in (b): $\lambda =1130$ nm, $1184.4$ nm, $1312.2$ nm and $1336.2$ nm from left to right. In (c)–(j) the silicon disk is outlined by the rectangle.
The zeroth-order reflectance and transmittance spectra, $R_0(\lambda )$ and $T_0(\lambda )$, as well the near-field distributions of the designed metasurface were numerically calculated with a home-built package for the fully vectorial rigorous coupled-wave analysis (RCWA), which was developed following [39–41]. In all the simulations, the wavelength-dependent refractive indices of silicon are tabulated in [42].
3. Results and discussion
3.1 Excitation of out-of-plane Mie ED-SLR
We first investigate the oblique incidence with TM polarization. Figure 1(b) depicts the simulated zeroth-order reflectance and transmittance spectra of the silicon metasurface under oblique incidence of angle $\theta =15^\circ$. Results show that there exist four reflectance peaks or transmittance dips locating at $\lambda _0=1130$ nm, $1184.4$ nm, $1312.2$ nm, and $1336.2$ nm from the left to the right, respectively. Among these resonances, two locate on the left side of the $(-1,0)$ RA line, which is determined by
Remarkably, the second resonance from the left has the narrowest linewidth of 8.2 nm. Thus the quality factor, which can be calculated by the ratio of the resonance wavelength to the linewidth, is estimated to be $Q=144$.In order to understand the physics underlying the above four resonances, the near-field optical pictures are required. Figures 1(c)(g) show that for the leftmost resonance at $\lambda =1130$ nm, the greatly enhanced electric field extending over large regions forms two circulations inside the silicon disk, and the energy flux propagates along the $-x$ direction. These features suggest the excitation of the in-plane MQ-SLR.
For the second resonance at $\lambda =1184.4$ nm, Fig. 1(d) shows that the electric field aligned with the $z$ axis is significantly enhanced around the top and bottom edges of the silicon disk, and is extended over large regions outside the disk, featuring an out-of-plane electric dipolar field pattern. Figure 1(h) shows that the energy flux propagating along the $-x$ direction passes through the silicon disk. These features suggest the excitation of the out-of-plane Mie ED-SLR. We note that the electric field distributions and the Poynting vector maps are distinct from those of the out-of-plane plasmonic ED-SLR supported by periodic metallic nanodisks [19,22]: the electric fields are highly confined to the left-top and right-bottom edges of the metallic nanodisk, and the energy flux propagating along the $+x$ or $-x$ direction (determined by the diffraction order) bypasses the metallic nanodisk. These differences arise because the dipole field for Mie ED-SLRs is induced by displacement currents, but by free electron gases for plasmonic ED-SLRs. Therefore, we have shown that the out-of-plane Mie ED-SLR can be excited in periodic silicon disks under oblique incidence with TM polarization, and can have distinct near-field distributions from the plasmonic counterpart.
For the third resonance at $\lambda =1312.2$ nm, the greatly enhanced electric field in Fig. 1(e) is aligned with the incident field showing an in-plane dipolar field pattern, and is also extended outside the silicon disk. These are typical features of the in-plane Mie ED-SLR. Correspondingly, the energy flux in Fig. 1(i) shows a circulation in the silicon disk. For the rightmost resonance at $\lambda =1336.2$ nm, the electric field in Fig. 1(f) forms a circulation around the disk, and the energy flux in Fig. 1(j) propagates along the $+x$ direction. These features suggest the excitation of the in-plane MD-SLR. Since the in-plane ED-SLR and MD-SLR have been well investigated in the literature [24,26,31–33,35,36,43], we will not elaborate on these resonances in this work.
In Figs. 1(c)–(f), we find that the out-of-plane Mie ED-SLR has the largest electric field intensity enhancement, which reaches as high as 250. This is consistent with its narrowest linewidth and largest quality factor. This enhancement is even comparable to those of plasmonic SLRs excited in periodic metallic disks [17,19,22]. The significantly enhanced and extended electric fields, together with the high quality factor, make the out-of-plane Mie ED-SLR very attractive in applications such as nanolasers, nonlinear optics, and optical sensing.
