Wave propagation in photonic bandgap slab waveguides
一維帶隙波導的光子傳輸
Student thesis: Doctoral Thesis
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Award date  15 Jul 2008 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(119dd68f21e54c6aacf5ab4e5fbad86b).html 

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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, onedimensional 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 lowindex layer sandwiched between two semiinfinite identical
PBG slab structures. To analyze the guided waves of the waveguide, a rayoptics
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
modefield 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 rayoptics
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 singlemode transmission with an ultrathick
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 rayoptics 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 twodimensional
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 indexguiding effect and the bandgapguiding 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
transfermatrix 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.
 Photonics, Optical wave guides, Wavemotion, Theory of