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Nonlocal metasurfaces for spectrally decoupled wavefront manipulation and eye tracking

Jung-Hwan Song, Jorik van de Groep, Soo Jin Kim, Mark L. Brongersma

arXiv:2102.05790v1physics.opticscs.LGeess.IV

TL;DR

Existing high-Q nonlocal metasurfaces have difficulty decoupling optical functions across wavelengths, limiting spectral control needed in applications such as eye tracking. This work uses absorption-tuned, atomically thin elements in a high-Q GMR metasurface; it redirects NIR light while preserving visible transmission, and demonstrates eye tracking with the prototype.

  • Problem

    High-Q nonlocal metasurfaces have had difficulty providing independent optical functions at different wavelengths, despite the need for greater spectral control.

    Source: Main text · pp. 2–3

  • Method

    The authors use spectrally dependent absorption in atomically thin pSi elements to control light flow in a high-Q, nonlocal GMR metasurface.

    Source: Main text · pp. 5–6

  • Results

    The measured TE GMR at 864 nm delivers 12.8% diffraction efficiency, while visible diffraction is 0.07% on average and zeroth-order transmission remains >85%.

    Source: Main text · p. 7

  • Takeaways & Limitations

    The metasurface provides independent optical functions across wavelength bands, and its resonant selectivity allows layers to be stacked.

    Source: Main text · p. 8

    The design requires a grating-height compromise: NIR diffraction rises with height, while visible diffraction and transmission constrain the acceptable range.

    Source: Main text · p. 6

Abstract

from arXiv · show

Metasurface-based optical elements typically manipulate light waves by imparting space-variant changes in the amplitude and phase with a dense array of scattering nanostructures. The highly-localized and low optical-quality-factor (Q) modes of nanostructures are beneficial for wavefront-shaping as they afford quasi-local control over the electromagnetic fields. However, many emerging imaging, sensing, communication, display, and non-linear optics applications instead require flat, high-Q optical elements that provide notable energy storage and a much higher degree of spectral control over the wavefront. Here, we demonstrate high-Q, nonlocal metasurfaces with atomically-thin metasurface elements that offer notably enhanced light-matter interaction and fully-decoupled optical functions at different wavelengths. We illustrate a possible use of such a flat optic in eye tracking for eye-wear. Here, a metasurface patterned on a regular pair of eye-glasses provides an unperturbed view of the world across the visible spectrum and redirects near-infrared light to a camera to allow imaging of the eye.

Main text

Conventional local metasurfaces provide limited spectral and angular control, while high-Q nonlocal designs have struggled to decouple functions at different wavelengths. This work uses absorption in atomically thin elements to let a high-Q GMR metasurface redirect near-infrared light while largely passing visible light, and demonstrates the approach in eye tracking.

Main text

  • Spectrally varying pSi absorption lets the high-Q GMR store and redirect NIR light but dissipates visible-range resonant energy before it leaves as diffraction.At 870 nm, the simulated Q is 8700 with internal-field intensity buildup of about 2500; in the visible, first-order diffraction is <0.1%.Source: Main text · pp. 5–6
  • The approximately 3-nm grating height balances stronger NIR scattering against visible diffraction and the requirement for >80% transmission.Visible diffraction falls below 0.1% for heights under 4 nm; at 2 nm, NIR diffraction already exceeds 4%.Source: Main text · p. 6
  • Atomic-scale thickness control was implemented over a 4 cm2 metasurface, but fabrication imperfections lowered measured Q to 540 from the simulated 1400.The authors attribute the lower measured Q to increased radiative decay from fabrication imperfections.Source: Main text · pp. 6–7
  • The fabricated GMR redirects 12.8% of 864-nm TE light while suppressing visible diffraction to 0.07% on average and retaining >85% zeroth-order transmission.The measured TE resonance was at 864 nm.Source: Main text · p. 7
  • Measured gaze directions from diffracted eye images agree with actual directions to within ~1°, while the prototype resolves key eye features at large rotations.The calculated image resolution is ~40 μm; the authors also report resolving 0.63-mm bars on a USAF target.Source: Main text · pp. 7–8
  • Because only resonant wavelengths are manipulated, the authors propose that these metasurfaces can be stacked to provide independent optical functions across wavelength bands.The conclusion identifies imaging, sensing, communication, display, and nonlinear optics as possible application areas.Source: Main text · p. 8

Supplementary Information

The supplementary analysis connects grating coupling and material absorption to the GMR’s wavelength-dependent behavior, while the dielectric stack is designed to maintain visible transmission.

