Direct characterization of states and modes in defect grating structures

For one-dimensional optical structures consisting of gratings surrounding a defect region, optical field solutions inside the bandgap are investigated that are steady states or fully transmitted modes. The observation that a mode is a suitable combination of two states, and that each state is a resonant phenomenon, implies that an accidental degeneracy condition has to be satisfied in order that two states occur at the same frequency. Finding the conditions, and thereby the required design-parameter s of the structure, makes it possible to characterize the modes without the necessity to scan the whole bandgap for transmission properties. The mathematical formulation is based on the optical transfer map and leads to a non-standard, not well-studied, eigenvalue problem on the defect region with effective boundary conditions that simulate the surrounding gratings.