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Abstract
Metasurfaces play a crucial role in trapping electromagnetic waves with specific wavelengths, serving as a significant platform for enhancing light–matter interactions. In all kinds of dynamic modulation metasurfaces, electro-optic modulation metasurfaces have attracted much attention due to its advantages of fast, stable and high efficiency. In order to respond to the extremely weak refractive index change of the electro-optical effect of the materials, the metasurfaces are required to support optical signals with high Q values. The quasi-bound state in the continuum (Q-BIC) is often used to enhance the light-field modulation capability of metasurfaces and to improve the modulation sensitivity of electro-optic modulators due to its ability to generate high Q-factor resonances. However, the design of an electro-optic modulation metasurface that facilitates the application of voltage and achieves modulation efficiency of nearly 100% is still in urgent need of development. In this study, single-crystal BTO metasurfaces are modeled using finite-difference time-domain method, and the structural symmetry is broken to introduce a Q-BIC resonance to generate a high Q-factor optical signal of 2.45 × 104 for high-depth electro-optic modulation. By simulating an applied electric field of 143 V/mm on the metasurface, a slight refractive index change of BTO of 8 × 10−4 was produced, leading to an electro-optical intensity modulation depth of 100%. Furthermore, the nanostructure of the metasurface was carefully designed to facilitate nano-fabrication and voltage application, and it is ideal for the development of low-power, CMOS-compatible, and miniaturized electro-optic modulation devices. Although the results of this study are based on simulations, they provide a crucial theoretical basis and guidance for the realization of efficient and realistic design of dynamic metasurfaces.
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1. Introduction
In recent years, metasurfaces made of sub-wavelength-sized materials have become a new platform for manipulation of optical signals. These metasurfaces can trap photons at subwavelength scales, resulting in the generation of localized electromagnetic energy with high density [1–4], which can be utilized to manipulate light fields in tiny spaces. Metasurface has been used in many fields for multi-dimensional optical signal manipulation, such as complex wavefront modulation [5,6], structural colors [7–9], and metalens [10–12]. In order to promote the practical application of metasurfaces and improve its application field, it is necessary to study the simple and feasible methods for the dynamic modulation of metasurfaces.
Currently, dynamic modulation of metasurfaces is achieved by various approaches: using phase-change materials [13,14], mechanical tuning [15–17], environmental modifications [18,19], and electrical modulations [20–23]. Among the above modulation methods, electrical modulation, especially electro-optical modulation, has the outstanding advantages of high speed, high stability, and ease of nanophotonics integration. Electro-optic modulation utilizes the characteristic that the refractive index of material changes under the applied electric field to realize the dynamic modulation of the optical signal. The Pockels and Kerr effects are the two most commonly used physical mechanisms for electro-optic modulation. The Pockels effect exhibits a linear relationship between the refractive index and applied voltage [24], whereas the Kerr effect demonstrates a quadratic nonlinear relationship [25]. In this study, the Pockels effect is selected for electro-optic modulation research due to its significantly larger refractive index change compared with the Kerr effect under CMOS-compatible voltage. The metasurfaces constructed by the fabrication of electro-optical materials into nanostructures exhibit dynamic optical properties that can be tuned by the applied voltage.
However, due to the extremely weak refractive index change caused by the electro-optical effect, it is necessary to introduce a high Q-value mode to maximize the optical modulation efficiency of the electro-optical metasurface at a low applied voltage. In various ways to achieve high Q, bound states in the continuum (BIC) possess remarkable capabilities for localizing light and exhibit an “infinite Q-factor” and no energy leakage [26,27]. The quasi-bound state in the continuum (Q-BIC) can be understood as BIC in the energy leakage mode. The BIC mode transitions into the Fano resonance with a finite ultra-high Q-factor, which is referred to as the Q-BIC when the system undergoes slight deviations from the ideal scenario [28,29].
