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Abstract
Spatial wavefront control and high-Q spectral filtering are both of great importance for various optical applications, such as eye-tracking for eyewear, planar optical modulators, and optical sensing. However, it is a great challenge to simultaneously satisfy these two functionalities in a metasurface due to the inevitable conflicts of local and non-local modes, where local modes of a single meta-atom manipulate the wavefront in a broadband range, while non-local collective modes of extended meta-atoms only support high-Q resonances at certain characteristic wavelengths. Here, we demonstrate a low-contrast dielectric non-local meta-grating that provides both spatial and spectral control of light in a broadband range of 700-1600 nm, offering elaborate wavefront shaping only for narrow-band resonances. Such counterintuitive functionality is supported by spatially tailored dark modes (quasi-bound states in the continuum) encoding with spatially varying geometric phases, while low-contrast dielectric provides broadband non-resonant transmission. Moreover, a broadband transparent polarization meta-grating with two resonance wavelengths is presented. Non-local geometric-phase metasurfaces open an exciting avenue for wavefront shaping and spectral manipulation, and may have potential applications in sensing, lasing, and spectral filtering.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurface is a kind of planar nano-optical device with a thickness less than or comparable to the wavelength of light [1–4], wherein orderly arranged independent (local) meta-units are capable of wavefront shaping. Such devices provide a compact platform for versatile spatial light field manipulation, such as meta-lenses with high numerical apertures or small aberrations [5–11], vortex beam plates [12,13], holographic display [14], and tunable metasurfaces [15,16]. As the optical interactions within meta-units of nano-scatterers are localized in deep subwavelength structures, light waves in the proximity of central wavelength or angle always inevitably exhibit similar optical responses (i. e., local responses with low Q-factor), leading to a great difficulty of spectral de-coupling regulations. In contrast, non-local metasurface characterized by long-range coherent coupling effects between meta-atoms could introduce sharp high-Q spectral responses, resulting from a collective mode-dominated optical response on many meta-units. Typical realizations of non-local metasurface include periodic photonic crystal slabs (PCS) [17,18], guided-mode resonators (GMR) [19–21], and symmetry-breaking PCS dimer perturbation structures that can access symmetry-protected bound states in the continuum (BIC) [22–24]. Such non-local structures are able to manipulate selectively specific wavelengths and angles to excite the non-local modes, promising applications ranging from structural color [25], sensing [26], modulation [27], enhancing optical nonlinearity [28,29], to optical analog computing in time and space [30,31]. However, these non-local metasurfaces with high-Q modes cannot spatially tailor the optical wavefront simultaneously.
Most recently, geometric phase was introduced into non-local metasurfaces to implement wavefront-shaping, which could shape the wavefront exclusively at resonance wavelength while keeping other undistorted [32–34]. In contrast with phase-modulated local metasurfaces or collective resonant metasurfaces, non-local geometric-phase metasurfaces offer an excellent ability to arbitrarily control the spatial and spectral properties of light, leading to promising applications in augmented reality [21], dynamic wavefront optical modulation [35], and thermal emission control [36]. However, one major limitation of these non-local geometric-phase metasurfaces is that the optical functionalities at different wavelengths are strongly tied together through their band structure [37]. How to implement multifunctional non-local geometric-phase metasurface at different operating wavelengths is still elusive. Note that a cascaded non-local metasurface with distinct resonant wavelengths has been proposed for hyperspectral wavefront-shaping, but with limited spectral selectivity [35]. Moreover, such a cascaded meta-optics system has stricter requirements in terms of transparency and bandwidth for single-layer metasurface.