3.2 Dispersion relationship
We further study the dispersion relationships of the above four Mie SLRs. Figure 2 shows the reflectance and transmittance spectra of the silicon disk array under incidence with $0^\circ \leq \theta \leq 40^\circ$ and TM polarization. Dashed curves represent the $(-1,0)$ and $(0,\pm 1)$ RAs, which are determined by Eq. (1) and by
respectively. Results show that the out-of-plane ED-SLR is excited together with the in-plane MQ-SLR, ED-SLR, and MD-SLR, as labelled in the figure. The out-of-plane ED-SLR always has smaller linewidth and thus larger quality factor than the in-plane one. To be more specifically, for $\theta \geq 5^\circ$ the quality factors of the out-of-plane ED-SLR are about $Q\sim 140$, whereas those of the in-plane ED-SLR are only $Q\sim 30$ for $\theta \geq 20^\circ$. In other words, the quality factors of the out-of-plane ED-SLR are about four times of those of the in-plane one.Fig. 2. Simulated (a) reflectance and (b) transmittance spectra of the periodic silicon disks under oblique incidence with TM polarization. The white dashed lines indicate the $(-1,0)$ RA, and the red dashed curves indicate the $(0, \pm 1)$ RA.(c) Zoom-in of the dashed box in (a). (d)(e) Extracted quality factors (balls) of (d) in-plane MQ-SLR and (e) out-of-plane ED-SLR as functions of the small incidence angle $\theta$. Solid curves denote theoretical fittings using inverse square functions.
We note that the dispersion relationship of the out-of-plane Mie ED-SLR is distinct from that of the plasmonic counterpart in periodic metallic nanodisks. For the out-of-plane Mie ED-SLR, the reflectance shows a narrow peak while the transmittance shows a narrow dip, as shown by Figs. 1(b) and 2(a)(b), whereas for the out-of-plane plasmonic ED-SLR, the reflectance shows a narrow dip within a broad peak [19]. Therefore, the out-of-plane Mie ED-SLR can be treated as a bright mode, whereas the out-of-plane Mie ED-SLR was referred to as a dark or subradiant plasmon mode [19].
In Fig. 2 we note that the in-plane ED-SLR and MD-SLR have different dispersion relationships following the $(0,\pm 1)$ and $(-1,0)$ diffraction orders, respectively. Their spectral overlap occurs around $\theta =9^\circ$ and $\lambda =1305$ nm, as indicated by the red circle. This leads to the resonant lattice Kerker effect, which corresponds to the zero reflectance and unitary transmittance.
Interestingly, we also find that both the in-plane MQ-SLR and the out-of-plane ED-SLR gradually disappear as the incidence angle approaches $0^\circ$, as highlighted by the pink dashed box in Fig. 2(a). In order to understand the underlying physics, we replot this region in Fig. 2(c) with a smaller step of the incidence angle ($0.1^\circ$). Results show that as the incidence angle decreases from $\theta =5^\circ$ to $0^\circ$ (normal incidence), the linewidths of both resonances decrease dramatically, and both resonances become dark at normal incidence (the $\Gamma$ point). These suggest the occurrence of two BICs emerging from the in-plane MQ-SLR and the out-of-plane ED-SLR, respectively. We note that these behaviors are similar to that of the out-of-plane MD-SLR excited under oblique incidence with TE polarization [34,37,38], and thus can be explained similarly: both symmetry-protected BICs at the $\Gamma$ point stem from the fact that the in-plane MQ-SLR and the out-of-plane ED-SLR are not allowed to emit at normal incidence.
