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Abstract
We numerically investigate linear and nonlinear optical responses in metasurfaces consisting of Au double-gap split ring resonators (DSRRs). Symmetry-protected dual bound states in the continuum (BICs) in such plasmonic metasurfaces are observed at the near-infrared optical regime. Efficient second harmonic generation (SHG) is obtained at the quasi-BIC models due to the symmetry broken. The optimized SHG responses are obtained at the critical couplings between radiation and nonradiation processes at the linearly x- and y-polarized light, respectively. High conversion efficiency of SHG of a value 10−6 is arrived at the fundamental intensity of 10 GW/cm2 at the quasi-BIC wavelength under the y-polarized illumination. Large extrinsic and tunable chirality of linear and nonlinear optical responses empowered by quasi-BICs is acquired in asymmetry metasurfaces at oblique circularly polarized incidence. The results indicate that the plasmonic metasurfaces of symmetry-protected BICs at the near-infrared optical regime have great potential applications in the on-chip efficient frequency conversion, and the linear and nonlinear chiral manipulation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nonlinear optical effects have important applications in modern photonic functionalities, including frequency conversion, ultrafast-pulse generation, all-optical signal processing, and switching [1,2]. Among various nonlinear optical effects, second harmonic generation (SHG) is one of the most fundamental nonlinear phenomena for frequency conversion, nonlinear microscopy, and interface structural detection [1,2]. Usually, the phase matching between the fundamental beam and SHG beam in the bulk materials must be satisfied for the efficient output of nonlinear signal. While the development of integrated photonics which require the miniaturization of optical components and devices down to the (sub)wavelength scales, the enhancement of local field is preferred and becomes an effective method for the efficient nonlinear response [3–6].
Metallic nanostructures of plasmonic resonances have drawn intensive attention and interest in nonlinear optics [6,7]. First of all, metals possess intrinsic nonlinear susceptibilities that are orders of magnitude larger than those in dielectric materials [1], which is the foundation of plasmonic nanostructures for nonlinear applications. Plasmonic excitations in the metal nanoparticles called localized surface plasmon resonance (LSPRs) can confine light to sub-wavelength scales, and thus the significantly enhanced local field, to boost nonlinear optical effects [6,7]. Since the local field enhancement occurs around the surface of metal nanostructures, plasmonic systems can have much smaller sizes, as well as smaller mode volumes, than dielectric resonators for ultra-compact integrated devices. Besides the miniaturized structure and large enhancement of local field, the plasmonic nanostructures have other advantages, including sensitively tunable resonances via the size, shape, and surrounding environment, which make the plasmonic nanostructures form flexible components with desired geometries. Plasmonic excitations have the ultrafast response on the timescale of femtosecond for ultrafast optical processing [6]. SHG from metal nanostructures is a surface nonlinear effect, so it can be acquired at oblique incidence or by breaking the centrosymmetry of nanostructure at normal incidence. And thus, the geometry design, along with the corresponding LSPRs, is crucial to SHG response in plasmonic nanostructures. Though the metal nanostructures have the aforementioned advantages, there exist serious drawbacks, i.e., the low damage threshold and intrinsic Ohmic losses at the optical frequency compared with dielectrics. Especially, the $Q$-factor in metal nanostructures of LSPRs is usually much lower than 10, which limits the light-matter interaction time along with the enhancement factor of electric field [8,9]. Typically, the theoretical conversion efficiency of SHG in the representative U-shaped nanoparticles at normal incidence is around $7\times 10^{-10}$ at the intensity of 55 MW/cm$^2$ [10], and is comparable to the experimental data around $2\times 10^{-11}$ at the intensity of 110 MW/cm$^2$ [11,12]. Surface lattice resonances (SLRs), a collective resonance happened in periodically arranged nanoparticles due to the coupling of LSPRs of individual nanoparticles, is found to an effective way to suppress the Ohmic loss, and to raise the $Q$-factor on the order of 10$^3$ [9,13]. Significant enhancements of a factor 20-2400 for the second- and third-order nonlinear processes via SLRs in metal nanoparticle arrays were theoretically reported [14].