Here, based on a configuration of low-contrast dimers, we demonstrate a non-local polarization meta-grating that is capable of steering light at selective wavelengths while keeping broadband transparent for near-infrared (NIR) light from 700-1600 nm. The low-contrast index meta-units are adopted to inhibit extraneous resonances and increase transparency. Specifically, the meta-units can support quasi-BIC (q-BIC) resonance modes that merge and degenerate in parameter space, which allows tunable resonance points to appear in the broadband transparent band. Under the excitation of circularly polarized light, the dimer meta-units assign a geometric phase of $\Phi \simeq 4\alpha$ to the cross-polarized transmitted light only on narrow-band resonances, providing a complete library for non-local geometric-phase metasurfaces. Moreover,we present a broadband transparent polarization meta-grating with two operating wavelengths. The non-local geometric-phase metasurfaces provide a compact and versatile platform for promising applications ranging from transparent displays, and optical communications, to optical sensors.
2. Results
2.1 Principle of dimerized meta-units
As shown in Fig. 1(a), a non-local geometric-phase metasurface enables the optical wavefront modulation only at the resonant wavelength while exhibiting transparency at other wavelengths. Based on the dimerized PCS selection rules [33], a typical dimerized meta-unit of a $p2$ space group consisting of two rectangular holes in a dielectric film is obtained, as illustrated in Fig. 1(b). In principle, the meta-units with parametric perturbations lead to leakage of the excitable q-BIC mode into free space with a finite radiative Q-factor (controlled by the structural asymmetry perturbation [23], $Q\propto 1/\delta ^2$, where $\delta =L-W$ ), resulting in sharp and high Q spectral responses.
Fig. 1. Non-local geometric-phase metasurface for broadband transparent wavefront shaping. (a) Functional schematic of the non-local geometric-phase metasurface, which spatially shapes wavefront only at resonant frequencies with sharp spectral features, while leaving that unchanged at other frequencies. (b) Schematic diagram of a meta-unit structure of the $p2$ space group of rectangle holes in the dielectric film with dimerized orthogonal perturbation. Meta-units with distinct rotation angles of $\alpha$ introduce a special geometric phase of $\Phi \simeq 4\alpha$ and can be tiled to shape the wavefront only on narrow-band resonances.
According to the in-plane symmetries [33,38], the in-plane rotation angle $\alpha$ and $\alpha +90^{\circ }$ determine whether q-BIC mode excitation from normal linear incidence with polarization angle $\phi$ into free space is allowed. As shown in Fig. 2(a), the $y$-polarization $\phi =90^{\circ }$ is allowed to couple into the space group when $\alpha =0^{\circ }$ and the $x$-polarization $\phi =0^{\circ }$ is allowed to couple when $\alpha =45^{\circ }$. Such linearly continuous behavior between the polarization angle of the q-BIC mode and the in-plane rotation angle of the hole, observed from Fig. 2(b), is similar to the well-known geometric phase [39]
Fig. 2. (a) Evolution of the transmission spectra versus polarization angle $\phi$. Spectra are relatively shifted by one unit. (b) Amplitude spectra of q-BIC with respect to polarization angle $\phi$ and rotation angle $\alpha$.
For right circularly polarized (RCP, $\Phi _{\phi +90^{\circ }} = \Phi _{\phi }+90^{\circ }$) incident light, the phase $\Phi$ of output light at resonance wavelength is attributed to two parts (shown in Fig. 3(a)): one from input light coupled from the spin state into the linear polarization $\Phi _1=\theta$, and the other from coupled radiation light decomposed from the linear polarization into the spin state $\Phi _2=\pm \sigma \theta$, where the signs correspond to LCP and RCP respectively. On the one hand, only the linearly-polarized light with half of the circularly polarized free-space incident light power in the polarization angle of $\theta ^r=\phi$ is fully coupled to the q-BIC mode that is resonantly reflected, while the orthogonally polarized light (polarization angle $\theta ^t=\phi +90^{\circ }$) is transmitted. On the other hand, the linearly-polarized resonant output light can be decomposed into its constituent LCP and RCP components carry the opposite geometric phase, i.e., $\Phi _2=\pm \sigma \theta$, where $\sigma =\pm 1$ for transmission and reflection, respectively. Thus, the total phase of the LCP and RCP components in reflection and transmission can be finally determined associated with the rotation angle, i.e., $\Phi =\Phi _1+\Phi _2$:
Fig. 3. Characterization of dimerized meta-units. (a) The phase of the non-local resonant light comes from the two parts of the coupling and radiation, where $\phi$ is the polarization angle coupled by the perturbation at the $p2$ interface. (b) Effect of different materials on the number and Q-factor of resonance points. The selected materials are ${\rm Si}(n=3.45), {\rm TiO_2}(n=2.47)$ and ${\rm Si_3N_4}(n=2.01)$, respectively, where $P=350$ nm, $H=250$ nm, $W=40$ nm, $L=150$ nm. (c) Out-of-plane electric field profiles on resonance for meta-units with different in-plane rotation angles. The rectangles denote the structure geometry.
