A two-color time-resolved Kerr rotation spectroscopy system was built, with a femtosecond Ti:sapphire laser and a photonic crystal fiber, to study coherent spin transfer processes in an InGaAs/GaAs quantum well sample. The femtosecond Ti:sapphire laser plays two roles: besides providing a pump beam with a tunable wavelength, it also excites the photonic crystal fiber to generate supercontinuum light ranging from 500 nm to 1600 nm, from which a probe beam with a desirable wavelength is selected with a suitable interference filter. With such a system, we studied spin transfer processes between two semiconductors of different gaps in an InGaAs/GaAs quantum well sample. We found that electron spins generated in the GaAs barrier were transferred coherently into the InGaAs quantum well. A model based on rate equations and Bloch-Torrey equations is used to describe the coherent spin transfer processes quantitatively. With this model, we obtain an effective electron spin accumulation time of 21 ps in the InGaAs quantum well.