Identifying and tracking magnetically induced polarization in Fe$_2$Mo$_3$O$_8$ by static and time-resolved second harmonic generation

Multiferroic materials offer a promising platform for ultrafast optical control of coupled magnetic and polar orders. However, a prerequisite for such control is to precisely identify how the magnetically induced polarization manifests itself on the ultrafast timescale, and then to probe its dynamics upon external perturbations. Here, we address this issue in the polar magnet Fe$_2$Mo$_3$O$_8$ by combining static and time-resolved second harmonic generation (SHG). Temperature-dependent static SHG reveals that, among the symmetry-allowed tensor elements, only $\chi^{(2)}_{ccc}$ exhibits a pronounced anomaly at the antiferromagnetic transition ($T_{\rm N} \approx 60$ K), identifying the $c$-axis polar response as the primary degree of freedom coupled to the magnetic order. Guided by this result, time-resolved SHG selectively tracks the dynamics of this tensor element following ultrafast photoexcitation. We observe a rapid enhancement of the $\chi^{(2)}_{ccc}$-related SHG signal, followed by biexponential recovery. The response is independent of the pump polarization, consistent with an ultrafast thermal origin, but is achieved at a fluence significantly below that required for conventional lattice heating. These results establish SHG as tensor-selective probe of ultrafast magnetoelectric dynamics and demonstrate the high sensitivity of the magnetically induced polarization in Fe$_2$Mo$_3$O$_8$ to optical excitation.