Wavefront shaping and modulation with resonant electro-optic phase gradient metasurfaces

Optical technologies spanning light detection and ranging (LIDAR), augmented and virtual reality (AR/VR), and light fidelity (LiFi) rely on generating or sensing wavefronts of light in a deterministic, controllable manner. Widespread adoption of these technologies requires high efficiency, large fields of view, and rapid modulation or tuning of the device response. Conventional spatial light modulators, which rely on mechanically actuated components or electrically tuned liquid crystals, can struggle to simultaneously satisfy each of these criteria. For example, mechanical modulation methods with MEMS devices exhibit a large dynamic range but suffer from slower operating speeds, stiction, and limited applicability when space and weight are constrained. Correspondingly, liquid crystal-based spatial light modulators are lightweight and fast, but pixel sizes span several micrometers, yielding limited resolution and field of view. Alternative approaches are needed to enable solid-state, high-resolution, and rapidly reconfigurable optical devices.¹ 

Modern developments in nanophotonics have achieved significant scaling of optical components while maintaining high efficiency. Metasurface lenses,^2,3 beamsteerers,⁴ holograms,⁵ and other wavefront shaping devices^6,7 promise to replace bulk optics with nanoantennas. Each antenna is designed to impart a specific amplitude, phase, and polarization to transmitted or reflected light; spatial variation of each antenna allows for complex operations. The sub-wavelength thickness of metasurfaces makes them particularly amenable for mobile devices.⁸ However, dynamic modulation of the optical transfer function is necessary for display and detector technologies.^9,10

Electrical modulation is a particularly attractive route for reconfigurable devices, promising dense integration and manufacturability similar to conventional electronic devices. However, the refractive index, a key parameter in metasurface design, weakly changes with an applied electric field. Conventional Pockels cells operate with millimeter path lengths, inconsistent with subwavelength antenna design. Other methods such as Stark tuning,¹¹ phase change materials,^12–16 carrier effects in semiconductors,¹⁷ epsilon-near-zero plasmonic devices,^18–20 and others²¹ can increase the effective index change, but face tradeoffs in efficiency and tunability. A design methodology that produces high-efficiency (> 50%), high-contrast, and high-resolution optical devices with modest electric fields is an area of active research within the metasurface community.

Here, we design highly resonant, electro-optic metasurfaces based on Si-on-lithium niobate (LiNbO₃, LNO) for efficient and dynamic electrical modulation of diffraction, without sacrificing the subwavelength size of the tuning elements. High-quality factor (high-Q) phase gradient metasurfaces create narrow-band diffractive responses in the far field²² and increases the photon lifetime in the near-field, desirable attributes for nonlinear²³ and reconfigurable^24,25 devices. First, we show how the beamsteering response at the design wavelength can be modulated with an applied bias. Next, we show full-field simulations for a metasurface beam splitter device, where an applied voltage between two contacts is used to turn on and off the beamsplitting response. This work provides a framework for high-efficiency metasurfaces with reasonable voltages, an enabling technology for deployable sensor arrays and wavefront shaping devices.

Lithium Niobate (LNO) possesses a large electro-optic coefficient. For an applied electric field polarized along the optical axis of the LNO (here, x-cut), the permittivity changes as

Here, the change in permittivity is directly related to both the electro-optic coefficient and the strength of the DC electric field along that direction. While $r33(≈31 pm/V)$ and $r13(≈9 pm/V)$²⁶ are stronger in LNO than other typical electro-optic materials, the associated permittivity change is still small. For example, with a uniform x-polarized electric field with the strength of E_x = 1 V/μm, the largest change is $Δϵxx≈7×10−4$⁠. To overcome the weak modulation associated with this, the electro-optic effect is typically magnified by highly resonant structures, as pioneered by the on-chip photonics community.^27–29 Here, a reasonable voltage of order 10 V shifts the resonant frequency the same order as the linewidth, leading to a significant (> 10 dB) change in transmission.

