Optically-induced spin currents have proven to be useful in spintronics applications, allowing for sub-ps all-optical control of magnetization. However, the mechanism responsible for their generation is still heavily debated. Here we use the excitation of spin-current induced THz spin-waves in noncollinear bilayer structures to study optically-induced spin currents in the time domain. We measure a significant laser-fluence dependence of the spin-wave phase, which can quantitatively be explained assuming the spin current is proportional to the time derivative of the magnetization. Measurements of the absolute spin-wave phase, supported by theoretical calculations and micromagnetic simulations, suggest that angular momentum transfer via the s...