Theory of phase and amplitude bunching in traveling wave interactions

A solution is given for the parametric interaction of two traveling waves ( ) and ( ) where . The strong pump wave at frequency υkis not affected but modulates the index of refraction for the weak wave at ω. The solution is for an arbitrary initial time dependence of the weak wave and can describe a large change in the fractional bandwidth of the modulated wave. If the phase velocities υ and are equal, the cycles of the high-frequency wave bunch together in every other half-cycle of the modulation wave and draw apart in the alternate half-cycles. This strong phase bunching is accompanied by strong amplitude modulation that is found to have the same and dependence as the instantaneous frequency.