Coupled-mode theory (CMT) has been widely utilized in the analysis of electromagnetic wave coupling and conversions due to its mathematical simplicity and physical intuitiveness. The early version of CMT was proposed by Pierce and Miller in 1950s to study the microwaves, and was mathematically formulated by Schelkunoff using mode expansion, and by Haus with variational principles.1-5 The CMT was later introduced to the investigation of optical waveguides by Marcuse,6 Snyder,7 and Kogelnik8 in the early 1970s. Since then, a series of formulations and applications in optical waveguide has been proposed and studied. The physical model of CMT is expanding the total field inside and optical waveguide in terms of the field of a reference waveguide structure, by applying the orthogonality condition, a set of ordinary differential equations are obtained. Generally, the CMT focused on

the guided modes with the assumption that only a limited number of guided modes (usually one or two) close to phase matching play significant roles in the interaction of the modal fields. In situations of applications involving the radiation mode coupling, the application of CMT becomes cumbersome due to the continuous spectrum of radiation modes. One possible solution to circumvent the problem of radiation modes is to introduce leaky modes to approximate the radiation modes.9-20 The leaky modes are, however, not orthogonal and normalizable in real domain. For this reason, it is difficult to deal with leaky mode formulations analytically and even more so numerically for practical applications.Recently, a new computation model was introduced to the mode-matching method in which the waveguide structure is enclosed by a perfectly matched layer (PML) terminated by a perfectly reflecting boundary conditions (PRB).21-23 This seemingly paradoxical combination of PML and PRB leads to a somewhat unexpected yet remarkable result: It creates an open and reflectionless environment in a close and finite computation domain. A set of complex modes can be derived from this waveguide model that are well behaved in terms of orthogonality and normalization and can be readily solved by standard analytical and numerical techniques. By utilizing the complex modes as an orthogonal base functions to represent the radiation fields, the CMT can be applied as if all the modes are discrete and guided. The complex CMT was subsequently applied to simulation and analysis of slab, circular, and channel optical waveguide structures and shown to be highly accurate and versatile. 2.1 Modal Analysis with Perfectly Matching

We make the following assumptions: The medium in the waveguide structure is lossless, linear, and isotropic. The permittivity and permeability of vacuum are denoted as e0 and m0, respectively. The permeability m in the medium is equal to the free space value m0 throughout this chapter. The time dependency is expressed as exp( jwt). The wave is propagating along z, and the z dependency is expressed as exp(–j gz), which refers to the propagation in the positive z direction, or exp(–j gz) in the negative z direction.