  Monolayer transition metal dichalcogenides (TMDs) exhibit high nonlinear
optical (NLO) susceptibilities. Experiments on MoS$_2$ have indeed revealed
very large second-order ($\chi^{(2)}$) and third-order ($\chi^{(3)}$) optical
susceptibilities. However, third harmonic generation results of other layered
TMDs has not been reported. Furthermore, the reported $\chi^{(2)}$ and
$\chi^{(3)}$ of MoS$_2$ vary by several orders of magnitude, and a reliable
quantitative comparison of optical nonlinearities across different TMDs has
remained elusive. Here, we investigate second- and third-harmonic generation,
and three-photon photoluminescence in TMDs. Specifically, we present an
experimental study of $\chi^{(2)}$, and $\chi^{(3)}$ of four common TMD
materials (\ce{MoS2}, \ce{MoSe2}, \ce{WS2} and \ce{WSe2}) by placing different
TMD flakes in close proximity to each other on a common substrate, allowing
their NLO properties to be accurately obtained from a single measurement.
$\chi^{(2)}$ and $\chi^{(3)}$ of the four monolayer TMDs have been compared,
indicating that they exhibit distinct NLO responses. We further present
theoretical simulations of these susceptibilities in qualitative agreement with
the measurements. Our comparative studies of the NLO responses of different
two-dimensional layered materials allow us to select the best candidates for
atomic-scale nonlinear photonic applications, such as frequency conversion and
all-optical signal processing.