For the spectrum response of dimerized meta-units at resonant and non-resonant, the widely used temporal coupled-mode theory (TCMT) is an effective way to analyze the performance. The dynamic equation for the resonant mode amplitude a of an optical resonator formed by dimerized meta-units can be written as [40]
Particularly, such meta-units structure assigns a geometric phase factor of twice the conventional geometric phase to transmit light with opposite chirality and reflected light with the same chirality on a narrow-band resonance. Figure 3(c) shows the twisted mode profiles of dimerized meta-units with a $90^{\circ }$ change in a rotation angle for one period ($2\pi$). In the non-resonant region, the two subunits of the meta-unit rotate at an angle difference of $90^{\circ }$, producing cross-polarized circularly polarized light with a phase difference of $180^{\circ }$, which satisfies the interferometric phase extinction condition and presents a wavelength-independent geometric phase broadband transparent response.
Using the finite element solver of the commercial software CST STUDIO SUITE, we develop a library of dimerized meta-units composed of typical rectangular holes in a platform of silicon nitride on fused silica to achieve broadband transparent non-local geometric-phase metasurface with finite controllable resonance point. As shown in Fig. 4(a-b), the transmission spectra of meta-units demonstrate that, for RCP incident light, cross-circular polarization (LCP) exhibits peaks at resonance wavelength while distinct dips appear for co-circular polarization (RCP). Especially, it keeps complete transparency in the non-resonant region over a range of 700-1600 nm. Importantly, the resonance points appearing in the broadband transparency window are wavelength shifted, merged, and degraded for different parameters, leading to a tunable capability for single- or dual-wavelength resonance. As predicted, the Q-factors of the resonant peaks are inverse square with the difference between $W$ and $L$, i.e., $Q\propto 1/(L-W)^2$. Note that, as only a specific linearly-polarized light (half of circularly-polarized incident) could excite q-BIC mode, the sum of transmittance of RCP and LCP light on resonance is below 0.5. The library in Fig. 4(c-d) represents meta-units with different $\alpha$ but identical geometrical parameters, i.e., fixed periods $P$ and $2P$ along with the x and y directions and dimensions $L, W$ of the rectangular aperture. The simulation results confirm that the optical geometric phase of the transmitted cross-circular polarization (LCP) is the approximately linear $4\alpha$ dependence of varying the in-plane rotation angle $\alpha$ from $0^{\circ }$ to $90^{\circ }$ at the corresponding resonant wavelength in each library. Specifically, for RCP incidence, the phases of transmitted LCP and reflected RCP light vary as $\Phi =2\phi \simeq 4\alpha$, while the phases of transmitted RCP light and reflected LCP light remain constant. Such meta-units provide a complete phase range from $0-2\pi$ for the resonant light. Therefore, a non-local geometric-phase metasurface with an arbitrary phase profile can be constructed for wavefront shaping.