S1. Optical constants for RCWA simulations

  • Across 500–900 nm, pSi’s refractive index decreases from 4.5 to 3.7 while its extinction coefficient decreases from 0.2 to 0.Si3N4’s refractive-index magnitude remains virtually constant at 2.0 across the spectral region of interest.Source: S1. Optical constants for RCWA… · p. 16

S2. Anti-reflecting layer design

  • The SiO2 capping layer offsets a transmission dip near 580 nm, and the combined SiO2/Si3N4 layers provide high transmission across the visible spectrum.The capping layer’s optical-path difference is 290 nm, corresponding to an approximately π phase difference near the dip.Source: S2. Anti-reflecting layer design · p. 16

S3. Grating coupling of free-space radiation to the GMR

  • A 1-μm-periodic grating couples normally incident light to the TE GMR near 870 nm, consistent with the experimentally observed 864-nm resonance.The coupling transfers second-order grating momentum, 4π/p; guided light propagates along the Si3N4 slab before leaking into first-order diffracted beams.Source: S3. Grating coupling of free-space… · pp. 16–17

S4. Temporal coupled-mode theory and power flow analysis of the GMR metasurface

  • Visible-range pSi absorption makes the GMR non-radiative, reducing extinction efficiency below 4% and yielding transmittance above 80%.In the NIR, the GMR remains in the radiative regime and redirects light into six diffractive orders.Source: S4. Temporal coupled-mode theory and… · p. 18

S5. Transmission and diffraction efficiency spectra for the high-Q GMR metasurface

The spectra show high transmission alongside diffraction, with narrow GMR features distinguished from broad dielectric-stack resonances.

S5. Transmission and diffraction efficiency spectra for the high-Q GMR metasurface

S6. Transition of GMR from nonradiative to radiative regime

Grating height controls whether the NIR GMR is radiative, and the supplementary procedures describe fabrication and oxidation-based thickness control.

S6. Transition of GMR from nonradiative to radiative regime

  • For grating heights below 2 nm, absorption accounts for more than 50% of total decay; at 2 nm, it falls below 50% and peak diffraction efficiency exceeds 4%.Source: S6. Transition of GMR from… · p. 19

S7. Fabrication of GMR structures

  • Fabrication leaves a 3-nm pSi layer on Si3N4 after wet oxidation, patterns a 2 cm × 2 cm area, and finishes with a silica capping layer.The described process deposits 200-nm Si3N4 and 50-nm pSi before oxidation and pattern transfer.Source: S7. Fabrication of GMR structures · p. 19

S8. Precise control over the pSi grating height by wet oxidation

  • Wet oxidation at 950°C provides atomic-scale pSi thickness control; a 2.8-nm layer is obtained after 30 minutes.Thickness is characterized by spectroscopic ellipsometry on wafers oxidized for different durations.Source: S8. Precise control over the pSi… · pp. 19–20

S9. Effect of over-etching of the pSi grating elements

Over-etching changes resonance bandwidth, peak efficiency, and background diffraction, with different peak-efficiency responses depending on wavelength and absorption.

S9. Effect of over-etching of the pSi grating elements

S10. Angle-resolved spectroscopy to map the GMR dispersion properties

The angle-resolved spectroscopy setup characterizes GMR dispersion and diffraction efficiency using two measurement modes. The collection system has an effective numerical aperture of 1/400.

S10. Angle-resolved spectroscopy to map the GMR dispersion properties

  • Cross-polarized transmission spectra measured against incident angle map the dispersion of both TE and TM modes.The broadband source stays on the optical axis while the metasurface is rotated; crossed polarizers transmit light only after interaction with the metasurface.Source: S10. Angle-resolved spectroscopy to… · p. 20
  • In diffraction mode, rotating the sample and source sweeps the collected frequency to obtain the diffraction-efficiency spectrum.A 1-mm-wide aperture at the collection lens’s back focal plane selects the frequency reaching the detector.Source: S10. Angle-resolved spectroscopy to… · p. 20
  • The effective numerical aperture is 1/400, corresponding to a collection area of approximately 1 mm2.Source: S10. Angle-resolved spectroscopy to… · p. 21

S11. Image formation with the GMR metasurface

The coordinate transformation maps object points to diffracted image points near the GMR plane, where a camera aligned with the diffracted output can capture a front view. The image is compressed toward the surface, while only incident rays within the GMR’s angular acceptance efficiently excite it.