Photonic bound states in the continuum (BICs) are emerging platforms of ultra-high $Q$ factors and local field enhancements to enhance nonlinear responses [15–17]. However, mostly studied BICs are in lossless dielectric resonators since the lossy materials could severely reduce the $Q$ factors as well as the enhancement factors. Due to the intrinsic loss of metal at the optical frequency, investigations of BICs in plasmonic systems are seldom reported [18,19], but demonstrations on BICs in metal nanostructures are restricted to long wavelengths at the mid-infrared [20,21], THz [22], and GHz regimes [23], or plasmonic-dielectric hybrid systems [24–27]. The study on nonlinear responses in plasmonic systems of BICs is still lack [28], especially in the all-plasmonic nanostructures.
In this article, we numerically study the linear and SHG responses in all-plasmonic metasurfaces consisting of periodically arranged Au DSRR arrays. Symmetry-protected dual bound states in the continuum (BICs) in such plasmonic metasurfaces are observed at the near-infrared optical regime under both linearly and circularly polarized illuminations. Efficient SHG at the quasi-BIC models is obtained at the metasurfaces satisfying the critical coupling condition between radiation and nonradiation processes. Large and tunable chirality of linear and SHG responses empowered by quasi-BICs is acquired in the asymmetry metasurfaces at oblique circularly polarized incidence. The results show that the BICs can be realized in the all-plasmonic metasurfaces at the near-infrared optical regime, and efficient SHG can be arrived assisted by quasi-BICs.
2. Designed structure and numerical modeling
Schematic of the proposed structure is shown in Fig. 1. The array of periodic double-gap split ring resonators (DSRRs) made of Au is prepared on the fused silica substrate. The parameters of a unit cell are, the periodic $P_x=P_y$=500 nm, the length of DSRR $L$=300 nm, the width $w$=50 nm, the gap $g$=50 nm, $h=(L-g)/2$=125 nm, and the thickness is 50 nm. The geometry symmetry of a DSRR is broken by shifting one gap from the center to a distance $\delta$. The total length of a DSRR is kept as $L_{tot}=2(L-g)$=500 nm, and the degree of asymmetry is defined as $\alpha =2\delta /L_{tot}$.
The wave equations of fundamental and SHG waves in the frame work of harmonic in Au nanostructures are [1],
In the limit of $\beta \to 0$, the surface contribution to the $\mathbf{P}_{\mathbf{2}}$ in Au nanostructures can be written approximately by an effective nonlinear current sheet as [10],
The coupled equations can be solved by the finite element method (FEM) using the commercial software Comsol Multiphysics. The details of simulation are the similar as the description in our Refs. [29,30]. In a brief, a unit cell of the Au nanostructure on a semi-infinite thickness substrate is chosen for the simulation. One port in the air domain, named port 1, is used for the illumination of plane waves, and the other port in the substrate domain is denoted as 2. The light normally incidents on metasurfaces unless otherwise specified. The periodic boundary conditions are used for the surroundings of the simulation domain. Two steps are performed, e.g. the first step is to calculate the linear optical properties including the transmittance/reflectance/absorptance and the electric field $\mathbf{E}_{\mathbf{1}}$ at the fundamental wavelength, and the second step is to solve the Eq. (2) for SHG including the SH power and the SH electric field $\mathbf{E}_{\mathbf{2}}$ . The transmitted SHG conversion efficiency is defined as ${\eta} _{\textrm{SHG}} = \textrm{P}_{\textrm{SHG}}/\textrm{P}_{\textrm{in}}$ , where $\textrm{P}_{\textrm{in}}$ and $\textrm{P}_{\textrm{SHG}}$ are the power of incident fundamental light and transmitted SHG from the port 2, respectively. Next, we will separately present and discuss the results obtained under the linearly and circularly polarized light.
3. Results and discussion
3.1 Linear polarization
The linear transmittance of nanostructures at the $x$- and $y$-polarized incident light is shown in Fig. 2(a) and Fig. 3(a), respectively. The resonance wavelengths at the around 1200 nm ($x$-polarization) and 800 nm ($y$-polarization) in the nanostructure $\delta =0$ nm are ascribed to the dipole resonance modes. When the right gap is displaced ($\delta \ne 0$ nm), the quasi-BICs spectra of Fano and electromagnetically induced transparency (EIT) lineshapes are presented at the $x$- and $y$-polarization, respectively. The distributions of local electric field |$\mathbf{E}_1/\mathbf{E}_{\mathbf{in}}$| at the quasi-BICs in nanostructures of some particular $\delta$ are shown in Fig. 2(b) ($x$-polarization), and Fig. 3(b) ($y$-polarization), respectively. The maximum enhancement factor of local electric field arrives around 34 ($x$-polarization) and 39 ($y$-polarization) in the metasurface of $\delta$ =60 nm and 30 nm at the corresponding quasi-BIC wavelength, respectively.
