Study of Phase-Matched Normal and Umklapp Third-Harmonic- Generation Processes in Cholesteric Liquid Crystals

Using the model of Oseen and de Vries, we show that phase matching of optical third-harmonic generation can be achieved in a cholesteric liquid crystal with the help of the lattice momentum. Many different collinear phase-matching conditions exist. In some cases, the phase-matched third harmonic is generated in the same direction as the fundamental, and in some other cases, it is generated in the opposite direction. In many other cases, phase-matched third-harmonic generation requires the simultaneous presence of fundamental waves propagating in opposite directions. Analogous to electron-electron interaction in a periodic lattice, these processes can be identified as coherent, normal, and umklapp third-harmonic-generation processes. Experiments using a mode-locked Nd laser as the fundamental source verify the existence of most of the predicted phase-matching conditions. Our results agree well with the theoretical predictions.