Asymptotic theory of mode coupling in a space-time periodic medium—Part II: Unstable interactions

A singular perturbation analysis of the coupling of plane electromagnetic waves in an unbounded medium having its permittivity modulated by a progressive sinusoidal wave is carried out for the waves progressing parallel to the modulation direction. The phase velocity of the wave modulation is greater than that of the electromagnetic waves in the unmodulated medium. The interactions of the fundamental wave progressing in the direction of the modulating wave with the first four harmonics progressing in the reverse direction are investigated. The periodic inhomogeneity gives rise to self and mutual interactions. The self effect, in general, produces a phase shift and causes the mutual effect to occur for smaller wavenumbers and higher frequencies than those predicted by the Bragg conditions. The mutual interaction introduces a complex change in the frequencies resulting in absolute instability of the interacting waves. Analytical expressions for the shifts of the wavenumber and the frequency as well as the other characteristics of the waves and instabilities in the interaction region are deduced. The asymptotic results are compared with the numerical results evaluated from the mode theory.