Numerous researchers have investigated lithium niobate (LN) in the study of dynamic electro-optic metasurfaces [21,22]. For instance, grating-type LN electro-optic metasurfaces capable of realizing the phase-delayed dynamic modulation of the transmitted light were proposed and experimentally demonstrated by Bofeng Gao et al. Q-BIC was introduced by changing the incident angle of the light source. The refractive index of the electro-optic crystal was changed under the applied simulative electric field, and effective phase modulation of the transmitted light near the Q-BIC frequency point was realized [30]. In 2022, an ultrathin Mie-based free-space electro-optic modulator was designed and developed by Benea-Chelmus et al. This modulator was achieved by applying a voltage through the crossed electrodes to a mixed organic material covering an asymmetric periodic ellipsoidal metasurface. The experiments revealed that the electro-optic metasurface they designed and developed achieved a maximum modulation depth of 67% in optical phase modulation [20].
At the same time, the design scheme used to realize electro-optic modulation by covering the block of the metasurface with organic electro-optic material is not conducive to the portability of electro-optic modulators and their CMOS integration requirements. In addition, the scheme for realizing electro-optic modulation by directly applying a voltage to the block of the metasurface suffers from the fact that the resonance Q-factor generated by the metasurface is not high enough to achieve a high modulation depth, and also leads to the difficulty of uniformly applying the voltage when the unit cells of the metasurface are not connected to each other.
In this paper, an electro-optic modulation BTO metasurface was designed to address these issues mentioned above. The metasurface supports the Q-BIC mode and generates a high Q-factor resonance to realize the high efficiency of dynamic modulation under a low applied voltage, and a continuous layer of BTO connects each unit cell to satisfy the application of voltage. As the building material, the higher electro-optical coefficient makes the BTO more sensitive to refractive index changes under an applied electric field [24,31,32], which can improve the responsivity of the electro-optic modulator compared with that building of LN. The metasurface's high Q resonance can significantly enhance the modulation depth of the electro-optic modulator, increase its sensitivity, and facilitate the development of device miniaturization, low-power operation, and spatial freedom [22].
2. Design
Metasurfaces serve as nanoscale resonator arrays capable of modulating the properties of light, such as its propagation direction, intensity, and wavelength. This study meticulously designed a metasurface to achieve ultra-high Q-factor resonance with incident light, the nanostructure of the metasurface as illustrated in Fig. 1(a). Investigating this resonance will facilitate subsequent research on electro-optical modulation. The yellow segments at the ends of the metasurface in Fig. 1(a) indicate the electrodes that can modulate the signal by applying a voltage to the electrodes. Finite-difference time-domain simulation software was used as the research tool. The unit cell of the metasurface comprises three layers: the top pair of BTO pillars, the middle layer of the BTO continuous layer for voltage application, and the bottom layer of the SiO2 substrate, as illustrated in Fig. 1(b). The metasurface’s dimensional parameters are specified as follows: Ay = 515 nm, By = 475 nm, w = 420 nm, h = 445 nm, Hsub = 75 nm, Px = 1000 nm, and Py = 900 nm. The resonance peaks of the transmission spectrum generated by the BTO metasurface and incident light exhibit low Q values when the unit cell structure of the BTO metasurface is symmetric. However, this does not meet our requirement for studying ultrahigh-Q metasurfaces. In order to solve this problem, the BIC state is destroyed by changing the structural parameters, so that it can produce a high Q-factor Q-BIC resonance to enhance the light-matter interaction and improve the modulation performance of the electro-optical modulator [33–35].
Fig. 1. BTO metasurface structural design. (a) Schematic representation of the periodically arranged BTO array constituting the BTO metasurface; the yellow part represents the electrode applying an electric field. (b) Diagram of the unit cell structure of the BTO metasurface, comprising SiO2 with a BTO layer as the substrate and a pair of BTO columns in the top layer. The dimensional parameters of the metasurface are Ay = 510 nm, By = 475 nm, w = 420 nm, g = 75 nm, Px = 1000 nm, Py = 900 nm, H = 445 nm, and Hsub = 75 nm.
