Evolution of pulses in parametric wave interactions utilized in the initial-value mode

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Abstract
Traveling‐wave parametric devices operating in the initial‐value mode are modeled and analyzed. In the initial‐value mode of operation, a signal pulse is injected into a nonlinear medium and allowed to propagate into the medium before the parametric pump is turned on. Examples of such operation have been based on magnetoelasticity and on photoelasticity with microwave sources and lasers, respectively, as the pump, and elastic wave pulses as the signal. Such devices are characterized here by a generic model of coupled transmission lines, with time‐space periodic (parametric) coupling. Rigorous (Floquet‐mode) solutions of the periodic model are utilized for initial values of small‐signal perturbances having pulse envelopes. Unstable responses to the initial excitation are calculated and shown to be predicted on the basis of the technique of asymptotic response previously used in the theory of plasma instabilities. The Fourier‐La Place representation is evaluated here using the (ω‐first) order of integration best suited to the asymptotic response, in which the instability characteristics can be displayed directly in terms of saddle‐point location and branch‐point separation of the instability dispersion function in k space. Characteristics such as thresholds of convective and absolute instabilities are found in terms of pump amplitude and losses. Growth rates—temporal or spatial as appropriate—and instability (crest) velocities are found in terms of the transmission line parameters. Finally, as a practical matter, it is noted that these unstable ``natural responses'' of the pumped medium introduce a distortion of the initial pulse envelope which would be unwanted in applications for delay lines and echo memories.