A coupled-mode formulation is described in which the radiation fields are represented in terms of discrete complex modes. The complex modes are obtained from a waveguide model facilitated by the combination of perfectly matched boundary (PML) and perfectly reflecting boundary (PRB) condition. By proper choice of the PML parameters, the guided modes of the structure remain unchanged, whereas the continuous radiation modes are discretized into orthogonal and normalizable complex quasi-leaky and PML modes. The complex coupled-mode formulation is identical to that for waveguides with loss and/or gain and can be solved by similar analytical and numerical techniques. By identifying the phase-matching conditions between the complex modes, the coupled mode formulation may be further simplified to yield analytical solutions. The complex coupled-mode theory is applied to Bragg grating in slab waveguides and validated by rigorous mode-matching method. It is for the first time that we can treat guided and radiation field in a unified and straightforward fashion without having to resort to cumbersome radiation modes. Highly accurate and insightful results are obtained with consideration of only the nearly phase-matched modes.
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