1. Introduction

Active optical modulation plays an important role in developing high-performance reconfigurable nanophotonic-devices [1–3]. The epsilon ($\varepsilon$)-near-zero (ENZ) materials, whose real part of the dielectric constant is zero or near-zero at specific wavelengths, becomes an exciting platform to achieve the tunability of the device. Due to the fascinating capabilities, nanodevices integrating ENZ materials exhibiting interesting optical coupling effects has enabled many advanced optical applications [4]. Indium tin oxide (ITO) is one of the most widely used ENZ materials, many potential applications of ITO have been proposed or demonstrated by harnessing the ENZ behavior of ITO [4,5]. The ENZ wavelength of ITO is closely related to the electron density and can be obtained by applying an electrostatic field [6–8] or optical pumping [9,10] to tune the electron density, which can change its dielectric constant in the near-infrared state from positive (dielectric-like) to negative (metal-like) [11,12]. When the dielectric constant of the ITO material is close to zero, which is the ENZ state, the intrinsic absorption loss inside the ITO material can be effectively changed. This may open the door of realizing strong electro-optical effects [13–17]. ITO is often combined with metal nanocavities for efficient optical modulation. For example: Alam et al. exploited an optical metasurface coupled with the dipole resonant mode of the gold antenna and the ENZ mode of the ITO thin film, thereby exhibiting a broadband and ultrafast nonlinear response [18]. Kuttruff et al. exploited linear photon absorption of ENZ-mode modulation systems of metal-insulator-metal (MIM) nanocavities, and achieved nondegenerate all-optical ultrafast modulation [19]. Tao et al. realized an electrically tunable ultra-compact plasmonic modulator with high modulation intensity using plasmonic-induced transparent structures [20]. Guo et al. proposed a hybrid metasurface composed of indium tin oxide (ITO) nanolayers and plasmonic metasurfaces, which can support two different coupling modes to realize double transparent windows in the communication frequency band [21]. Xie et al. proposed a tunable optical switch based on an ENZ metasurface, which can function as an electro-optical switch and an all-optical switch [22]. However, metal metasurfaces have large ohmic losses in the visible and near-infrared range, so metal devices based on localized surface plasmon resonance have lower Q-factor, and the optical modulation based on hybrid plasmonic metasurfaces and ENZ materials has been limited.

In recent years, dielectric metasurfaces have received extensive attention due to their low loss, CMOS-compatible fabrication process, and abundant resonant modes, providing a new platform for the realization of high Q-factor [23–27]. The perturbation of optical bound states in the continuum (BICs) is a simple and effective method to obtain high Q-factor resonators [28–41]. BIC has an infinite Q-factor, and is not coupled to free space [42,43]. It usually appears at highly symmetric dielectric metasurfaces such as lattice with $C_{\rm 4v}$ group symmetry. When perturbing the symmetry of the structure, the higher order group $C_{\rm 4v}$ evolves into lower order group $C_{\rm s}$, which converts non-radiative optical BICs modes into quasi-BICs (QBICs) [44–48]. The QBICs in metasurface can be usually accompanied by the generation of high Q-factor Fano resonances, which can greatly enhance the light-matter interaction. For example: QBICs can reduce the threshold of the laser [49], improve the nonlinear conversion efficiency [50–54], realize the high-efficiency electro-optic modulators [55,56], improve the sensitivity of optical sensors [57,58], etc. Moreover, the Q-factor of QBIC in the dielectric metasurface can be arbitrarily regulated by the asymmetric parameter, which can control the light-matter interaction strength. Among the BIC family, the symmetry-protected BIC are is the most common and are is easy to realize experimentally. For practical application, one should break the unit cell’s symmetry by removing or adding part of the geometric structure in the metasurface to transform such a BIC into a QBIC with a high Q-factor. Some typical examples of broken-symmetry in unit cell include split rings [59], asymmetric nanorods [52,60], notched cubes [61,62] and nanodisks [39,63]. It is worth noting that the smaller the asymmetric parameter, the larger the Q-factor. However, it is very challenging to fabricate such nanostructures with an ultra-small asymmetry parameter in the experiment due to the limitation of fabrication, thus limiting the actual Q-factor of QBIC. In addition, to date, there are few studies on active control on QBICs by either optical pumping, thermal-tuning or nano-electromechanical tuning [56,64–68]. In real applications, it is also highly desired to achieve tunable QBICs by electrical gating. Given the salient property of ENZ ITO that can be controlled by external bias, a large spatial modulation and fast response should be expected by integrating ENZ ITO thin film with dielectric metasurface supporting QBICs.

