Saturation effects in a travelling-wave parametric amplifier

The equations governing the voltage distribution along a parametric travelling-wave amplifier can, under certain conditions, be solved in terms of elliptic functions. The solution shows that there is a spatially periodic transfer of energy between the signal, idling and pumping waves. The maximum saturated power gain that can be achieved is 1 + ¿1/¿3 pump input power/signal input power where ¿1 and ¿3 are the angular frequencies of the signal and of the pump, respectively. The positions of these gain maxima can be calculated explicitly without invoking small-signal conditions.