several microns.

![(a) White light image of a WS$_2$ flake covering the plasmonic hole array, and (b) its wide field PL. (c) Normalized image of the diffraction-limited laser spot on the structure (indicated by the white dashed rectangle in (c) and (d)). The logarithmic scale cross-cuts (red curves in (c) and (d)) are taken along the vertical axis, through the intensity maximum. (d) Normalized chiraliton PL image. \[fig:S2\]](FigureS2.png){width="0.8\columnwidth"}

We note that the PL of the WS$_2$ monolayer also extends further away from the flake above the OSO array (Fig. S\[fig:S2\](b)). We extract from this result the $1/e$ decay length of the plasmon to be $\sim 3.4 ~\mu$m. This value nicely agrees with that obtained in section A from the linear dispersion analysis.

C: Resonant Second Harmonic generation on a WS$_2$ monolayer
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As discussed in the main text, TMD monolayers have recently been shown to give a high valley contrast in the generation of a second harmonic (SH) signal resonant with their $A$-excitons. We obain a similar result when measuring a part of the WS$_2$ monolayer sitting above the bare metallic surface, i.e. aside from the plasmonic hole array. In Fig. S\[fig:S3\] we show the SH signal obtained in left 