Parametric Amplification and Frequency Mixing in Propagating Circuits

When two circuits such as filters or transmission lines are coupled together by means of distributed reactances which can vary sinusoidally in time and space, energy can be converted between waves of different frequencies in a variety of ways. If the waves on the two transmission lines are characterized by frequencies ω1 and ω2 and phase constants β1 and β2, while the coupling reactances vary as ω=ω1−ω2 and β=β1−β2+Δβ, power P1 at ω1 is converted into power P2 at ω2 and vice versa in a manner reminiscent of waves on coupled passive circuits, except that a relationship (P2/ω2)=(P1/ω1) is obeyed. If the group velocities on both transmission lines are in the same direction, the direction of power transfer reverses periodically with distance. If they are not in the same direction, the power transfer increases monotonically with distance for small Δβ but reverses periodically for Δβ larger than a certain limit.When the coupling reactance varies as given by ω=ω1+ω2 and β=β1+β2, parametric amplification is possi...