Fig. 2. (a) Linear optical transmittance of the metasurfaces at the $x$-polarized incident light (the dash lines show the resonance dips and are just guides for the eye), (b) electric field around Au DSRRs and (c) surface charge on Au DSRRs of different $\delta$ at the resonance wavelength around 1800 nm. The white double arrow in the inset of (b) indicates the direction of incident polarization along $x$-axis. The black arrows in (c) present the schematic of electric dipole arrangements.
Fig. 3. (a) Linear optical transmittance of the metasurfaces at the $y$-polarized incident light (the dash lines show the resonance dips and are just guides for the eye), (b) electric field around Au DSRRs and (c) surface charge on Au DSRRs of different $\delta$ at the resonance wavelength around 790 nm. The white double arrow in the inset of (b) indicates the direction of incident polarization along $y$-axis. The black arrows in (c) present the schematic of electric dipole arrangements.
The quasi-BICs can be explained by the effective electric dipole moment $\textrm{P}_{\textrm{eff}}$. When the nanostructure of $\delta =0$ nm, the eigenmode analysis was conducted to obtained the BIC modes around 1800 nm and 800 nm [31]. The distributions of electric field and surface charge are particularly boxed in Fig. 2(b)(c) and Fig. 3(b)(c), respectively. At the $x$-polarization, the electric dipole with opposite direction and equal magnitude is formed in the top and bottom SRR, respectively, and thus the net contribution is zero. At the $y$-polarization, there are four mirror symmetric branches excitation, which can be deduced from the direction of electric field and the distribution of surface charge, also leading to zero net polarization. And thus the net $\textrm{P}_{\textrm{eff}}$ =0 will not introduce the additional radiation, but only the Ohmic loss (nonradiation process) exists due to the imaginary part of permittivity of Au, called the BIC state. However, when the symmetry of the nanostructures is broken, the dipole symmetry is also broken in terms of the orientation and magnitude (as shown in Fig. 2(c) and Fig. 3(c) for the nanostructure of $\delta \ne 0$ nm), and leads to the net $\textrm{P}_{\textrm{eff}}$. The radiation of $\textrm{P}_{\textrm{eff}}$ causes the leakage of BICs, i.e., the quasi-BICs. The resonant modes of quasi-BICs are analysed using a multipole decomposition under the Cartesian coordinate [32]. The electric dipole (ED) moment and electric quadrupole (EQ) moment are the dominate response at the $x$- and $y$-polarization, respectively (not shown here). The radiative quality factor $Q_r$ follows the inverse quadratic law for the small asymmetry parameter $\alpha$ ($\alpha \le 0.1$) as ${Q_r} = {Q_0}{\alpha ^{ - 2}}$ using the perturbation theory, where $Q_0$ is a constant determined by the metasurface being independent of $\alpha$ [33]. For the larger $\alpha$, the $Q_r$ decreases faster since the weak perturbation is not valid and the radiative rate $\gamma _{rad}$ becomes larger. The total quality factor $Q$ satisfies $1/Q=1/Q_r+1/Q_{nr}$, where $Q_{nr}$ is nonradiative quality factor due to the Ohmic loss of materials. The dependence of the total quality factor $Q$ on the asymmetry parameter $\alpha$ can be written as [34],
Fig. 4. $Q$ factors in metasurfaces of different asymmetry parameters at the $x$- and $y$-polarized incident light, respectively. The structures satisfy the critical couplings at different polarizations are labeled separately.
