Group-Velocity-Matched Three Wave Mixing in Birefringent Crystals

We show that the combination of pulse-front slant, k-vector tilt, and crystal birefringence often permits exact matching of both phase and group velocities in three wave mixing in birefringent crystals. This makes possible more efficient mixing of short light pulses, and it permits efficient mixing of chirped or broad bandwidth light. We analyze this process and present examples. Differences in the group velocities of the three interacting waves in a nonlinear crystal often limits the effective interaction length. For example, in mixing very short pulses, temporal walk off can stretch the pulses in time unless the crystal is very short. Efficient mixing with such short crystals requires high irradiances, but the irradiances are limited by higher order nonlinear effects such as intensity-dependent refractive index and two-photon absorption. Improved matching of the group velocities can alleviate this problem, allowing longer crystal and lower irradiances. Similarly, for high energy pulses, practical limits on crystal apertures mandate temporally stretching the pulses to reduce irradiances. For the resulting chirped pulses, temporal walk off restricts the chirp range unless the group velocities are well matched. In addition to perfectly matching the group velocities of all three waves, it is sometimes useful to match two velocities, such as the signal and idler in parametric amplification, permitting broadband parametric amplification, or to arrange the velocities of two inputs to bracket the generated sum frequency pulse, giving pulse compression under suitable circumstances