For small incidence angles, the excitation field symmetry is slightly broken. As a result, both symmetry-protected BICs transit into quasi-BICs. We extract the angle-dependent quality factors of both quasi-BICs and plot them in Figs. 2(d)(e). Results show that the quality factor decreases significantly from $Q=7.8\times 10^4$ (or $4.6\times 10^4$) at $\theta =0.2^\circ$ to $Q=146$ (or 80) at $\theta =5^\circ$ for the in-plane MQ quasi-BIC (or the out-of-plane ED quasi-BIC). For both quasi-BICs, the dependence of the quality factor on $\theta$ is shown to be inverse quadratic,
as demonstrated by the nearly perfect fitting.3.3 Effects of the silicon disk’s size
We now investigate the effects of the silicon disk’s diameter and height on the above four Mie SLRs excited under oblique incidence with $\theta =15^\circ$ and TM polarization. Figures 3(a)(b) show that, as the disk diameter decreases from $d=500$ nm to 300 nm, the in-plane MQ-SLR gradually disappears, the in-plane ED-SLR gradually mixes with the out-of-plane ED-SLR, and the in-plane MD-SLR narrows down. On the other hand, as the disk height decreases from $h=200$ nm to 50 nm, the in-plane MQ-SLR becomes narrower, the out-of-plane ED-SLR also becomes narrower but gradually disappears when approaching the $(0,\pm 1)$ RA wavelength (the red dashed line), whereas the in-plane ED-SLR and MD-SLR become mixed. All the four Mie SLRs are blue-shifted as the silicon diameter or height reduces. Therefore, we have shown that the in-plane MQ-SLR exists only when the silicon disk diameter is large enough in order to support two electric field circulations, and that the out-of-plane ED-SLR can be excited provided the silicon disk is tall enough to support an out-of-plane electric dipole.
Fig. 3. Simulated (a)(c) reflectance and (b)(d) transmittance spectra of the silicon disk array with different (a)(b) diameters or (c)(d) heights. The calculations were performed with oblique incidence of $\theta =15^\circ$ and TM polarization. The red and white dashed lines indicate $(0,\pm 1)$ and $(-1,0)$ RA wavelengths.
Additionally, we have also found that by adopting smaller silicon diameter or height, we can achieve much narrower linewidth and thus much larger quality factor for the in-plane MD-SLR or the out-of-plane ED-SLR, respectively. Figure 4(a) shows that the quality factor of the in-plane MD-SLR can be increased by an order of magnitude, from $Q=75$ for $d=500$ nm to $Q=850$ for $d=300$ nm. As the silicon disk’s height decreases from $h=200$ nm to 60 nm, Fig. 4(b) shows that the quality factor of the out-of-plane ED-SLR increases by two orders of magnitude, from $Q=144$ to $1\times 10^4$.
Fig. 4. (a) Extracted quality factor of in-plane MD-SLR as a function of silicon disk’s diameter. (b) Extracted quality factor of out-of-plane ED-SLR as a function of silicon disk’s height.
3.4 Oblique incidence with TE polarization
While until now we considered oblique incidence with TM polarization, hereafter we will investigate the scenario of TE polarization for the sake of completeness. Figure 5(a) depicts the simulated zeroth-order transmittance spectra under oblique incidence with TE polarization. Results show that there exist three branches of resonances: the leftmost one has the narrowest linewidths, and the other two become mixed as the incidence angle approaches $0^\circ$.
Fig. 5. (a) Simulated transmittance spectra of the periodic silicon disks under different incidence angles with TE polarization. The red dashed curve and the white dashed line indicate $(0,\pm 1)$ and $(-1,0)$ RAs, respectively. (b) Reflectance and transmittance spectra for $\theta =15^\circ$. The dashed line denotes the $(-1,0)$ RA wavelength. (c) Top-viewed and (d)(e) side-viewed near-field electric field distributions $|E|^2$ (color for intensity and arrows for directions) at the resonance wavelengths of $\lambda _0=1193.2$ nm, 1264 nm, and 1335.4 nm, respectively. (f) Extracted quality factors of the out-of-plane MD-SLR as a function of the small incidence angle.