SHG responses in nanostructures under the $x$- or $y$-polarized fundamental light of an intensity 10 MW/cm$^2$ are shown in Fig. 5. When the nanostructure of $\alpha$=0, there is no SHG signal at both the $x$- and $y$-polarization of fundamental light due to the symmetry of nanostructure. In the nanostructure of $\alpha \ne 0$ nm, SHG signals become stronger as the broken of symmetry of nanostructures. The maximum conversion efficiency of SHG appears around the $\delta$= 60 nm ($\alpha$=0.24) and 30 nm ($\alpha$=0.12) when the fundamental light is of $x$- and $y$-polarization, respectively. This can be explained by the local field enhancement as well as the flow of second-order surface current $\mathbf{J}_{\mathbf{NL}}$ on the Au DSRR. According to the temporal coupled mode theory, the local field enhancement factor |$\mathbf{E}_{\mathbf{1}}/\mathbf{E}_{\mathbf{in}}$| at the quasi-BIC mode is proportional to |$\alpha Q(\alpha )$| [34]. The maximum enhancement occurs exactly at the critical coupling condition, i.e., $Q_r=Q_{nr}$ when $\alpha =\alpha _{cr}$. That explains why the maximum conversion efficiency of SHG occurs in the nanostructure of $\delta$ around 60 nm ($\alpha$=0.24) and 30 nm ($\alpha$=0.12) at the $x$- and $y$-polarized fundamental light, respectively. The SHG field distributions are presented in Fig. 5(c) and (d). However, it is noting that the SHG response in the nanostructure of $\delta$=30 nm at the $y$-polarized fundamental light is much larger than that in the nanostructure of $\delta$=60 nm but at the $x$-polarized fundamental light, though the local field enhancements at the two conditions are quite close to each other. The fact can probably be explained by the flow of second-order surface current $\mathbf{J}_{\mathbf{NL}}$ on the Au DSRR, as shown in Fig. 5(e) and (f). Since the surface current $\mathbf{J}_{\mathbf{NL}}$ on the interface of Au/substrate is much larger than that on the Au/air interface, we rotate the Au DSRR to mainly show the current distribution on the Au/substrate interface (Fig. 5(e) and (f)). First, it is clear that the maximum nonlinear surface current on the Au DSRR of $\delta$=30 nm at the $y$-polarized fundamental light is much larger than that on the Au DSRR of $\delta$=60 nm at the $x$-polarized incident light. Second, since the SHG radiation is mainly contributed by the net current on the Au DSRR, the direction of nonlinear surface current is crucial. The domain nonlinear surface current on the Au DSRR of $\delta$=30 nm locates on the right arm of the top U-shaped Au (Fig. 5(f)), and has a downward direction. Though the direction on the right arm of the bottom U-shaped Au is upward, the net nonlinear current is still of a downward direction considering the difference of magnitude, which has the same direction as that on the left arms of the top and bottom U-shaped Au to strength the total SHG radiation. While the nonlinear currents on the arms of the top and bottom Au DSRR of $\delta$=60 nm is always reversed with each other (Fig. 5(e)), even the remnant nonlinear currents due to the different magnitudes finally have a downward direction on both sides and have yield SHG radiation, but the SHG is subsequently weak. Besides the currents on the arms of U-shape Au are emphasized above, the currents on the bottom side of a U-shaped Au are exist, but they are almost cancelled due to the inverse directions and have the negligible contribution of SHG.
Fig. 5. SHG conversion efficiency calculated at the (a) $x$- and (b) $y$-polarized fundamental light, respectively. SHG field distributions at the optimized DSRR of (c) $\delta$ = 60 nm and (d) $\delta$ = 30 nm, and surface currents $J_{NL}$ on the DSRR of (e) $\delta$ = 60 nm and (f) $\delta$ = 30 nm, at the $x$- and $y$-polarized fundamental light, respectively. The intensity of fundamental light is 10 MW/cm$^2$. The units of (c) (d) are V/m, and (e) (f) are A/m$^2$.
