Theory of phase and amplitude bunching in traveling wave interactions

A solution is given for the parametric interaction of two traveling waves (
\upsilon_{k, k}
) and (
\omega, k
) where
(\upsilon_{k, k}) \ll (\omega, K)
. The strong pump wave at frequency υ
k
is 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
\omega/K
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
x
and
t
dependence as the instantaneous frequency.