One of the simplest yet most fundamental phase gradient elements is a beamsteerer, which re-directs light to a pre-defined direction. A linear phase profile that spans $2π$ within a period p will re-direct light according to³⁰ 

Here, p is the supercell period, λ is the incident wavelength, n_i (n_t) is the refractive index of the incident (transmitted) medium, and θ_i (θ_t) is the incident (transmitted) angle relative to the metasurface normal. Using this metasurface design method, the period is subdivided equally into n metasurface elements that approximate the phase gradient as n blocks of the exiting phase. The designs here are based on commercially available thin-film Lithium Niobate wafers, with the addition of deposited or wafer-bonded Silicon on top. We first consider a beamsteering metasurface whose unit cell is composed of three 600 nm tall Silicon nanobars sitting on a 200-nm-thick layer of Lithium Niobate on an oxide substrate. The nanobar center-to-center difference is 600 nm and designed around a free-space wavelength of 1400 nm. First, we sweep the width of the bar using full-field simulations (COMSOL Multiphysics) using experimentally retrieved refractive indices for LNO.³¹ Figure 1(a) shows the unit cell of the finalized metasurface. Figure 1(b) shows the transmitted amplitude and phase exiting the metasurface as a function of the element width. We achieve the $2π$ phase while maintaining high transmittance with this design structure due to the overlap of electric and magnetic dipolar modes.³² Requiring three elements that differ in the exiting phase by $2π/3$⁠, we choose widths of 175 nm, 236 nm, and 350 nm.

Next, we simulate the beamsteering properties by creating a periodic supercell of the three bars. Figure 1(c) shows the diffraction efficiency of the allowed orders within 50 nm of the design wavelength. An efficiency approaching 70% transmittance is observed in the designed +1 diffraction order (⁠$θt≈51°$⁠), while less than 10% diffracts into either the 0th or −1st diffraction orders within the entire range.

In order to generate a high Q, we consider the in-plane guided modes supported by the metasurface elements. The index contrast between Silicon and the sub/superstrate allows the nanobars to act as waveguides.^33–35 Figure 1(d) shows the guided mode dispersion of the metasurface near 1400 nm, with the lowest six modes shown. Perturbations with periodicity matching the guided mode wavelength selectively couple to these modes.³⁶ Recent experimental work has demonstrated that phase gradient elements with translational invariance along one direction can be used as high-Q dipolar resonators for wavefront shaping.²² Here, coupling to a guided mode forms a guided mode resonance in the diffraction spectrum. The Q and the phase gradient response can be tuned based on the perturbation geometry, symmetry, and element used in the design³⁵ and can exceed thousands. Figure 1(e) shows the six lowest energy guided modes in this structure. As seen, the modes are generally confined within individual nanoantennas and are horizontally or vertically polarized. As the guided wavelength increases, higher order modes begin to appear. For our electro-optic application, modes that leak into the LNO are most desirable. For this reason, we choose mode 5, a higher order, vertically polarized mode in the largest bar, to actively modulate the diffraction.

To operate near 1400 nm, we choose a period Λ of 601 nm, slightly higher than the theoretical value due to the perturbation size. Figure 2(a) shows a top-down schematic of the geometry used here. The perturbation is a rectangular section (25 nm × 100 nm) removed from the Silicon. We plot the high-Q diffraction features of our metasurface in Fig. 2(b). We observe a Lorentzian-like resonance in the +1st diffracted order with an extracted quality factor of ∼31 000 at 1396.3 nm. The transmitted intensity into the +1st order varies from approximately 70% to 7% (⁠$≈$ 10 dB) across the resonance. We also observe a Lorentzian-like increase in the diffraction efficiency to the −1st order on resonance and a small, Fano-like feature in the 0th order. We confirm the high-Q nature of the mode within the largest bar by plotting the near-field electric field enhancement in Fig. 2(c). Here, the cross section shows field enhancements approaching 120 times within the silicon nanoantenna and mode shape matching the one selected from Fig. 1(e). Importantly, a large portion of the electric field penetrates into the LNO region.