In this work, we demonstrate an active control on the QBICs with ENZ materials. We consider a silicon metasurface comprising five square holes in a unit cell. Such a metasurface hosts multiple BICs realized by moving the position of center hole, where the disturbance parameter of QBICs can be accurately controlled in the experiment by this way. By integrating this metasurface with an ENZ ITO film, we demonstrate that multiple QBICs can be tuned by the carrier density of ITO film, evidenced by transmission spectra, far-field radiation, and near-field distributions. Thanks to the high-Q nature of QBICs, the transmission intensity can be modulated from 0 to >90% at certain wavelength by controlling electron density of ITO, corresponding to modulation depth of 14.8 dB. Our results may pave the way toward developing high-performance spatial optical modulators.

2. Results and discussion

2.1 Multiple QBICs excited by Si metasurfaces

The Si metasurface is composed of a periodic array of Si thin films embedded with five compound square air-holes, as shown in Fig. 1(a). The top view of unit cell is shown in Fig. 1(b), the five square air-holes are marked as H1, H2, H3, H4 and H5, respectively. If we set the centre of unit cell as origin, the coordinates of these four holes are H1=(-200 nm, 200 nm), H2=(200 nm, 200 nm), H3=(-200 nm, -200 nm) and H4=(200 nm, -200 nm). The period of the device is $P_x=P_y=600$ nm, the side length of the square hole is $l=120$ nm, the depth of the hole is 220 nm, and the substrate is ${\rm SiO_2}$. When H5 is located at the center of the device, namely H5(0, 0), the device satisfies the $C_{\rm 4v}$ rotational symmetry group, and there are multiple non-radiative modes, also known as BICs. When the air-hole H5 is moved up and down along the $y$-axis, the high-order group $C_{\rm 4v}$ degenerates into the low-order group $C_{\rm s}$, thus opening the radiation channel, so that these non-radiative BICs transform into leaky mode quasi BICs. In addition, when the device parameters are changed, interference cancellation occurs between the multi-dipoles, which drastically reduces the far-field radiation of light, and another BICs mode appears. The complex eigenfrequency ($\omega =\omega _0-i\gamma$) can be calculated by commercial software Comsol Multiphysics based on the finite-element method (FEM) (Study$\to$Eigenvalue or Eigenfrequency). The Q-factor is calculated by $Q=\omega _0/2\gamma$. The transmission spectra, field distributions, scattered power of different multipoles, and average field enhancement factor are calculated by the finite difference-time-domain method (Lumerical-Ansys FDTD Solution). In the simulation, the $x$- and $y$-directions are set as periodic boundary conditions, the $z$-direction is set as a perfectly matched layer boundary condition, the $x$-polarized incident light is perpendicular to the metasurface along the $z$-axis, and the optical constants of Si and ${\rm SiO_2}$ are taken from the Palik Handbook [69].

 

Fig. 1. (a) Schematic diagram of the nanostructure array. (b) Geometry of the unit cell, the off-set distance of the air-hole H5 from the center of unit cell is marked $d$. (c) Transmission spectra for different $d$ values. The three modes are marked with Mode I, Mode II, and Mode III, respectively, and A, B, C, and D corresponds to the four BICs, respectively. (d), (e), (f) Q-factors of the three modes as the function of off-set distance $d$, respectively.

