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

Complete dispersion characteristics are exhibited for electromagnetic waves propagating in a time-space periodically-modulated medium which is all-pass and dispersionless in the absence of the (pump) modulation. A rigorous solution, including all of the infinite number of time-space harmonics, is obtained for the linearized model, which also is representative of a traveling-wave parametric circuit. In contrast with previous work relating to traveling-wave parametric circuits, however, no assumptions are imposed on the present model regarding mode coupling. The dispersion relation is represented graphically in a form which is a generalization of the familiar one-dimensional Brillouin diagram for stationary periodic structures. From the repeated cell pattern one ascertains that for this medium only two types of interaction are possible, 1) stable interactions, of the stop-band type, associated with frequency conversion effects and 2) potentially unstable interactions, characterized by a stop band in wavenumber rather than in frequency. These types of interaction occur, respectively, when the phase velocity of the pump modulation is less than or greater than that of a signal in the unmodulated medium. Separating these two types of interaction is a "sonic" region, which appears whenever these two phase velocities are approximately equal The characteristics of the stable interactions and the sonic region are discussed in the present paper; the unstable interactions are considered in Part II (to be published at a later date). The amplitudes of several of the time-space harmonics have also been calculated and are shown to satisfy the Manley-Rowe relations in the stop bands; of particular interest is the role of the minor harmonics and the error introduced by using only two harmonics, as in coupled-mode theory.