S11. Image formation with the GMR metasurface

  • Equation S8 maps object points P (x0, z0) to image points P’ (xi, zi) using the incident and diffracted ray geometry.The derivation assumes the GMR mode is at the Γ-point (θ = 0°), so normally incident light excites it optimally, as in the experiments.Source: S11. Image formation with the GMR… · p. 21
  • For Q = 1400, the acceptance angle θc is ±0.25°; incident angles beyond θc hardly excite the GMR because they lie outside its spatial bandwidth.Source: S11. Image formation with the GMR… · p. 21
  • The transformation rotates the surface-normal vector toward the diffracted output and reduces its magnitude by cos2θo, squeezing the image toward the GMR surface.A camera oriented along the diffracted ray captures a front view even from an oblique direction; the image remains virtually attached to the surface regardless of eye distance.Source: S11. Image formation with the GMR… · pp. 21–22

S12. Diffractive ray imaging affords a large depth of field

Diffractive ray imaging places the eye image close to the GMR surface across different eye-to-surface distances. Artificial-eye images were clear at both tested distances, although the farther image was slightly blurrier.

S12. Diffractive ray imaging affords a large depth of field

S13. Q-factor of GMR and spatial resolution of diffractive ray imaging

The GMR’s propagation length and spatial bandwidth link its Q-factor to the angular acceptance and resolving power of diffractive ray imaging. The tested resolution suffices to distinguish relevant human-eye components, with greater resolving power requiring a lower Q-factor.

S13. Q-factor of GMR and spatial resolution of diffractive ray imaging

  • The incident wave couples to a GMR, propagates as a quasi-guided wave along the Si3N4 slab, then couples out through the grating as diffracted beams.The propagation length L is proportional to the GMR Q-factor and to the number Q/2π of optical cycles.Source: S13. Q-factor of GMR and spatial… · p. 22
  • The tested diffractive images resolve discs sufficiently to distinguish relevant human-eye components; greater resolving power requires lowering Q to widen the accepted angular range.The effective numerical aperture is limited by sinθc = πng/Q.Source: S13. Q-factor of GMR and spatial… · p. 23
  • Incident plane waves excite the GMR when their kx magnitude is below Δkx; the resulting acceptance angle θc sets the angular components collected from the object.The finite propagation length L sets the available spatial-frequency bandwidth in the x-direction.Source: S13. Q-factor of GMR and spatial… · pp. 22–23

S14. Functional glasses for Augmented Reality applications

The proposed high-Q guided-mode-resonance metasurface combines a thin, lightweight form factor with visible transparency and reduced rainbow effects. Resonance damping can switch off beam redirection, and the structures use Si-compatible fabrication processes.

S14. Functional glasses for Augmented Reality applications

  • Compared with conventional optics, diffractive elements reduce bulk and can provide high optical power efficiency, but gratings bring visible rainbow effects.Conventional optics trade optical power efficiency against transparency and can be bulky; darkened glasses reduce rainbows at the cost of transparency.Source: S14. Functional glasses for Augmented… · p. 23
  • The high-Q GMR approach is presented as addressing bulkiness, manufacture complexity, optical power efficiency, and rainbow challenges.Its wavefront manipulation is mediated by high-Q resonances, so damping a resonance can turn off diffraction even with minute absorption.Source: S14. Functional glasses for Augmented… · p. 23
  • High-Q resonance damping lets minute optical absorption switch off beam redirection while retaining visible transparency.The few-nanometer Si grating on a half-λ-thin dielectric film is compact, and the Si3N4 structures can be used on curved surfaces.Source: S14. Functional glasses for Augmented… · p. 23
  • Standard Si-compatible optical lithography and top-down dry etching are compatible with mass production schemes.The authors also numerically demonstrate potential asymmetric GMR designs with approximately 53% diffraction efficiency in full-field RCWA simulations.Source: S14. Functional glasses for Augmented… · p. 24

S15. Demonstration of accurate object tracking using diffracted images

The tracking demonstration maps diffracted images of a target to measured positions and compares those positions with known rotation-stage angles. The measured polar and azimuthal angles agree with the set angles to within about 1°.

