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

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.