A rigorous modal approach based on the transmission-line description has developed to explore effectively the filtering characteristics of planar optical DFB guiding structures. Using the modal transmission-line theory, the leakage and filtering characteristics of metal-strip gratings and dielectric gratings with gain or loss are first evaluated in details at the first- and third-order Bragg regimes. It can thus serve as a powerful template for computational algorithms to determine systematically and rigorously the optical effects of multilayered periodic guiding structures, which are not readily obtained by other methods.

The field supported by multilayered periodic waveguides is well characterized by only one or two discrete leaky waves, rather than by a more complicated field representation that includes continuous spectra. The rigorous leaky-modes coupled in multilayered geometry can be then treated by relatively simpler and analytic model that describes the operation of practical optoelectronic devices in terms of leakage effects. To complement our modeling, we discuss and emphasize novel mathematical formulations based on the field orthogonality conditions of TE and TM modes coupled in multilayered periodic structures. In addition, to show the validity of our approach we numerically evaluate new physical meanings to illustrate quantitatively and rigorously the coupling efficiency of grating-assisted directional couplers (GADCs). The results reveal that the systematic and effective technique yields phenomenologically useful interpretations.