Wave propagation in photonic bandgap slab waveguides

Abstract

Photonic bandgap (PBG) structures in the form of periodic arrangements of dielectric or metallic materials have received extensive attention in recent years because of their many potential applications in the realization of modern photonic devices. The most important feature of a PBG structure is the existence of forbidden bands or bandgaps, which are frequency bands within which light is prohibited from propagation. By enclosing a dielectric medium with a PBG structure, light within the bandgaps of the PBG structure can be confined and guided along the medium. Such a PBG waveguide can take the form of a slab waveguide, a channel waveguide, or a circular fiber, depending on whether the light confinement is one dimensional or two dimensional in space. In this thesis, one-dimensional PBG waveguides or PBG slab waveguides are investigated theoretically with the objective to elucidate various unique physical phenomena arising from the bandgap effects. The advantage of studying slab waveguides instead of the more practical but complicated channel waveguides is the tremendous simplification in the mathematical treatment. As demonstrated in this thesis, a study of slab waveguides often leads to analytical solutions and thus provides a much clearer physical insight into the understanding of the transmission characteristics of general PBG waveguides. The PBG slab waveguides studied in this thesis include symmetric waveguides, asymmetric waveguides, waveguides with truncated PBG structures, and coupled waveguides. The refractive index of the guiding layer can be lower or higher than that of the PBG structures. The thesis starts with a detailed analysis of a symmetric PBG slab waveguide, which consists of a low-index layer sandwiched between two semi-infinite identical PBG slab structures. To analyze the guided waves of the waveguide, a ray-optics model is developed, which allows the dispersion characteristics of the waveguide to be calculated from an analytical condition, known as the transverse resonance condition. With this model, normalized dispersion curves for both the TE and TM waves are calculated and a nomenclature system for the guided modes based on the mode-field patterns in the waveguide is proposed. The cutoff conditions and the confinement factors of the modes are discussed. The effects of varying the physical parameters of the waveguide on the dispersion characteristics of the modes are also analyzed. Some new modal phenomena are highlighted, which include the disappearance of some specific modes due to the shrinking of the bandgaps, the existence of modes with a minus mode order, and the change in the mode distribution as a result of eliminating the Brewster angle by lowering the refractive index of the guiding layer. These results provide a basic understanding of the bandgap effects in a PBG waveguide. The study is next generalized to an asymmetric waveguide, where the guiding layer is sandwiched between two different PBG structures. The ray-optics model is extended and so are the transverse resonance condition and the nomenclature system for the guided modes. For the wave to be guided in an asymmetric waveguide, the optical frequency must fall within the overlapped bandgaps of the two PBG structures. As a result, the number of modes supported by an asymmetric waveguide can be much smaller, compared with a similar symmetric waveguide. One can engineer the asymmetry of the two PBG structures to control the modal properties of the waveguide. Some potential applications are discussed, including polarization filtering and single-mode transmission with an ultra-thick guiding layer. The thesis also considers a practical PBG waveguide, where the PBG structure is finite. Because of the finiteness of the PBG structure, the guided modes suffer from leakage losses. By means of the ray-optics model together with a perturbation analysis, the leakage losses of a symmetric waveguide that consists of two identical truncated PBG structures are evaluated. In particular, an approximate explicit expression is derived, which shows how the number of the periods in the PBG structures affects the propagation constants and the leakage losses of the modes. The results can facilitate the design and fabrication of practical PBG waveguides. The study is further extended to a class of PBG slab waveguide that has a higher refractive index in the core, which mimics the popular two-dimensional photonic crystal fibers and waveguides formed by introducing air holes around a solid guiding core. An eigenvalue equation is derived to calculate the modal characteristics of the waveguide including the cutoff conditions. The results show clearly the presence of both the index-guiding effect and the bandgap-guiding effect in such a waveguide. The effects of truncating the PBG structures on the transmission characteristics of the waveguide are also analyzed. Much physical insight can be gained from this study for the understanding of wave propagation in photonic crystal fibers and waveguides. Finally, the thesis discusses a PBG coupler formed by two parallel identical PBG slab waveguides. A class of leaky modes in such a coupler is identified, which are originated from the modes of the array structure that separates the two PBG waveguides and represent a new source of optical loss. With the help of the transfer-matrix method, an eigenvalue equation is derived for the analysis of both the guided modes and the leaky modes of the coupler, and an approximate formula for the calculation of the losses of the leaky modes is obtained. Furthermore, with the knowledge of the leaky modes, the condition that determines whether the fundamental guided mode of a PBG coupler is a symmetric mode or an antisymmetric mode is found.