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When H5 is moved along the $y$-axis, the two symmetry-protected quasi-BICs are excited because the in-plane mirror symmetry is broken with respect to $xoz$ plane. In Fig. 1(c), we calculate the transmission spectra of metasurface when H5 is located at different off-set distances $d$. When $d$=0, there is only one leaky mode (Mode II) at the wavelength of 1455.5 nm. As $d$ increases, the symmetry of the structure is broken, and two leaky modes appear on both sides of Mode II, which we mark as Mode I and Mode III, they are governed by symmetry-protected BIC. Figure 2 shows the electric field distribution of three modes at different off-set distances. Interestingly, the evolution of these two modes differs greatly when $d$ increases gradually. For Mode III, it always conforms to the rules of normal symmetry-protected BIC, the larger the perturbation parameter, the wider the spectra, thus the smaller the Q-factor. However, for Mode I, as $d$ increases, the resonance spectrum first gradually broadens, then narrows, and then broadens again. The resonance spectrum disappears at $d$=70 nm, implying that it is a non-radiative BIC mode but different from symmetry-protected BIC, which is formed by completely destructive interference between multiple dipoles. Therefore, Mode I can be evolved into two types of BICs as $d$ varies. For Mode II, it can be also evolved into BIC when center hole H5 moves along $y$ axis. When $d$=0, Mode II has a finite Q-factor and manifests itself as a Fano resonance in transmission spectrum. As $d$ increases, the resonance linewidth becomes narrower. At $d$=86.5 nm, the resonance disappears, then reappears and gradually broadens. The vanished resonance linewidth marks the appearance of BICs. The same BIC can be found if H5 is moved in the opposite direction. In addition, the same optical response can be observed when H5 is moved along the $x$-axis under $y$-polarized light.

 

Fig. 2. Electric field distribution and field vector distribution at the $x-y$ plane for the three modes, (a)-(c) $d=30$ nm, (d)-(f) $d=170$ nm. Here, the vector arrows are normalized.

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To prove that multiple BICs indeed exist at these critical positions, we calculate the Q-factors of each mode under different $d$, respectively. For free-standing metasurface, the Q-factors of the four BICs are all over $10^9$ (see Fig. 7). The Q-factor can reach infinity for the extremely enhanced numerical resolution, confirming the existence of these BICs. The Q-factors of accidental BICs (Modes I and II) for a photonic crystal slab embedded in a symmetric environment with respect to $z$=0 plane are calculated (see Supplement 1, Fig. S1), which is set as ${\rm SiO_2}$. Indeed, the Q-factor of accidental BICs (B and C) can be Q>$10^8$. If only the bottom ${\rm SiO_2}$ substrate is introduced, the Q-factor of accidental BICs will drop significantly due to the out-plane mirror symmetry-breaking, as shown in Figs. 1(d)-(f). The larger the refractive index of substrate, the smaller the Q-factor of accidental BICs. In a word, to achieve an ultrahigh Q-factor of accidental BICs, it is required to create a symmetry environment. It is noted that the other two BICs (A and D) are maintained because they are normal symmetry-protected BICs which are not influence by the substrate. For Mode I, there is an infinite Q-factor when $d$=0. As $d$ increases, the Q-factor first decreases gradually, and then increases to maximum $1.48\times 10^6$ at $|d|$=68.5 nm ($2.8\times 10^9$ for free-standing metasurface). Further increase $d$ results in the decreased Q-factor. The maximum of Q-factor is also confirmed by transmission spectra in Fig. 1(c). Similar trend is expected when H5 moves in an opposite direction. For Mode II, when $d$=0, it has a moderate Q-factor, and with the increase of $d$, the Q-factor gradually increases, and a high Q-factor appears at $|d|$=86.5 nm, and its Q-factor is up to $4.2\times 10^6$. Then, the Q-factor gradually decreases again, agreeing with transmission spectra evolution in Fig. 1(c). It is noted that BICs B and C are accidental BICs, which arise from the destructive interference of the multipoles response. For Mode III, as $d$ increases, the Q-factor decreases, it is an ordinary symmetry-protected BIC. Therefore, the Q-factor of three modes and transmission spectra at different $d$ confirm each other. It is noted that the Q-factor of the quasi symmetry-protected BIC satisfies the formula $Q\propto \alpha ^{-2}$, $\alpha$ is the asymmetry degree [30]. In this work, $\alpha =2d/P$. For Mode I, the Q-factor is proportional to $\alpha$ within the quasi symmetry-protected BIC range. In the accidental BIC region, the inverse quadratic law for the Q-factor is not satisfied. For Mode III, there exists only symmetry-protected BIC, and the inverse quadratic law for the Q-factor of the quasi-BIC is satisfied (Supplement 1, see Fig. S2).