We demonstrate the reconfigurability of this metasurface platform by applying an electric field in the LNO layer. We simulate an infinitely periodic system with a uniform x-polarized electric field within the lithium niobate layer and compute the diffraction through the structure at 1396.31 nm. The electric field changes the refractive index of the LNO, shifting the mode resonant frequency (see the supplementary material for a brief discussion).^37,38 Figure 2(d) shows the diffraction efficiency of $λ=1396.1$ nm light as the applied electric field is varied from −15 V/μm to +10 V/μm. Here, the diffraction efficiency into the +1st order changes by approximately an order of magnitude within this range. This result shows that the electro-optic effect can sweep out the entire range of the high-Q resonance within reasonable applied fields.

We further visualize this by plotting the E_x component of the optical field as a function of applied DC bias in Fig. 2(e). At a uniform applied field of −15 V/μm, we see a majority of diffraction goes into the +1st order, acting as a beamsteering metasurface. As the electric field changes to −2 V/μm, the +1st diffraction efficiency drops significantly, showing efficient modulation of this beamsteering structure. We also observe an increase in the -1st diffraction order, with equal diffraction into both orders with an efficiency of 18% each. This applied field, therefore, converts the device from a high-efficiency beamsteerer (70% absolute efficiency into the +1st order) to a lower-efficiency beam splitter (36% efficiency). Further modification of the perturbation geometry can potentially improve the overall efficiency.²² Finally, the −1st order dominates at 0.6 V/μm. These results show that the broad tunability enabled by high quality factor resonances in the diffraction spectrum of phase gradient metasurfaces and that efficient electro-optic modulation of particular diffraction orders can be designed.

Having demonstrated that an applied electric field can actively modulate the resonant diffraction from our metasurfaces, we now show the general applicability of this design. Using fully coupled electrostatic and electromagnetic simulations, we show the capability for dynamic beamsplitting. For our nanoscale beam splitter, we adapt one of the simpler design methodologies: two repeating unit cells radiate π out of phase with each other. The subwavelength periodicity means that the 0th order is canceled out, meaning that light evenly diffracts into the +1st and −1st orders.

We design our metasurfaces with a region with no metasurface element (the blank regions), while the other element should radiate with a π phase difference relative to the blank area at 1400 nm. A schematic of the unit cell is shown in Fig. 3(a). Here, the phase gradient element is two silicon bars with the same top width and a center-to-center difference of 400 nm. Figure 3(b) shows the phase variation as a function of bar top width. Figure 3(c) shows the beamsplitting efficiency as a function of top width for the entire supercell at 1400 nm. A top width of 235 nm maximizes the beam splitter efficiency.

Next, we compute the guided mode dispersion for the bar system near 1400 nm in Fig. 3(d). Because the unit cell is composed of two identical and closely spaced bars, the modes of each bar couple, leading to symmetric and antisymmetric combinations for each polarization type, as visualized in Fig. 3(e). Figure 3(e) displays the electric field distributions for the six lowest order modes, showing that the modes are indeed equally distributed between the bars. Like in the beamsteering metasurfaces, we additionally find that a higher-order, vertically polarized mode leaks more light into the lithium niobate region. We choose a perturbation period of 710 nm to couple into this mode.

We introduce a high quality factor mode by inserting 10 nm × 200 nm long perturbations into both silicon bars. Figure 4(a) shows the top-down view of the supercell describing the geometry. The period of 710 nm allows incident light to couple to guided mode resonances near 1400 nm, which we confirm in Fig. 4(b). Here, a resonant decrease in the ±1 diffraction orders at 1397 nm occurs within a resonance with extracted Q $≈$ 28 000. There is also a marked increase in the directly transmitted light (0th order). Consistent with our design, on resonance, we additionally observe resonant amplification of light within the nanoantennas. Figure 4(c) shows field enhancements approaching 110 on resonance, with the field additionally leaking into the lithium niobate.

