Three-mode coupling in a passive dielectric waveguide with index modulation

The characteristics of a three-mode coupling in a passive dielectric waveguide are investigated for the case of even TE modes. The index is assumed to have a doubly periodic harmonic variation in the propagation direction. By a singular perturbation procedure using multiple space scales, the coupled-mode equations governing the Bragg interactions are systematically deduced. The three-mode coupling causes a stop band in frequency and the characteristics of the resulting evanescent mode are analyzed. For a periodic dielectric waveguide of finite length, the three coupled-mode equations are solved subject to suitable boundary conditions to obtain the reflection and the transmission coefficients. The frequency response of the reflection and the transmission coefficients are investigated and compared with the results of the pair of corresponding two-mode couplings. Representative numerical results are presented to illustrate the characteristics of the interaction of the three modes.