Fig. 4. Low-contrast index meta-unit library design. (a) Cross-circularly polarized transmission spectra of the dimerized meta-unit library for RCP incident light, with dimensions $P$ = 350 nm, $H$ = 400 nm, $W$ = 70, 110, 150 nm and $L = 20-330$ nm. The inset shows the Q-factor corresponding to the resonant peaks. (b) Circularly polarized transmission spectra of cross-circular polarization (red) and co-circular polarization (blue) of the dimerized meta-unit with $L = 300$ nm, which corresponds to the meta-units marked by the white dashed line in (a). Cross-circularly polarized transmission spectra of the dimerized meta-unit with different in-plane rotation angle when $L$ = 260 nm and (c) $W=70$ nm or (d) $W=150$ nm. Second rows are the transmission amplitudes and geometric phases on resonance. The geometric phase is approximately four times function (indicated by the dashed line) of in-plane rotation angle $\alpha$.
2.2 Broadband transparent single-wavelength polarization meta-grating
The twisted mode exhibited by the $p2$ space group demonstrates the existence of a geometric phase. By spatially arranging these meta-units according to the generalized Snell’s law [42], a constant deflected phase gradient for anomalous refraction light can be easily assigned only on resonance. We demonstrate a proof-of-principle implementation in the form of a resonant polarization-grating that deflects the light of the selected wavelength and to selected directions. A metasurface is designed to create the desired phase profile by arranging meta-unit libraries with spatially varying $\alpha$ but with the exact aperture dimensions and the lattice constants. Figure 5 depicts the spatial distribution of the in-plane rotation angles for one period of the non-local geometric-phase metasurface. To mitigate the resonance frequency dispersion to the incident and deflection angle, the phase gradient is arranged along a direction orthogonal to the dimerization direction [33]. We devise a non-local metasurface with periodic phase gradient for NIR light transparency on a platform of rectangular apertures (with $L$ = 260 nm and $W$ = 150 nm) in 400-nm thick silicon nitride thin films on the glass, which deflects the portion of transmitted converted cross-polarization light to $\sim 43.5^{\circ }$ at a single resonant point. Unlike the conventional circularly polarized grating [43–45], the polarized grating designed using resonant non-local meta-units performs wavefront shaping only at resonance and only for cross-circularly polarized (LCP) transmitted and co-circularly polarized (RCP) reflected light (shown in Fig. 5(a)), while maintaining transparent at non-resonance (shown in Fig. 5(b)).
Fig. 5. Schematic of one super-period of the non-local geometric-phase metasurface on (a) resonance and (b) non-resonance, which consist of nine dimerized meta-units with spatially varying angles $\alpha$ from $0$ to $\pi$ to cover $4\pi$ phase range across 3.15 µm.
As shown in Fig. 6(a), the simulated transmission spectra show the high transmission of unconverted light and the minimal transmission of converted light at offset resonance. The non-local geometric-phase metasurface shows a very high 0th-order transmission amplitude (above 90$\%$) of the co-circularly polarized transmitted light at normal incidence across the NIR broadband range of 700-1600 nm. Notably, the transmission spectrum of this non-local geometric-phase metasurface shows a sharp spectral signature near 1.0846 µm, i.e., the resonance peak of the light of converted chirality cross-circularly polarized transmission with the corresponding decrease of the light of unconverted chiral co-circularly polarized transmission. The spectral features of the high-Q factor resonance depend on the collective resonance mode of an infinite extending number of coherent units transversely under the periodic boundary. This resonance mode, i.e., first coupling and then redirecting light, provides a second indirect pathway for the light to pass through the non-local metasurface structure. About 14$\%$ of the light is removed from the incident beam at the resonant wavelength and redirected into the second-order diffracted beam. Simulated optical wavefronts of the polarization meta-grating reveal that only on resonance the converted handedness light is angled by the gradient, while no deflection for unconverted RCP light or non-resonance. In contrast to the GMR metasurface [37], the non-local geometric phase metasurface enables decoupling of resonant amplitude light from direct transmission light.
