Perfect all-angle asymmetric transmission via normal susceptibilities: exact spatial derivative by local meta-atoms and nonlocal metasurfaces

TL;DR

This work tackles the challenge of achieving accurate asymmetric all-angle transmission in nonlocal metasurfaces by deriving closed-form susceptibility conditions for exact first-order spatial differentiation with unity numerical aperture. Using grazing-angle Huygens' condition and GSTC-based analysis, it prescribes a specific set of local susceptibilities that yield $t(k_x) \\propto -j k_x$ across all propagating angles. The authors validate the theory with two realizations: a conceptual local meta-atom (rotated loop and loaded line) and a nonlocal PCB metasurface (misaligned loaded strips), both demonstrating near-ideal, high-NA spatial differentiation in simulations and full-wave studies. The methodology provides a universal, modular path to high-resolution asymmetric nonlocal metasurfaces and points toward extensions to TM polarization and optical-frequency implementations with loss-aware design. Overall, the paper delivers a rigorous framework linking abstract susceptibility balances to physical inclusions for all-angle, high-fidelity optical analog processing.

Abstract

We present a systematic methodology for realizing accurate asymmetric all-angle transmission in nonlocal metasurfaces. As a representative example, we derive closed-form susceptibility conditions for exact first-order spatial differentiation of unity numerical aperture, clarifying the role of each underlying balance. We provide rigorous and detailed designs of physically meaningful structures that directly feature such susceptibilities: a conceptual local meta-atom and a realistic nonlocal multilayered printed circuit board (PCB). Importantly, the latter leverages an intricate system of nearfield coupling beyond standard homogenization. Validated in simulations, our results provide a general and modular route to high-resolution asymmetric nonlocal metasurfaces for optical analog processing.

Figures (3)

• Figure 1: (a) Physical configuration of a spatially deriving (asymmetric) MS: the spatial profile of the incident field $e_{y}^{\mathrm{inc}}(x)$ is derived through transmission $\propto\partial_{x}e_{y}^{\mathrm{inc}}(x)$. In the spectral domain, the incident plane wave [$E_{y}^{\mathrm{inc}}(\vec{r})$] of tangential wavenumber $k_{x}$ is specularly reflected [$E_{y}^{\mathrm{ref}}(\vec{r})$] and directly transmitted [$E_{y}^{\mathrm{tran}}(\vec{r})$] with the desired transmission coefficient $t(k_{x})\propto -jk_{x}$. (b) All-angle meta-atom design (one period) consisting of a subwavelength loaded loop rotated by angle $\alpha$ around the $y$-axis and a loaded line (Sec. \ref{['Subsec:MetaAtom']}). (c) Realistic all-angle trilayered PCB design (one period) of misaligned printed loaded strips (Sec. \ref{['Subsec:PCB']}). Finalized parameter values are provided in the main text as a part of their careful tuning process.
• Figure 2: (a) Rotated loop geometry and coordinate system for manifesting anisotropy (Sec. \ref{['Subsec:MetaAtom']}). (b) Full-wave results (circle markers) for the 20-GHz transmission magnitude [$|t(k_{x)}|$, black, left abscissa] and phase [$\angle t(k_{x})$, red, right] versus tangential wavenumber (angle) through the rigorously tuned meta-atom in Fig. \ref{['Fig:Setup']}(b) in comparison to transmission through a perfect spatial differentiator $t(k_{x})=-j\widetilde{k}_{x}$ (dashed traces).
• Figure 3: (a) Analytical (solid traces, evaluated via an FB analysis Rabinovich2018AnalyticalRabinovich2018ArbitraryYashno2023) and full-wave (circle markers) results for the 20-GHz transmission magnitude [$|t(k_{x)}|$, black, left abscissa] and phase [$\angle t(k_{x})$, red, right] versus tangential wavenumber (angle) for the rigorously tuned PCB in Fig. \ref{['Fig:Setup']}(c) in comparison to transmission through a perfect spatial differentiator $t(k_{x})=-j\widetilde{k}_{x}$ (dashed traces) (b) Full-wave results (circle markers) for $f=20.11$ GHz compared to ideal values (dashed traces).