Anisotropic below bandgap harmonic generation in $\beta$-gallium oxide

Harmonic generation in wide bandgap semiconductors and dielectrics has attracted a surge of interest as compact solid-state light sources and a method of mapping the band structure and crystal symmetry in the material itself. Longer driving wavelengths can support longer electron trajectories and suppress interband transitions, yielding spectra that are sensitive to crystal orientation and band shapes. Here, we combine a longer wavelength ($\approx$ 3.0 $\mu$m) laser to drive harmonic generation in a wide band-gap material with wide-ranging applications in photonics and power electronics, beta-gallium oxide ($\beta$-Ga$_2$O$_3$). We generate harmonics up to the ninth order (H9=3.7 eV) in air, with a $\approx$ 3.0 $\mu$m, 100 kHz, 40 fs mid-infrared (MIR) driving laser in (100) and (010) oriented single crystals of $\beta$-Ga$_2$O$_3$. We observe odd and even harmonics, spectral interference, and the strong anisotropic response of both odd and even harmonics to laser polarization. Our results are explained in terms of the microscopic crystal structure and macroscopic optical properties of $\beta$-Ga$_2$O$_3$, and reveal the high-order non-linear optical properties of this material.