S15. Demonstration of accurate object tracking using diffracted images

  • The experiment binarizes a diffracted image of a 4-mm circular target and extracts its centroid before rotating the target on polar and azimuthal stages.The target represents an eye and could correspond to the pupil or the boundary between the iris and sclera.Source: S15. Demonstration of accurate object… · p. 24
  • Target-to-axis offsets and initial stage angles determine the modelled x and y centroid coordinates as the polar and azimuthal angles rotate.The equations express the x and y coordinates as functions of those rotations and displacements.Source: S15. Demonstration of accurate object… · p. 24
  • A model incorporating target and rotation-axis offsets matches measured centroid coordinates when δp = 3.3 mm, δa = 2.5 mm, θo = 105°, and φo = 6°.Deviations at θ = 20° and 25° are attributed to non-ideal illumination that hinders binary-image extraction.Source: S15. Demonstration of accurate object… · p. 24
  • Measured azimuthal and polar angles agree with rotation-stage angles within ~1° deviation.The authors note that this is consistent with what is generally required in commercial eye-tracking systems.Source: S15. Demonstration of accurate object… · p. 25
  • The target’s measured horizontal diameter follows the expected cosine behavior during azimuthal rotation, with distortion visible only at large positive and negative angles.The authors state that distortion can be corrected by calibration and that the method can also measure pupil size and identify surface-normal gaze direction.Source: S15. Demonstration of accurate object… · p. 25

S16. Guided-mode resonance designs for asymmetric diffraction

The section presents two GMR designs for asymmetric diffraction: matching even and odd Huygens contributions, and asymmetric waveguiding. Simulations report peak efficiencies above 50% for the first design and above 40% for the second.

S16. Guided-mode resonance designs for asymmetric diffraction

  • The design principle is to increase efficiency into a desired output channel by suppressing fractional decay into non-desired channels.The section introduces two approaches to achieve highly asymmetric diffraction.Source: S16. Guided-mode resonance designs for… · p. 25
  • The first design uses 3-nm Si gratings as weak out-couplers and treats their diffraction with first-order Born approximation.The high-Q GMR guides coupled light in the Si3N4 slab and establishes the surface field that interacts with the gratings.Source: S16. Guided-mode resonance designs for… · p. 26
  • The Huygens amplitude into a diffractive order is determined by integrating polarization-current contributions across the unit cell.The contributions interfere with a kernel wave function for the output order.Source: S16. Guided-mode resonance designs for… · p. 26
  • 53% peak diffraction efficiency into the +1st order is obtained at the 907.3-nm GMR frequency, with asymmetric diffraction sustained at incident angles up to 7°.The matching condition is also reported to hold with several tens of nanometers of position or width error in the Si grating elements.Source: S16. Guided-mode resonance designs for… · pp. 27–28
  • Matching the even and odd local Huygens contributions makes the +m amplitude large while the -m amplitude nearly vanishes.The Si elements are placed around the two matching points, x = ±112.5 nm, where the even and odd integrands are equal.Source: S16. Guided-mode resonance designs for… · p. 27
  • Above 40% peak diffraction efficiency into the -1st order is simulated at 946 nm using asymmetric waveguiding with a 6° incident angle.The condition is stated to hold for incident angles larger than 2.8°.Source: S16. Guided-mode resonance designs for… · p. 28

S17. Independent control of spectrally-decoupled optical functions in wavelength-division multiplexing

The section numerically examines stacked high-Q GMR layers that operate at closely spaced wavelengths and can be controlled independently through selective absorption. The simulations show that one layer’s resonance can be damped while the other layers’ resonance locations and linewidths remain largely unchanged.

S17. Independent control of spectrally-decoupled optical functions in wavelength-division multiplexing

  • Stacking GMR layers is proposed as a way to multiplex or demultiplex beams at closely spaced frequencies while retaining narrow spectral bandwidths.Different optical functions can be integrated by stacking layers that operate at different wavelengths.Source: S17. Independent control of… · p. 28
  • The top-layer GMR in the simulated stack has Q = 12,000 and a 0.06-nm bandwidth at 710 nm.The three layers are designed to interact with normally incident beams at 710 nm, 705 nm, and 700 nm.Source: S17. Independent control of… · p. 29
  • The three-layer design provides spectrally decoupled optical functions at 710 nm, 705 nm, and 700 nm, with independent control through absorbers allocated to each grating layer.The functions are also spatially separated in the simulated local electric-field profiles.Source: S17. Independent control of… · p. 29
  • The numerical study uses plausible Lorentzian oscillators whose absorption can be switched on and off to represent control of each layer.The assumed oscillator linewidth is 10 nm and its amplitude is set to one twentieth of a reported monolayer WS2 exciton amplitude.Source: S17. Independent control of… · p. 29
  • Damping the top-layer GMR with absorption nearly restores the 710-nm response to that of a stack without the top layer.The authors use the natural optical absorption of pSi to erase visible-range beam-redirection functions and suppress rainbow effects.Source: S17. Independent control of… · pp. 29–30
  • The other two layers’ resonance locations and linewidths remain unchanged when one layer is independently controlled.The reported approximately 1.5% changes in transmission-dip depth or reflection-peak height are attributed to altered background spectra.Source: S17. Independent control of… · p. 30
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