The dependence of SHG conversion efficiency in the optimized nanostructures on the input intensity of fundamental light at the respective quasi-BIC wavelength is shown in Fig. 6. Considering the damage threshold of Au nanostructure, the maximum intensity is limited to be 100 MW/cm$^2$, which was used for experimental measurements of SRRs [11,12]. The maximum SHG conversion efficiency around $10^{-8}$ is arrived at the $y$-polarized fundamental light, and is three orders magnitude larger than the experimental value [11,12]. Though the peak intensity up to 40 GW/cm$^2$ was reported to be used for 40-nm Au colloids for third harmonic generation [35], the Au nanorods were tested to be deformed or even destructed at the high temperature or the higher laser intensity [36]. The damage threshold of light intensity for ultrathin alumina covered Au nanobars can be significantly improved to be over 10 GW/cm$^2$ [36]. And thus the conversion efficiency of SHG in the current plasmonic metasurfaces with protection layers still has much room to be enhanced at higher laser intensity. It is worth noting that, even the nanostructures of the protection layer can sustain the intensity of 10 GW/cm$^2$, the maximum conversion efficiency in the optimized plasmonic nanostructure is around $10^{-6}$, which is still orders of magnitude lower than that in the dielectric resonators of BICs, such as GaAs or AlGaAs resonators [16,30,37,38]. Further, the multiple BICs can also occur in the dielectric metasurfaces for nonlinear harmonic generation and mixing [39–41]. The much lower losses, bulk nonlinear responses, and the higher damage threshold in all-dielectric resonators, comparing with plasmonic nanostructures, indeed enable the larger nonlinear responses and efficient conversion efficiency. However, it has to be notice that, the characteristic size of the dielectric nanostructures is about $\lambda /n_{\textrm{eff}}$, where $\lambda$ is the resonant wavelength and $n_{\textrm{eff}}$ is the effective refractive index of the dielectric [42]. In order to reduce the size of dielectric resonators and confine the light tightly inside the cavity, dielectrics of high refractive index are required. Even the refractive index is up to 3.5 such as silicon or GaAs, the typical size of resonators of resonance wavelength which is far away the absorption regime of dielectrics is still several hundreds nanometers [16,30,37–42]. While the plasmonic nanostructures can tightly confine the energy at metal surfaces to as small as a few nanometers and allows for deep subwavelength localization [6,7], they are more suitable for ultra-compact integrated photonic elements. The aforementioned qualities, including the sensitivity of LSPR on geometries and ultrafast response, are the advantage of plasmonics nanostructures comparing with dielectric resonators in photonic devices. All in all, the large conversion efficiency, along with the high sensitivity of the nonlinear response on the asymmetry, shows the quasi-BICs in all-plasmonic systems of potential applications in integrated nonlinear devices.
Fig. 6. Dependence of SHG conversion efficiency on the input intensity of fundamental light. The slopes are 1 in the log-log graph.
3.2 Circular polarization
The proposed nanostructures with $C_2$ symmetry ($\alpha$=0) or with broken $C_2$ symmetry ($\alpha \ne$0) are achiral metasurfaces at normal incidence, since the symmetry of out-of-plane is still preserved [23]. The nanostructures with $C_2$ symmetry ($\alpha$=0) can also not give rise to the chirality at whatever oblique incidence or normal incidence due to the symmetry of in-plane. However, the extrinsic chirality can be realized and empowered by quasi-BICs in nanostructures of broken $C_2$ symmetry ($\alpha \ne$0) at oblique incidence [23]. In this part, we take the nanostructure of $\delta$ = 60 nm ($\alpha$=0.24) as an example to demonstrate the chirality of linear and nonlinear responses at quasi-BICs modes. The other nanostructures of broken symmetry possess the similar results, and the chirality at the other resonance modes are not discussed here.
The linear transmittance of the nanostructure at different angles of incidence under the incident light of left-handed circularly polarization (LCP) and right-handed circularly polarization (RCP) is shown in Fig. 7. For comparison, the linear transmittance from the nanostructure of the $C_2$ symmetry ($\alpha$=0) is also given. It is clear that the chirality can only be realized when the symmetry of the in-plane structure and the out-of-plane incidence is both broken, and the parameter of chirality CD which is defined as $\textrm{CD}=|\textrm{t}_\textrm{R}|^2-|\textrm{t}_\textrm{L}|^2$ is empowered by quasi-BICs with the increase of incident angles, where $t_R$ and $t_L$ are the transmission coefficients of RCP and LCP light, respectively. The CD value at the quasi-BIC I (as marked in Fig. 7(a)) is around 0.17 at angle of incidence ${15}^{\circ}$, and increases to be 0.32 and 0.62 at ${30}^{\circ}$ and ${60}^{\circ}$ (not shown here), respectively. However, the