GaAs quantum wells are simultaneously illuminated with near-infrared (NIR) radiation at frequency&ohgr;nirand intense far-infrared (FIR) radiation from a free-electron laser at&ohgr;fir.Magnetic fields up to 9 T are applied. Strong and narrow sidebands are observed at&ohgr;sideband=&ohgr;nir±2&ohgr;fir.The intensity of the sidebands is enhanced when either&ohgr;sidebandor&ohgr;niris near the onset of NIR absorption in the quantum well, or when&ohgr;firis near the free-electron cyclotron frequency. We attribute these sidebands to four-wave mixing of NIR and FIR photons whose energies differ by more than a factor of 100. ©1997 American Institute of Physics.