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

A singular perturbation analysis of the coupling of plane electromagnetic waves in an unbounded medium having its permitivity modulated by a progressive sinusoidal wave is carried out for the waves progressing parallel to the modulational direction. The phase velocity of the wave modulation is less 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 introduces self-effect and mutual effect. The self effect, in general, produces a phase shift and causes the mutual effect to occur for different frequencies and wavenumbers from those predicted by the Bragg conditions. The mutual effect introduces a complex change in the wavenumber resulting in the evanescence of the interacting waves. Analytical expressions for the shifts in the frequency and the wavenumber as well as the other characteristics of the waves in the interaction region are deduced. The asymptotic results are compared with the numerical results evaluated from the mode theory.