Fig. 6. Transmission spectra and electric field distribution of single-wavelength non-local geometric-phase metasurfaces. (a) Simulated transmission spectra of the non-local geometric-phase metasurface for RCP (first row) and LCP (second row) for normal RCP incidence. The full width at half-maximum of the transmission resonance is only $\Delta \lambda$ = 1 nm, corresponding to a Q-factor of Q = 1072. (b) Simulated electric field (real part) profiles for both RCP and LCP both on resonance and non-resonance ($\lambda _r$ = 1.0846 µm and $\lambda _{nr}$ = 1.06 µm, 1.10 µm, respectively), which shows that beam steering (to a $43.5^{\circ }$ angle) only occurs on resonance for cross-circular polarization and remains largely transparent for non-resonant light. Both color scales are normalized to the maximum.
2.3 Broadband transparent dual-wavelengths polarization meta-grating
To clarify the wavelength decoupling capability of non-local geometric-phase metasurfaces, we extend the single-wavelength design to a dual-wavelengths scenario. The resonance peak is split in the parameter space ($W$ and $L$) of the dimerized meta-units, providing a tunable capability for dual-wavelength resonance. We design a resonant polarization meta-grating on a platform of rectangular apertures in a 400-nm thick film on glass to produce a gradient phase distribution with geometric-like phases. It is consistent with the in-plane rotation angles of Fig. 5, except that the dimensions of the meta-units structures are changed to $L$ = 260 nm and $W$ = 70 nm. Successive orthogonal chiral perturbations enable us to implement similar wavefront control on different resonances. As shown in Fig. 7(a), the diffraction energy distribution of non-local geometric-phase metasurface at specific dual-wavelengths is similar to that of the single-wavelength one. However, the resonant spectrum is split into two peaks, wherein about 20$\%$ of the energy is capable of anomalous resonant wavefront modulation. Moreover, the electric field distribution in Fig. 7(b) shows that the cross-circular polarization light only radiates at resonant wavelengths of 1074 nm and 1121 nm, deflecting the light to 43$^{\circ }$ and 45.4$^{\circ }$ direction, respectively. On the contrary, the co-circular polarization light normally transmits vertically at both resonant and non-resonant positions.
Fig. 7. Transmission spectra and electric field distribution of non-local geometric-phase metasurfaces for dual-wavelengths. (a) Simulated transmission spectra for RCP (first row) and LCP (second row) light for normal RCP incidence. The converted LCP light has resonant peaks with a Q-factor of $\sim$470 and $\sim$1120, respectively. (b) Simulated electric field (real part) profiles for both RCP and LCP light both on resonance and non-resonance ($\lambda _r$ = 1.0736 µm, 1.1206 µm and $\lambda _{nr}$ = 1.05 µm, 1.11 µm, 1.145 µm, respectively), deflect light at $43^{\circ }$ and $45.4^{\circ }$ for different resonant wavelengths. Both color scales are normalized to the maximum.
3. Conclusion
In summary, we demonstrate that low-contrast non-local geometric-phase metasurfaces could deflect light only at narrowband resonances and are otherwise transparent in the NIR broadband. It enables independent modulation in both spatial (by introducing successive orthogonal perturbations) and frequency (by tunning meta-unit geometry and perturbation intensity) domains, extending the capabilities of multifunctional meta-optics with high Q-factor wavefront shaping resonances. Moreover, the presented high-efficiency and wide-band non-local metasurface could implement wavefront shaping for dual-wavelengths in a single-chip configuration. Note that resonance detuning due to the variation of the in-plane rotation angle and the coupling between different meta-units makes extensive tuning calculations required in the device design. Such resonance detuning could further lead to deterioration of device performance especially when structural parameters vary from design or incident light is off-normal. The latest topology optimization by introducing inverse design can avoid complex calculations and achieve better performance with robust and compact structures. In conclusion, the non-local geometric-phase metasurfaces capable of wavelength-selective phase-modulation have promising applications, including resonant lenses, resonant holographic plates, active resonant wavefront shaping, and secure optical communications.
Funding
National Natural Science Foundation of China (12104509, 62105338); International S and T Cooperation Program of Sichuan Province (2020YFH0002).
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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