As with the reconfigurable beamsteerer, we apply a uniform electric field in the +x direction to observe the electro-optic response. Operating at 1397 nm, we clearly see in Fig. 4(d) that the diffraction can be tuned with an electric field. The contrast is lower than that in the beamsteering example, but there is, nevertheless, a strongly observable signal that can be modulated on or off with the electric field.

To demonstrate the practicality of our design, we perform coupled simulations in a realistic device design. First, we choose to operate with only 2 supercells, a finite sized device that is close to previously demonstrated finite-sized metasurfaces. This further allows the Aluminum contacts³⁹ to be placed in close proximity, decreasing the required applied voltage. The contacts are applied laterally and spaced with a distance of $4 μm$⁠, acting as both an aperture and a biasing mechanism. We simulate this structure by first determining the local, spatially varying, electric field profile when a voltage difference of 1 V is applied to the contacts. Then, we perturb the refractive index of lithium niobate using Eqs. (1) and (2), linearly adjusted as a function of desired voltage. The contacts are applied such that the dominant electric field direction is in the x-direction, consistent with the previous, infinitely periodic, simulations. The local description of the electric field, therefore, defines a spatially dependent refractive index that would be observed in experiment. Finally, we simulate the scattering properties for a wave incident from below and record the AC optical field exiting the metasurface. A schematic of the device design and uniform DC electric field distribution is shown in the left panel of Fig. 4(e).

Operating at 1397 nm and with no applied voltage [Fig. 4(e)], we observe no discernable beamsplitting. The simulated transmitted power is approximately 47%, similar to the expected value in Fig. 4(b) (50%). However, applying a voltage of −30 V, we clearly observe beamsplitting in the far field. Here, two general lobes transmit at equivalent but opposite angles relative to the incident direction (i.e., to the +1 and −1 orders). Further, the field intensity drops off significantly in the direct (i.e., 0th) order direction, meaning that most of the power incident on the material is not directly transmitted. The transmitted power (76%) is similar to the value in Fig. 4(b) (78%) at −7.5 V/μm. The difference likely comes from the metasurface's finite size and finite illumination, leading to unintended diffraction of the input beam. This fully rigorous calculation demonstrates that electro-optic modulation can be used in finite size metasurfaces at reasonable voltages. We note that the required frequency shift, and hence, required voltage, in these designs is dictated by Q. For a lower applied voltage, a design with a higher Q can be used. Modifying the gate geometry or mode overlap with the electro-optic material can further reduce the required voltage.

We have shown in this work that electro-optic modulation is a simple scheme to realize reconfigurable optics in phase gradient metasurfaces. Using full-field coupled simulations, we have shown directly that highly resonant metasurface elements can be efficiently modulated with schemes that can, in principle, operate at high speeds, potentially useful in future sensing and communication platforms. Modifying the gate geometry and electro-optic mode overlap could realize highly efficient, subwavelength, and individually addressable phase pixels for fully dynamic metasurfaces. Finally, we have performed simulations where the metasurface element is LNO (see the supplementary material) to push the designs to the visible and to increase the mode overlap with the electro-optic material. The design methodology presented here, thus, represents a general route to transfer function modulation in subwavelength footprints, an enabling technology in the design of lightweight, reconfigurable, and efficient wavefront shaping devices.

See the supplementary material for a brief discussion regarding the frequency shift induced by electro-optics and preliminary simulations of device designs for visible light wavefront modulation using LNO nanoantennas.

We gratefully acknowledge support from an AFOSR grant (Grant No. FA9550-20-1-0120), which supported the work and the salaries of D.B., M.L., and J.A.D. The authors would like to thank Seagate Technology PLC for their support of this work, especially Zoran Jandric and Aditya Jain for their useful feedback and advice on this manuscript. D.B. would like to thank Elissa Klopfer for help in generating the schematic images in the figures.