and right circular polarization for an incident femto-second pump beam (120 fs pulse duration, 1 kHz repetition rate at 1.01 eV) in (a) left and (b) right circular polarization. This result confirms that the SH signal polarization is a good observable of the valley degree of freedom of the WS$_2$ monolayer, with a contrast reaching ca. $80\%$. In Fig. 3 (c) and (d) in the main text, we show how this valley contrast is imprinted on the chiralitonic states.

![Resonant SH spectrum for left and right circular analysis, for (a) left and (b) right circular excitation at $1.01$eV. \[fig:S3\]](FigureS3.png){width="\columnwidth"}

D: Resonant SH generation in the strong coupling regime
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The resonant SH signal writes as [@Heinz1982; @ChervyNano2016]: $$I(2\omega)\propto (\rho_{\omega}I_{\omega})^2\cdot |\chi^{(2)}(2\omega) |^2 \cdot \rho_{2\omega}$$ where $I_{\omega}$ is the pump intensity, $\chi^{(2)}(2\omega)$ the second order susceptibility, $\rho_{\omega}$ the optical mode density of the resonator related to the fraction of the pump intensity that reaches WS$_2$ and $\rho_{2\omega}$ the optical mode density of the resonator that determines the fraction of SH intensity decoupled into the far field. While $\rho_{\omega}$ can safely be assumed to be non-dispersive at $\hbar\omega = 1$ eV, the dispersive nature of the resonator leads to $\rho_{2\omega}$ strongly dependent on 