Parametric coupling between modes of propagation in nonconservative systems

The small-signal theory of the parametric coupling of two waves in the presence of attenuation is summarized. The example chosen to illustrate the effects of losses on parametric coupling is two transmission lines coupled by a temporally and spatially varying capacitance. Included are analyses of contra- and co-flow parametric coupling, i.e., group velocities of the waves in the opposite and the same directions, respectively, with the pumping frequency either the sum or difference of the frequencies of the two waves. Contra-flow coupling with sum frequency pmnping is discussed in detail and it is found that the solutions for the mode amplitudes with loss are exponential, while without losses they are sinusoidal.