  The second-order $\chi^{2}$ process underpins many important nonlinear
optical applications in the field of classical and quantum optics. Generally,
the $\chi^{2}$ process manifests itself only in a non-centrosymmetric
dielectric medium via an anharmonic electron oscillation when driven by an
intense optical field. Due to inversion symmetry, group-IV semiconductors like
silicon (Si) and germanium (Ge) are traditionally not considered as ideal
candidates for second-order nonlinear optics applications. Here, we report the
experimental observation of the second-harmonic generation (SHG) in a
Ge-on-insulator (GOI) sample under femtosecond optical pumping. Specially, we
report the first-time measurement of the SHG signal from a GOI sample in the
telecom S-band by pumping at $\sim$$3000$ nm.