In order to understand the physics of these three resonances, we plot the reflectance and transmittance spectra for $\theta =15^\circ$ in Fig. 5(b), and the near-field electric field distributions at the corresponding resonance wavelengths of $\lambda _0=1193.2$ nm, 1264 nm, and 1335.4 nm in Figs. 5(c)–(e). Figure 5(c) shows that the greatly enhanced electric field extending over large regions forms a circulation in the top-viewed cross section at the middle of the silicon disk. This suggests that the leftmost resonance is an out-of-plane MD-SLR. The side-viewed electric field in Fig. 5(d) shows a circulation whereas that in Fig. 5(e) shows an in-plane electric dipole field aligned with the incident field, which is polarized along the $y$ direction for TE polarization. These are typical features of the in-plane MD-SLR and ED-SLR, respectively.
In Fig. 5(a) we find that the out-of-plane MD-SLR always has the narrowest linewidths and thus the largest quality factors compared with the in-plane ED- and MD-SLRs. Furthermore, as $\theta$ approach $0^\circ$, the out-of-plane MD-SLR narrows down and becomes dark at normal incidence. Therefore, the out-of-plane MD-SLR defines another symmetry-protected BIC at the $\Gamma$ point, which occurs because the out-of-plane MD-SLR is not allowed to emit at normal incidence. For small incidence angles, this BIC also transits into a quasi-BIC due to the excitation field symmetry breaking. The extracted quality factors of the quasi-BIC as a function of the small incidence angle can also be well fitted with a inverse quadratic function following Eq. (3), as shown by Fig. 5(f).
We note that the characteristics of the out-of-plane MD-SLR excited under TE polarization are similar to those of the out-of-plane ED-SLR launched under TM polarization, which were discussed previously. This analogue further confirms our results because of the transition between the ED-SLR and the MD-SLR as the linear polarization of incident light changes to the orthogonal direction.
Comparing with the literature on the out-of-plane MD-SLR [34,37,38], here we consider the infrared regime in which the absorption loss of silicon can be neglected. Therefore, the total quality factor is equal to the scattering quality factor, and thus its dependence on the small incidence angle can be well fitted with an inverse quadratic function following Eq. (3). The good agreement between simulation data and fitting results greatly facilitates the confirmation of the quasi-BIC.
4. Conclusions
In conclusions, we have numerically shown that high-$Q$ out-of-plane Mie ED-SLRs can be excited together with the in-plane ED-, MD- and MQ-SLRs in periodic silicon disks under oblique incidence with TM polarization. Results have shown that the out-of-plane Mie ED-SLR can have four times larger quality factors than the in-plane one, and can have distinct near-field distributions and dispersion relationship compared with the out-of-plane plasmonic ED-SLR. We have also shown that by adopting smallar disk height or diameter, the quality factor of the out-of-plane ED-SLR or of the in-plane MD-SLR can be enhanced by orders of magnitude. On the other hand, under oblique incidence with TE polarization, we have shown the excitation of the out-of-plane MD-SLR, consistent with the literature. Remarkably, we have also found that the out-of-plane ED-SLR, the in-plane MQ-SLR, and the out-of-plane MD-SLR can define three symmetry-protected BICs at normal incidence (the $\Gamma$ point). For small incidence angles, which slightly break the excitation field symmetry, these BICs transit into quasi-BICs, of which the quality factors have inverse quadratic dependence on the incidence angle. We therefore expect this work will provide a new approach for achieving high quality factors of Mie SLRs, and the obtained high-$Q$ out-of-plane Mie ED-SLR will find potential applications in light–matter interactions, especially metasurface-based nanolasers, nonlinear optics, and optical sensing.
Funding
Guangdong Guangxi Joint Science Key Foundation (2021GXNSFDA076001); Guangxi Major Projects of Science and Technology (2020AA21077007); Natural Science Foundation of Guangdong Province (2022A1515010086); Shenzhen Institute of Artificial Intelligence and Robotics for Society.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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