Next, we discuss the local field properties of the above three resonance. The calculation results are exhibited in Fig. 2, showing the electric field intensity and field vector distribution at the resonance wavelength of the three QBICs for $d$=30 nm and 170 nm. Here, the color scale corresponds to the field intensity, the black dashed area represents the square air-hole. The arrows describe the field vector distribution in the $x-y$ plane at $z$ = 110 nm. For Mode I, as shown in Fig. 2(a), it can be observed that there is an obvious electric quadrupole distribution in the $x-y$ plane. The electric field is mainly distributed around the square air-holes H3 and H4. In Fig. 2(d), for $d$=170 nm, the electric field distribution and field vector distribution have changed significantly, and the electric field is mainly distributed inside of the square air-hole and its edge area. In the region below the air-holes H3 and H4, the electric field vector exhibits an obvious circular distribution, so the magnetic dipole of $z$-direction dominates the response. Here, the near-field distribution is also greatly affected by other dipoles, and each dipole moment interferes each other, so the total field distribution is the result of the interaction of multiple dipoles. The significant multipole interference induces the non-radiative BIC mode. For Mode II, when $d=30$ nm and $d=170$ nm, as shown in Figs. 2(b) and (e), the distinct circular electric field distribution is located the region of below H5, exciting the magnetic dipoles oscillation along the $z$-direction. The electric field is mainly distributed in the square air-hole and its edge area, showing slight differences, and they are almost mirror-symmetrical along the $y$-axis. It is suggested that, with the change of structural parameters, Mode II are relatively stable. For Mode III, as shown in Figs. 2(c) and (f), when $d=30$ nm, a very obvious circular electric field distribution appears in the H5 region of the $x-y$ plane, which excites the magnetic dipole along the $z$ direction, the electric field mainly distributed in the square air holes H1, H2 and H3 and their edge areas. When $d=170$ nm, two opposite circular electric field distributions appear at the $x-y$ plane, and excite a circular magnetic field distribution on the $x-z$ plane, which eventually leads to a electric toroidal dipole response in the $x$-direction. We perform an in-depth discussion on the near-field distribution characteristics of the three QBICs, and these results are mutually confirmed with the results of multipole decomposition. It is noted that the eigenfield fields of the BIC mode (A, D) at $d$=0 are highly symmetrically distributed, corresponding to non-degenerate modes, which do not couple to the normal incident light, as shown in Supplement 1, Fig. S3.

In order to reveal the nature of these QBICs, we perform the multipole decomposition on the structure [70–72]. For Mode I, when $d=30$ nm, as shown in Fig. 3(a), the far-field scattering power of the nanostructure at the resonance wavelength is dominated by the electric quadrupole (EQ) at the $x-y$ plane. Also, a mild contribution comes from the magnetic dipole (MD). When $d=170$ nm, as shown in Fig. 3(d), the MD response along the $z$-direction dominates this mode, followed by EQ, the other dipole radiations are suppressed. With the increase of parameter $d$, the dominant multipole of Mode I evolves from EQ to MD. These results are consistent with the field distribution in Figs. 2(a) and (d). For Mode II, as shown in Figs. 3(b) and (d), when $d=30$ nm and 170 nm, the MD response in the $z$-direction is always dominant for this mode. Therefore, Mode II is relatively stable, the result is in good agreement with the field distribution in Figs. 2(b) and (e). For Mode III, as shown in Figs. 3(c) and (f), when $d=30$ nm, the dominant far-field scattering is MD along the $z$-direction, followed by toroidal dipole (TD). For $d=$170 nm, the strongest contribution to the resonances is provided by the TD along $x$-direction. As the parameter $d$ changes, the dominant dipole evolves from MD to TD. This result is consistent with the one in Figs. 2(c) and (f). Therefore, when the perturbation parameter is changed, the coupling between the multiple dipoles will change significantly, resulting in an obvious change in the resonance properties of Mode I and Mode III. Among them, the dominant contribution of Mode I evolves from EQ to MD, and Mode III evolves from MD to TD. Mode II is relatively stable and is always the MD response.