We use the linear electro-optical effect to study the electro-optical modulation of the BTO metasurface. This modulation method is embodied in the linear change law between the applied electric field and the refractive index of BTO. The expression for the correlation between the external electric field and the change in the refractive index is as follows:
Where the change in relative permittivity induced by the applied electric field E along the direction of light propagation is denoted by Δ(1/n2), the relationship between them can be expressed by the electro-optic coefficient tensor matrix γij. The three main electro-optic coefficients of BTO are γ13, γ33, and γ42. Among these, γ42 has the largest electro-optic coefficient and the most obvious change of refractive index under the applied electric field, γ42 = 1300 × 10−12 m/V [36,37]. As γ42 belongs to the transverse electro-optic coefficient, in order to utilise γ42 in the electro-optic modulation presented in this paper, the modulation mode employed in this paper is transverse electro-optic modulation. The incident light is incident through the z-axis direction, and the electric field is applied in the xy-plane parallel to the x-direction as shown in Fig. 1. Such modulation ensures that the electro-optic coefficient γ42 is the primary contribution to electro-optic modulation.3. Results and discussions
The Q-BIC introduced in this study, termed the symmetry-protected BIC, is an extensively investigated class of Q-BICs. Figure 2(a) illustrates a schematic of how structural symmetry is broken to introduce Q-BIC. The electric field direction of the incident light is set to coincide with the x-direction because the transmittance of the light signal across the metasurface is very low when the electric field direction is in the y-direction, which is unsuitable for electro-optical intensity modulation studies. Furthermore, the x-direction was not selected to break the structural symmetry for the introduction of Q-BIC because to achieve the high Q-factor required for electro-optic modulation in this study, a structural asymmetric parameter of 10 nm in the x-direction is needed, which implies finer processing techniques, higher micro-nano-processing cost and not conducive to practical applications, but the y-direction requires only 40 nm asymmetric parameter to achieve the high Q-factor resonance required for the study.
Fig. 2. Investigation of the Q-BIC on the metasurface. (a) Modification of the asymmetry parameter α to disrupt the structure’s symmetry and introduce the Q-BIC, (b) transmission spectra scanning across different asymmetry parameters of α, (c) transmission spectra at various α values, and (d) resonance Q-factor and resonance peak wavelength at various α.
The gap size in the metasurface unit structure was determined by combining the contribution of various gaps to the Q-factor during the metasurface design process and current fabrication techniques. As illustrated in the transmission spectrum scan in Fig. 2(b), to transform the BIC into a SP- BIC, we employed the symmetry breaking of Ay in the unit cell structure, transforming the BIC into a Q-BIC in leakage mode, and an asymmetry parameter α (α=Ay - By) was used to effectively modulate the Q-BIC excitation of the Q factor, which can be visualized in the metasurface as a localized effect on the energy of the light field. The BIC undergoes perturbation by adjusting the asymmetry parameter α, transforming it from an invisible BIC state to a visible Q-BIC. As α changes, the transmission spectrum changes from the invisible BIC state at α = 0 to the observable Q-BIC state. This change is depicted in Fig. 2(b) as a gradual widening and narrowing of the resonance band, which is attributed to the exceptionally high Q value of the Fano peak produced by the Q-BIC within this structure. In Fig. 2(c), the transmission spectrum variation plot illustrates that at α=0, no resonance effect is observed because the structure is in a perfectly localized state. However, the transmission spectrum exhibits ultrahigh Q resonance oscillations and shifts in the resonance peak frequency as α is varied. In Fig. 2(d), the Q value of the resonance peak progressively increases toward infinity as α tends to 0; however, as |α| increases, it decreases sharply. This trend arises from the decreasing localization effect of the Q-BIC energy in the leakage mode as the asymmetry factor increases, leading to a rapid decrease in the resonance peak’s Q-factor observed in the transmission spectrum. In addition, as α varies from small to large values, a linear change in the resonance peak position of the transmission spectrum was observed, as shown in Fig. 2(d).
