the in-plane wave vector $k_x$. The optical mode density being proportional to the absorption, $\rho_{2\omega}(k_x)$ is given by the angular absorption spectrum crosscut at $2\hbar\omega = 2$ eV, displayed in the lower panels of Fig. 3 (e) and (f) in the main text.

Under the same approximations of [@LinPRA1993], the resonant second order susceptibility can be written as $$\chi^{(2)}(2\omega)=\alpha^{(1)}(2\omega)\sum_{n}\frac{K_{eng}}{\omega_{ng}-\omega}$$ where $\sum_n$ sums over virtual electronic transitions, and $K_{eng}=\langle e|{\bf p}|g\rangle\otimes \langle e|{\bf p}|n\rangle\otimes\langle n|{\bf p}|g\rangle$ is a third-rank tensor built on the electronic dipole moments ${\bf p}$ taken between the $e,n,g$ states. The prefactor $\alpha^{(1)}(2\omega)$ is the linear polarizability of the system at frequency $2\omega$, yielding resonantly enhanced SH signal at every allowed $|g\rangle \rightarrow |e\rangle$ electronic transitions of the system.

With two populations of uncoupled and strongly coupled WS$_2$ excitons, the SH signal is therefore expected to be resonantly enhanced when the SH frequency matches the transition frequency of either uncoupled or strongly coupled excitons. When the excited state is an uncoupled exciton associated with a transition energy fixed at frequency $2\hbar\omega=2$ eV for all angles, the tensor $K_{eng}$ is non-dispersive and the SH signal is simply determined and angularly distributed from $\rho_{2\omega}(k_x)$.

When the excited state is a 