CONTRADIRECTIONAL COUPLING IN GRATING-ASSISTED GUIDED-WAVE DEVICES

Contra-directional coupling in the grating-assisted guided-wave devices is studied by applying a vector nonorthogonal coupled-mode formulation. The coupled-mode equations are solved by a transfer matrix method, which is much more efficient and insightful than the direct numerical two-point boundary-value method. All the space-harmonics generated by the periodic grating are considered. The contra-directional power coupling can be understood in terms of the interference among the normal modes of the coupled waveguides with the grating perturbation. Phase-matching grating periods for maximum reflections are therefore equal to the beat lengths between the two normal modes involved in the coupling process. The reflections are built up constructively (Bragg reflection), resulting in the stopbands in the spectral response. The expressions for these new grating periods are obtained and compared with those derived from the conventional phase-matching conditions based on the modes of the individual waveguide (waveguide modes). The stopbands in the spectral response are precisely predicted by the new phase-matching conditions. The shift of the spectral peak from the central wavelength, reported by unexplained previously, is due to the improper choice of the phase-matching conditions.