Finite-Size Effects in Nonlocal Metasurfaces

Abstract

Metasurfaces leveraging nonlocal resonances enable narrowband spectral control and strong near-fields, with applications spanning augmented reality, biosensing, and nonlinear optics. However, the large spa- tial extent of these modes also poses new challenges: finite-size effects often deteriorate the performance of practical, footprint-limited devices. Here, we develop a spatiotemporal coupled-mode theory model that intuitively and quantitatively captures how finite size affects the scattering response of nonlocal metasurfaces. This reveals that, when the modal propagation length becomes constrained by the phys- ical interaction length, the scattered field shows strong interference fringes and linewidth broadening. We derive an expression for the quality factor that incorporates an additional edge-loss channel, demon- strating that the stored energy and effective lifetime scale exponentially with the interaction length. We validate these predictions experimentally using position- and momentum-resolved spectroscopy on a 30-micron-wide metasurface. Overall, this work formalizes the impact of finite size on the scattering re- sponse of nonlocal photonic systems, and provides handles on how to minimize the impact of finite-size effects in metasurface design.

Figures (9)

• Figure 1: A nonlocal metasurface with finite-size effects. (a) An incident field $s_+(x',\omega)$ centered at $x_0$ excites a right-propagating quasi-guided mode with amplitude $a(x,t)$. The mode has an interaction length $L$, over which it decays through intrinsic loss ($\Gamma_\mathrm{int}$) and re-radiation ($\Gamma_\mathrm{ext}$). Power reaching the right edge, $x=W$, is irreversibly lost, giving rise to an additional edge loss channel $\Gamma_\mathrm{edge}$. The scattered field is denoted $s_-(x,\omega)$. (b) Modeled reflectance for an infinite, and (c) a finite metasurface array, respectively, showing the impact of finite size on the scattering response near the resonant frequency $\omega_0$ or in-plane wavevector $k_{x,\mathrm{res}}$.
• Figure 2: Semi-infinite guided-mode resonator. (a) Full-wave simulation of the electric-field profile $\mathrm{Re}(E_y)$ of the quasi-guided mode in a semi-infinite resonator comprised of a CSAR subwavelength grating on a hBN waveguide with an Au back-reflector. (b) Calculated TE0 guided-mode dispersion, where the grating is modeled as an effective medium. The grating folds the dispersion into the first Brillouin zone ($-\pi/\Lambda$ to $+\pi/\Lambda$), shown by vertical dashed lines, with the portion that lies within the light cone indicated by the gray shaded region. (c) Simulated momentum-resolved reflectance ($|r|^2$) showing the corresponding dispersion.
• Figure 3: Spatiotemporal coupled-mode theory of a finite-size guided-mode resonator. (a) Real-space input field $|s_+(x')|^2$ (salmon) and the corresponding scattered field $|s_-(x)|^2$ (purple), calculated for a $P_\mathrm{in} = 1$ W excitation at $x_0=22~\mu\mathrm{m}$ ($L=8~\mu\mathrm{m}$) and $\mathrm{NA}=k_{x,\mathrm{max}}/k_0=0.4$. The inset shows the abrupt truncation of the mode amplitude $a(x)$ at $x=W$. (b) The corresponding transverse-momentum representation of the input field $|S_+(k_x)|^2$. The resulting scattered field $|S_-(k_x)|^2$ shows broadening and fringes arising from edge truncation when compared to the semi-infinite response $|S_{-}^{\infty}|^2$ (blue, dashed). (c) Evolution of the $Q$-factor (top) and reflectance (bottom) as a function of $x_0$, for fixed $k_x/k_0=-0.13$, with the propagation length $L_p$ indicated with respect to the edge of the resonator (vertical, dashed). The $Q$-factor tends to the semi-infinite limit (horizontal, dashed) as $L$ increases. (d--h) The corresponding reflectance dispersion showing $x_0$-dependent interference fringes and linewidths.
• Figure 4: Position-dependent linewidth and interference pattern. (a) Optical micrograph of the fabricated resonator. Inset: atomic force micrograph of the subwavelength grating. The green star marks the excitation position ($x_0=9~\mu\mathrm{m}$). (b) Horizontal linecut through the center of the reflected beam imaged in real-space. Inset: same image recorded with a longer exposure to reveal the low-intensity features, scaled to match the intensity in the main panel. (c) The corresponding reflection imaged in momentum space (Fourier plane). (d--h) Measured momentum-resolved reflectance at different incident positions $x_0$, confirming the predicted interference fringes and linewidth variation.
• Figure 5: Comparison between model and experiment. (a) Measured reflectance versus excitation position $x_0$ and transverse momentum $k_x$, at $k_0=9.7~\mu\mathrm{m}^{-1}$. (b--d) Representative momentum-resolved reflectance linecuts (green) at selected $x_0$, overlaid with the global STCMT fit (purple). The fitted lengths $L$ are indicated. (e) Quality factor $Q(L)$ inferred from the global STCMT fit, with the extracted propagation length $L_p$ indicated (dashed line). Points represent fits performed independently at each linecut; the shaded band indicates the corresponding model discrepancy (one standard deviation) relative to these pointwise fits.