 

Fig. 3. Far-field scattered power and dominant multipole components of the multipole decomposition of Mode I, Mode II and Mode III; (a), (b), (c) are $d$=30 nm (d), (e ), (f) corresponds to $d$=170 nm. ED, MD, TD, EQ, MQ correspond to the response of electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, and magnetic quadrupole, respectively.

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2.2 Active modulation of QBICs metasurface by ENZ materials

After establishing the knowledge of multiple BICs supported by such a metasurface, we move to investigate the optical modulation of the metasurface by introducing ENZ material ITO. The ITO film is sandwiched between the Si metasurface and ${\rm SiO_2}$ substrate, as shown in Fig. 4(a). The off-set distance of air-hole H5 is $d=$170 nm, the thickness of the ITO film is 10 nm, such the total hole depth is 230 nm.

In the visible and near-infrared regions, the dielectric constant of ITO can be described by the Drude model [9,73]:

 

Fig. 4. (a) Schematic diagram of the Si-ITO metasurface. The thickness of Si is 220 nm, the thickness of ITO is 10 nm. (b), (c) The changes of the real and imaginary parts of the dielectric constant of ITO with the current density $N_{\rm ITO}$, and the green dotted line is the corresponding ENZ position. (d), (e), and (f) are the transmission spectra of the metasurface without and with ITO film, respectively. The inset is the electric field distribution on the $x-y$ plane at the resonance wavelength, and $N_{\rm ITO}$ corresponds to the overlap between the resonance wavelength and the ENZ wavelength.

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Next, we show how such an ENZ material combining with QBIC helps to modulate the transmission spectra. The ENZ wavelength will change significantly when the carrier density $N_{\rm ITO}$ is changed, and the resonance mode will be affected. Here, in order to achieve an excellent optimal modulation, we manipulate the three modes respectively, as shown in Fig. 5. As $N_{\rm ITO}$ increases, the resonances are all blue-shifted due to the decrease of the effective refractive index of the nanostructures. As demonstrated above, the Q-factor and field distribution of the three modes are different under the same parameters, thus the influence of different $N_{\rm ITO}$ on the transmission amplitude and resonance linewidth is different, as illustrated in Figs. 5(a)-(c). We evaluate the modulation depth by the following formula [74] :

 

Fig. 5. Modulation of transmission spectrum with different $N_{\rm ITO}$ values. (a)-(c) The transmission spectra and corresponding modulation depths of Mode I, Mode II, and Mode III under different $N_{\rm ITO}$, respectively.

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Above we explored the modulation of the transmission for the three QBIC modes, and now we calculate the corresponding total far-field scattering and near-field light trapping modulation. Figures 6(a)-(c) show the normalized scattering intensity of three QBICs for different $N_{\rm ITO}$, it can be clearly seen that the three modes are significantly modulated in the scattering peak position and intensity. In addition, the corresponding contributions of multiple dipoles are shown in Fig. 9, and the different $N_{\rm ITO}$ value does not affect the dominant dipole response.

 

Fig. 6. (a)-(c) QBICs normalized far-field scattering modulation and (d)-(f) near-field capture modulation for different $N_{\rm ITO}$ values.