CD is not changed obviously due to the mixing of higher-order plasmon resonances. So we mainly focus on the chirality on quasi-BIC I models. SHG responses in the nanostructure of $\delta$= 60 nm at different incident angles are shown in Fig. 7(b). The chirality of SHG at the quasi-BIC I model appears at the oblique incidences as the linear transmittance. For example, the conversion efficiency of SHG is around $1.29 \times 10^{-13}$ and $2.31 \times 10^{-14}$ at the illumination of LCP and RCP fundamental light of incident angle ${30}^{\circ}$, respectively. The value of the normalized SHG-CD varies between -2 and 2. The normalized SHG-CD can be described as $\textrm{SHG}_{\textrm{CD}}=(\textrm{P}_{\textrm{LCP}}-\textrm{P}_{\textrm{RCP}})/[(\textrm{P}_{\textrm{LCP}}+\textrm{P}_{\textrm{RCP}})/2]$ [43], where P is the power of SHG, and the subscripts LCP and RCP indicate the left- and right-hand circularly polarization of fundamental light, respectively. The SHG-CD at the quasi-BIC I is around 0 at normal incidence, and increases to be 1.26, 1.39, and 1.71 at the angle of incidence ${15}^{\circ}$, ${30}^{\circ}$ and ${60}^{\circ}$, respectively. The difference of SHG response at the LCP and RCP of fundamental light is ascribed to the local field enhancement and the distribution of nonlinear surface current $\mathbf{J}_{\mathbf{NL}}$. The distributions of fundamental- and SHG-field around the DSRR, and the second-order surface current on the DSRR, at the quasi-BIC I wavelength when the incident angle is ${30}^{\circ}$, are shown in Fig. 8. The maximum enhancement factor of fundamental field of LCP is around 36 (Fig. 8(a)), which is larger than that at the RCP condition (Fig. 8(d)). It is interesting that the maximum enhancement of local field occurs at different gaps when the fundamental light possesses the LCP and RCP, respectively, i.e. the maximum enhancement appears at the left gap of DSRR at the LCP of fundamental light, but at the right gap of DSRR at the RCP condition. Such difference gives rise to the different SHG field and nonlinear surface current, or the chirality of SHG. At the same intensity of fundamental light (10 MW/cm$^2$), the maximum SHG field is around $3.6 \times 10^3$ V/m at the LCP of fundamental light (Fig. 8(b)), and $1.4 \times 10^3$ V/m at the RCP of fundamental light (Fig. 8(e)). The surface current on the DSRR is also dominated on the Au/substrate interface, not the Au/air interface, as shown in Fig. 8(c) and (f). It is worth noting that the surface current on the two arms of bottom and top U-shaped Au structures almost has the same direction to enhance the SHG response when the polarization of fundamental light is LCP (Fig. 8(c)), while the surface current has the inverse direction on the two arms of bottom and top U-shaped Au structures at the RCP fundamental light (Fig. 8(f)). And the magnitude of maximum surface current is around 34 A/m$^2$ at the LCP fundamental light, which is twice of that at the RCP fundamental light. These could explain why the SHG response at the LCP condition is much larger than that at the RCP condition, and the chirality of SHG. Such large extrinsic and tunable chirality of linear and nonlinear responses in asymmetry metasurfaces empowered by Quasi-BICs has important potential applications in spin-selective linear transmission and frequency conversion, and on-chip linear/nonlinear chiral manipulation.
Fig. 7. (a) Linear transmittance and (b) SHG response in the metasurface of $\delta$ = 60 nm under the $LCP$ and $RCP$ incident light at the angles of incidence ${0}^{\circ}$, ${15}^{\circ}$ and ${30}^{\circ}$, respectively.
Fig. 8. (a) (d) Fundamental local field, (b) (e) SHG field, and (c) (f) SHG surface current on the metasurface of $\delta$ = 60 nm. (a) (b) (c) are obtained at the LCP fundamental light, (d) (e) (f) are calculated at the RCP fundamental light. The units of (b) (e) are V/m, and (c) (f) are A/m$^2$.
4. Conclusions
We numerically investigate the linear and nonlinear optical responses in metasurfaces consisting of Au DSRRs of symmetry-protected dual BICs under linearly and circularly polarized illuminations at the near-infrared optical regime. The optimized nanostructures for SHG response are respectively obtained at the quasi-BIC models in symmetry-broken nanostructures due to the critical coupling between radiation and nonradiation processes at the linearly $x$- and $y$-polarized incidences. Large extrinsic and tunable chirality of linear and nonlinear optical responses empowered by quasi-BICs is acquired in asymmetry metasurfaces at oblique circularly polarized incidence. The results indicate that the plasmonic metasurfaces of symmetry-protected BICs at the near-infrared optical regime can be used to modulate the linear and nonlinear signals sensitively, and have great potential applications in the on-chip chiral linear and nonlinear photonic devices.
Funding
National Natural Science Foundation of China (12174228, 12274271).
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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