Dispersion relations in time-space periodic media part II—Unstable interactions

The conditions sufficient to assert the existence of absolute (time-growing) instabilities in distributed parametric structures or media are shown. Treating the simple, linearized, time-space periodic transmission line model, it is shown that time-growing instabilities exist in an unbounded medium whenever the pump modulation velocity is greater then the characteristic velocity of the medium and is in the convergent ("nonsonic") range of operation. The existence of these instabilities is based upon the rigorous calculation of the dispersion relation for the model, accounting for all small-signal frequencies. Proof of the existence of the instability, using the dispersion relation of the active structure, is based upon the method of Briggs, previously used in electron-stream interactions. The waveform of these instabilities is that of the backward-wave oscillator and comparisons with backward-wave oscillator (electron-beam) tubes are made. The predictions of the theory hold only for the transient period, for which the linearized model is valid, before the fields reach the nonlinear regime. The analysis, however, forms the basis for certain simple calculations of oscillation frequency, growth rates, predictions of competing interactions, and identification of wave types--all useful for application. The method of analysis should be useful for application to various other types of parametric interactions involving elastic waves, spin waves, plasma waves, etc., in single or multimode coupling schemes. The frequency range of application of such traveling-wave parametric effects extends from the optical range down to the radio-frequency range.