光子晶體的能隙（energy band gaps）會限制可在光子晶體中傳播的電磁波的頻率，這是光子晶體的一項重要性質。在光子晶體中製造一些缺陷 (defects) ，例如在由周期排列的介電質柱子陣列所構成的光子晶體中移去一整排的柱子，形成線缺陷 (line defect)，會使得電磁波被侷限在線缺陷區域內傳播，這是使用光子晶體來構成光子晶體波導（Photonic Crystal Waveguide, PCW）的基本方法。
在此論文中，我們研究兩種較精巧的光子晶體波導特性，並探討它們的應用。我們發現藉著改變 A. 線缺陷兩旁柱子的半徑（後面以 來表示）或 B. 在線缺陷中填入一整排半徑不同的柱子（後面以 來表示此線缺陷內的柱子半徑），都會使波導模態的色散關係產生改變。應用這種性質，不管是在正方晶格還是三角晶格中，只要選擇適當的 與 ，我們就可以設計出符合需求的波導，將不同頻率的電磁波分開來，而達到分波的效果，並以此特性來製造分波器。

除了上述分波器，我們也研究在正方晶格與三角晶格下的其他幾種分波器。正方晶格的分波器可以分出比較大範圍頻率的電磁波，而三角晶格的分波器可分出比較窄頻寬的電磁波。因為這兩者特性的差異，我們預期它們將在奈米光學上具有不同的應用。

Photonic band gaps are defined as the frequency intervals that light is forbidden to propagate inside the photonic crystal (PC). This is the most important characteristic of photonic crystals. A photonic crystal waveguide (PCW) can be formed, taking the dielectric rod-array structure as an example, by making a line defect through removing one row of dielectric rods. This is the fundamental method to make a PCW.
In this thesis, we study the transmission properties of two sophisticated PCWs formed by the structures of “PCW of air defect” (PCWA) and “PCW of dielectric rods defect” (PCWD), respectively. Specifically, we design PCWA by taking out a line of rods and changing the radius of the “side-wall rods” of the line defect to , and design PCWD by replacing a line of rods of radius with a line of rods of radius , -being the radius of the rods consisting of the background PC. Both approaches change the dispersion relation of the guided modes, which gives us the possibility of designing nano-devices such like wavelength division multiplexers (WDM) through tuning the radius of the dielectric rods.
Besides the above mentioned wavelength division multiplexers we also design other kinds of WDMs in the square-lattice and triangular-lattice PCs. Based on our simulation results, we find that the WDMs in the triangular-lattice has higher resolution in distinguishing two waves of very close frequencies. These designs, we expect, after some elaboration, can serve as useful nano-devices in the future.

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