Mirror-Constrained Optical Hall Rectification in Asymmetric Gold Gratings:\\ Edelstein Conversion and a Phase-Seeded Kerr Four-Wave Mixing Instability

Transverse optical rectification has been reported in a one-dimensional asymmetric Au grating even though an ideal profile $h(x)$ retains the $xz$ mirror $\Me:y\to-y$. Because $J_y$ is odd under $\Me$, whereas the $p$-polarized bilinears $|E_x|^2$, $|E_z|^2$, and $\mathrm{Re}(E_xE_z^*)$ are even, th