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The QBICs can generate huge local field enhancement within nanostructures. We finally investigate the modulation of light trapping under different $N_{\rm ITO}$ values. It is calculated by integrating the enhanced electric field intensity within the nanostructure and normalizing its incident intensity within the same volume. The calculated average field enhancement factor (EF) is shown in the following formula [75,76]:

The trapping and manipulation of light at the nanoscale are the important aim of nanophotonics. Enhancing the dielectric constant contrast between different materials serves as an effective way of controlling the electromagnetic field at the nanoscale. The ITO is one of the most widely used ENZ materials. Its carrier density can be effectively modified with an external bias. ENZ can be easily realized by controlling its carrier density, inducing large variations of its optical properties, for which near-unity modulation, as well as short, subpicosecond, modulation times can be achieved. In contrast to the other active materials commonly used to achieve the active optical modulation, such as phase change materials [75,78,79] and two-dimensional materials graphene [80], the dielectric constant of ENZ material easily reaches zero, and it holds a relatively large dielectric constant contrast when adjust the carrier density, which can lead to a more significant optical modulation.

3. Conclusion

In summary, we investigated the multi-BIC modes in Si nanostructures, and discussed the modulation of the transmission, far-field scattering, and near-field trapping of QBICs based on ENZ (ITO) materials. We found that by changing the relative psitions of the hole in a highly symmetric composite air-hole nanostructure, silicon metasurfaces with unit cell consisting of five square holes in silicon slab host multiple BICs. The nature of these QBICs properties is fully discussed through the near field distribution and the far-field scattering contribution of multipole via multipole decomposition. By introducing ENZ material (ITO) thin film, we systematically investigated the modulation properties of the hybrid Si-ITO metasurface. The modulation of transmission, normalized far-field scattering intensity, and near-field enhancement factor of the three QBICs modes are calculated by changing current density $N_{\rm ITO}$. The different light modulation characteristics are shown for different resonance modes. It is noted that the disturbance parameter of these QBICs can be accurately controlled in experiment for this design, which provides a way for the realization of an ultrahigh Q-factor. Our results pave a solid avenue to achieve the high-performance light modulation devices.

Appendix

A. Q-factors of free-standing metasurfaces

 

Fig. 7. (a) Schematics of unit cell for free-standing metasurfaces. (b)-(d) The Q-factors of Modes I, II and III with respect to the offset distance, respectively.

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B. Permittivity of ITO film

 

Fig. 8. Real (solid) and imaginary (dashed) parts of the permittivity of ITO as a function of wavelength. Points A, B, and C represent zero-crossing at $\lambda _{\rm ENZ}=1373$ nm, $\lambda _{\rm ENZ}=1440$ nm, and $\lambda _{\rm ENZ}=1528$ nm, respectively.

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C. Far-field scattering power of multipole for ITO with different current densities

 

Fig. 9. Multipole contributions of Si-ITO metasurface from three QBICs at different current densities $N_{\rm ITO}$ at $d=$170 nm.

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D. Optical modulation of Si-ITO metasurfaces without etching air-holes in ITO films

In the main text, we calculate the modulation properties of transmission spectra, far-field scattering intensity, and near-field trapping of the Si-ITO metasurface with etched air-hole in ITO layer. Here, we discuss the optical modulation properties of Si-ITO metasurfaces without etched air-holes in ITO film. The calculated results are shown in Fig.  10. Similar modulation performance is shown, and the modulation effect is slightly worse than the case where ITO is etched with air-holes.

 

Fig. 10. (a)-(c) Transmission spectra and (a)-(c) corresponding modulation depths under different $N_{\rm ITO}$ values.

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Funding

National Natural Science Foundation of China (12004084, 12164008, 11564005); Guizhou Provincial Science and Technology Projects (ZK[2021]030, ZK(2022)198, ZK[2022]203); Science and Technology Talent Support Project of the Department of Education in the Guizhou Province (KY[2018]043); Natural Science Research Project of Guizhou Minzu University (GZMU[2019]YB29, GZMUZK[2021]YB06, GZMUZK[2021]YB08); Youth Science and Technology Talent Project of Guizhou Province (KY[2021]105, KY[2021]117); Key laboratory of Guizhou Minzu University (GZMUSYS[2021]03); Construction Project of Characteristic Key Laboratory in Guizhou Colleges and Universities (Y[2021]003); Central Guiding Local Science and Technology Development Foudation of China (QK ZYD[2019]4012); Shanghai Pujiang Program (22PJ1402900).

Disclosures

The authors declare that they have no conflict 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.

Supplemental document

See Supplement 1 for supporting content.

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