  We study high harmonic generation in altermagnetic metals with and without
spin-orbit coupling. The altermagnetism manifests itself in the magnetic field
dependence of the low harmonics associated with intra-band dynamics. Spin-orbit
coupling leads to additional higher energy peaks and plateau structures
originating from inter-band transitions. While the pure altermagnet or
spin-orbit system exhibits no circular dichroism in the high-harmonic response,
an altermagnetic system with spin-orbit coupling shows such a dichroism. We
also analyze the spin currents and their high harmonic spectrum.