Considering the existing metasurface fabrication technology, instrument fabrication tolerances, and the necessary electro-optical modulation depth for this study, α and the change in the resonance peak of the transmission spectrum were explored, and the asymmetric parameter α=+40 nm (Ay = 515 nm) was selected for electro-optical modulation research [38,39]. A comparison of the transmission spectra when α=0 and α=40 nm is shown in Fig. 3(a). The structure is in an unobservable BIC state and has an infinite Q factor value when α=0. The spectral resonance wavelength is 1541.5 nm, and the Q factor is 2.45 × 104 when α=40 nm. The scattering power of the magnetic toroidal dipole, electric toroidal dipole (ETD), magnetic dipole, electric dipole, electric quadrupole, and magnetic quadrupole (MQ) in the cartesian coordinate system after the multipole expansion of the quasi-BIC resonance in Fig. 3(a) is shown in Fig. 3(b). ETD and MQ dominate the contribution to the scattered power, but ETD is more than an order of magnitude higher than the contribution of MQ. The inset shows the decomposition of the ETD in different directions, where only the ETD in the z-direction contributes significantly. Therefore, this resonance is dominated by the component of the ETD in the z-direction. The computed electromagnetic field strengths in Figs. 3(c) and (d) show a significant toroidal displacement current distribution in the XZ plane and a toroidal magnetic field vector distribution in the XY plane, which is consistent with the findings that the ETD accounts for the major contribution to the resonance in multipole decomposition. The electric field under the ETD-Q-BIC resonance in Figs. 3(e) and (f) are highly concentrated in the y-direction gap and develop a higher field strength. This ETD-Q-BIC-induced energy localization effect plays a crucial role in the modulation performance enhancement of electro-optical intensity modulators.
Fig. 3. Transmission spectrum at α = 40 nm, (a) resonance peak map of the transmission spectrum at α = 40 nm, wavelength: 1541.52 nm, (b) far-field scattering multilevel expansion analysis at α = 40 nm, the inset shows the components of the ETD in the x, y, z directions, (c) plan view of the xy electric field at α = 40 nm, (d) distribution of magnetic field strength in the xy-plane at α = 40 nm, (e) plan view of the xz electric field at α = 40 nm, and (f) distribution of electric field strength in the yz plane at α = 40 nm, x = 0.
We investigated the simulated electro-optic modulation of the structure at α = 40 nm and calculated the change in the refractive index of the structure under the action of the external electric field via the linear electro-optic effect.
Through computation, the refractive index of BTO is shifted by ±0.001 when an electric field of ±12.7 V/mm is applied to the metasurface because of the electro-optic effect. This shift increases gradually as the voltage increases. Figure 4(a) shows the transmission spectra for the refractive index change (Δn) ranging from ±0.001 to ±0.005 under the applied electric field. The amplification of the resonance peaks of the transmission spectra corresponding to changes in the refractive index is shown in the inset. As shown in Fig. 4(b), with the change in Δn, the change in wavelength Δλ of the resonance peak corresponding to various Δn in the transmission spectrum shows a linear change.
Fig. 4. BTO electro-optic modulation using the electro-optic effect. (a) Transmission spectral changes corresponding to the refractive index of BTO under the effect of different external electric fields, (b) resonance peak offset Δ λ varies with refractive index Δ n, (c) schematic of the computation of the modulation depth for the electro-optic intensity modulator, (d) trend of modulation depth increase with variations in Δ n, (e) variation curves of modulation depth corresponding to different values of Δ n, and (f) distribution of electric field strength in the xy-plane when Δ n = 0 and Δ n = 8 × 10−4.
The modulation depth of the BTO metasurface modulator is determined by the amount of refractive index change under the applied electric field. This refractive index change results in the modulation of the transmission spectrum resonance peak, leading to a shift of the resonance peak (as shown in Fig. 4(d)). The principle of measuring the intensity of an optical signal is to integrate the intensity of the optical signal in an extremely narrow band distribution. Therefore, if the peak shift caused by dynamic modulation is too weak, the integrated intensity of the signal measured before and after modulation will not change significantly. To characterize the signal modulation capability of a dynamic metasurface, we introduce a criterion to calculate the overlapping integral area of the transmission spectrum before and after the refractive index change. A stronger modulation depth will further separate the transmission spectrum before and after modulation, and the overlap area will be smaller. Figure 4(c) shows a schematic representation of the modulation depth of the electro-optical intensity modulator. The modulation depth (D) is computed using the following formula:
The electric field strength E = 143 V/mm required for the BTO metasurface electro-optic modulator to achieve 100% modulation depth using formula (1) was calculated. The metasurface array designed in this paper is a rectangular array with 100 µm in the x direction and 90 µm in the y direction. Based on the distance of the metasurface electrode design and the relationship between E and voltage, the voltage required to reach 100% modulation depth on the BTO metasurface was computed to be only 14.3 V. This demonstrates that the BTO metasurface exhibits a notably low trigger voltage and high modulation efficiency during electro-optical modulation.
