incident light into a geometric phase $\Phi_g=- \Omega\sigma x$ [@bliokh2008coriolis]. The gradient of this geometric phase imparts a momentum ${\bf k}_g=-\sigma(\phi / \Lambda)\hat{x}$ added to the matching condition on the array between the plasmonic ${\bf k}_{\rm SP}$ and incidence in-plane ${\bf
  k}_{\rm in}$ momenta: ${\bf k}_{\rm
  SP}={\bf k}_{\rm in} + (2\pi / \Lambda) (n\hat{x}+m\hat{y})+{\bf
  k}_g$. This condition defines different $(n,m)$ orders for the plasmonic dispersions, which are transverse magnetic (TM) and transverse electric (TE) polarized along the $x$ and $y$ axis of the array respectively (see Fig. \[fig:2\](b)). The dispersive properties of such a resonator thus combines two modal responses: plasmon excitations directly determined on the square Bravais lattice of the grating for both $\sigma^+$ and $\sigma^-$ illuminations via $(2\pi / \Lambda) (n\hat{x}+m\hat{y})$, and spin-dependent plasmon OSO modes launched by the additional geometric momentum ${\bf k}_{g}$. It is important to note that the contribution of the geometric phase impacts the TM dispersions only. The period of our structure $\Lambda=480$ nm is optimized to have $n=+1$ and $n=-1$ TM modes resonant with the absorption energy of the $A$-exciton of WS$_2$ at $2.01$ eV for $\sigma^+$ and $\sigma^-$ illuminations respectively. This strict relation between $n=\pm
1$ and $\sigma=\pm 1$ is the OSO 