在本論文中，我們提出一個新的等效波導理論，並用以
研究在傳播方向上有變化的光波導。藉由保角映射的空間轉
換觀念，我們可以將長直狀光波導轉換成包含彎曲波導、錐
狀波導及S 形彎曲錐狀波導等各種不同的理想結構。這些理
想光波導的特性將在論文中被徹底地探討。我們發現理想光
波導的形狀及折射率變化是具有連續性的。在理想彎曲波
導，其特徵模的波前在彎角處必須能正確地偏折，以避免幅
射損失的發生。在理想錐狀波導，其特徵模必須能逐漸而適
當地轉變，以避免誘發高階模態或幅射損失的發生。這些特
殊的物理特性將被應用於實際光波導的設計。
等效波導理論將被更進一步地用於分析光波導。藉由考
慮波前演化，等效波導理論將結合光束傳播法成為一個新的
數值工具，並可精確地分析大角度光波導。我們將以此數值
工具來分析兩種常見的錐狀波導，包含錐狀橫切面波導及固
定V 值錐狀波導。
我們將以彎曲波導及Y 形岔狀波導為例，將理想光波導
的物理機制應用於實際波導的設計。首先，我們提出多重相
位補償的觀念，並用以設計具有雙尖頂連接狀圓形光柵的低
損耗彎曲波導。雙尖頂連接狀圓形光柵不僅可以補償在彎角
處的相位差異，並可避免干擾特徵模態。模擬結果發現，在
大角度彎曲波導，其特徵模場的波前仍可正確地偏折。在彎
曲角度達10°的例子中，其能量的傳輸效率仍可達89﹪。
Y 形岔狀波導在傳播方向上有一個分岔點，因此不可能
去設計具有連續波導路徑的理想結構。在本論文中，一個考
量波前演化的相位補償法則將被提出，並用以設計大角度、
低損耗的Y 形岔狀波導。模擬結果發現，在大角度Y 形分岔
區域，良好的模態分離及低損耗特性可如預期地達成。在分
岔角度達20°的例子中，其能量的傳輸效率亦可達89﹪。

A novel equivalent waveguide theory for studying the longitudinally varying
waveguides has been proposed in this dissertation. By adopting the space
transformation concept based on the conformal mapping, a straight optical
waveguide is transformed into various ideal structures including bends, tapers, and
S-shaped bent tapers. The characteristics of these ideal optical waveguides are
thoroughly discussed in the dissertation. The shape and refractive -index distribution
are continuous in these ideal waveguides. The phase front can be correctly tilted in
the ideal bent waveguides without any radiation loss. The mode size can also be
gradually converted in the ideal tapered structures without mode conversion to
higher-order modes or to radiation modes taking place. These special physical
properties can be applied to design the practical waveguides.
Furthermore, the equivalent waveguide theory can be applied to analyze the
optical waveguides. Considering the phase-front evolution, the equivalent waveguide
theory is combined the beam propagation method to explicitly analyze the
wide-angle waveguides. Two kinds of popular tapered waveguides, including the
cross-sectional dimension tapering and the constant V-number tapering, are used to
analyze by the combination of conformal mapping method and beam propagation
method.
In order to apply the design rule of optical waveguides obtained from the
analysis of ideal structures, bent waveguides and Y branches are taken as examples.
First, a low-loss waveguide bend wi th two apexes-linked circle gratings based on the
concept of multiple phase-compensation is designed. The apexes-linked circle
gratings not only compensate the phase-difference in the bend corner but also avoid
distorting the eigenmode. Simulation results predict that the phase front of the
modal field can be tilted correctly in the wide-angle waveguide bend. The transmitted
power efficiency can be as high as 89% for bending angle up to 10°.
Since the Y branch has a branching point in propagation, it is impossible to
design an ideal Y branch with continuous waveguide path. In the dissertation, a novel
phase compensation rule is derived by taking the phase front evolution into
consideration to design the low-loss and wide-angle Y branch. Simulation results
predict that a perfect mode-separation with low branching losses can be achieved in
the wide -angle Y-junction branch. The transmitted power efficiency can be as high as
89% for the branching angle is up to 20°

94
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