In the metasurface described in this study, the BTO substrate serves the purpose of providing electrodes that enable the applied electric field to act on top of the metasurface. Nevertheless, the electric field may not fully act on all BTO columns if the BTO substrate is excessively thin. The height of the BTO columns should be decreased and the thickness of the BTO substrate should be increased to ensure complete exposure of the metasurface to the applied electric field. This study investigates the optimal parameters for the height of the BTO by adjusting the height while preserving the existing modulation effect. As illustrated in Fig. 5(a), this study maintained a constant sum of the height of the BTO column and the BTO substrate, with H + Hsub = 520 nm. As the height of the BTO column decreased, the height of the BTO substrate increased. As depicted in Fig. 5(b), the resonance peaks of the BTO metasurface were observed to only redshift with increasing height of the BTO column, which did not substantially affect the resonance Q-factor of the Q-BIC. The transmittance spectra exhibit Q-BIC resonance peaks on the left side and Fano resonance peaks on the right side during the initial stage of decreasing the height of the BTO column.
Fig. 5. Impact of changes in the BTO pillar height on the modulation performance of the metasurface. (a) The sum of H and Hsub is fixed at 520 nm. Hsub increases as H decreases. (b) Changes in the transmission spectrum as H decreases from 445 nm to 320 nm. (c) Changes in the Q-factor within this interval as H decreases to 320 nm. (d) Transmission spectrum at H = 330 nm (Hsub = 190 nm).
Notably, the redshift observed in the Fano resonance peaks is not substantial. The Fano and Q-BIC peaks demonstrate varying degrees of redshift, with the Q-BIC peak demonstrating a more substantial change. As depicted in Fig. 5(c), when the height of the BTO column is gradually reduced, this phenomenon leads to the coupling of the two peaks, particularly around H = 330 nm. The coupling of the peaks leads to a significant change in the Q factor. The Q factor reaches its highest value of 2.53 × 105 at H = 330 nm. Although an increase in the Q factor significantly enhances the modulation performance of the metasurface, the ultra-high Q-factor when H = 330 nm reduces the modulation performance of the modulator. As illustrated in Fig. 5(d), a broad resonance peak on the left side considerably influences the Q-BIC resonance with the increase in H, and the transmittance of the metasurface to be reduced and unstable, resulting in a significant energy concentration within the metasurface. The energy concentration can weaken the modulation effect, preventing the modulation depth from reaching 100%. The modulation effect remains unaffected when the Q-BIC peak is unaffected by the ultra-broad resonance peak, rendering it more sensitive to the influence of an applied electric field. Consequently, the optimal values for H and Hsub were determined to be 340 and 180 nm, respectively.
4. Conclusion and Outlook
This study explored the electro-optical modulation characteristics of metasurfaces comprising asymmetric BTO pillars. High Q-factor Q-BIC resonance was introduced through the metasurface’s symmetry breaking substantially to enhance the constraints of light-field energy within the metasurface, thereby enhancing the modulation sensitivity and depth. It was observed that when a low voltage is applied to the metasurface, it leads to a high modulation depth, and the BTO metasurface exhibits sensitivity to variations of the refractive index when subjected to an external electric field. For instance, when a voltage of 14.3 V was applied to the BTO metasurface designed in this study, a modulation depth of 100% was achieved according to the judging criteria that innovatively proposed in this study. Moreover, incorporating a conductive layer of BTO in this study solves the problem of the electric field being difficult to apply to the discrete unit-cell of traditional metasurfaces to trigger the electro-optic effect of BTO. Although our conclusions are based only on simulation experiments, they still provide a theoretical basis and research direction for the study of BTO metasurface electro-optic modulators due to the realizability of structure parameters. Outstanding modulation capabilities in narrowband modulation, optical signal processing, quantum optics, and other aspects of optical communications are exhibited by this device. In addition, this device demonstrates higher sensitivity and modulation depth than other conventional waveguide electro-optical modulation devices, thus rendering it exceptionally promising for applications requiring low-power, miniaturized, and integrated electro-optical modulation devices. Furthermore, these results serve as a foundation for developing ultra-compact spatial light modulation techniques.
Funding
National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (62305075).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon reasonable request.
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