近年來對於光子晶體異常折射現象的研究是很熱門的研究課題，因為若左手物質和負折射現象存在，這代表光可以突破繞射極限或反司乃爾定律•••等不尋常光的現象，使很多人著手於模擬與實驗來驗證負折射和左手物質存在的可能性，但現今仍有許多的困難存在。
現今很少人去研究光在光子晶體中的走向，因為光射入光子晶體後，可能會出現反射、散射的情況發生，所以現今研究光在光子晶體的走向，都是計算出光子晶體的能帶結構後，得到等頻率面，對等頻率面取梯度，計算出群速度的方向，再依群速度的走向，來判斷光進入光子晶體中的走向。
本論文是提出一個新的想法來定義光在光子晶體中的走向，相關討論詳見第三章。在第二章裡介紹我們所使用的模擬方法，分別為有限時域差分法和平面波展開法。第三章分為三個部分，第一部分是用部分能帶的理論去看光在光子晶體中的走向，發現平面波大角度入射進光子晶體達穩態後，會有類似負折射的現象產生。第二部分是用計算等頻率面去預測光在光子晶體中的走向，也是在平面波於大角度入射光子晶體達穩態後，發現光在光子晶體中所走的方向不吻合等頻率面所算出的角度。第三部分是平面波入射進光子晶體內部達穩態後，計算在光子晶體內的能流分佈，在用光線追跡的方法，去預測當點光源入射時，光會在光子晶體中的走向，與有限時域差分法所計算出的點光源的場圖去做對
照，發現結果是相當吻合的。因此本論文的方法可有效應用於分析光子晶體中光傳播的異常折射現象。

Recently, the abnormal refraction phenomena of the light propagating in the photon crystals have attracted many attentions. The so-called left-handed materials (LHMs) possess some peculiar properties for the electromagnetic
waves such as the inverse Snell’s law, the reversed Doppler shift, and the reversed Cherenkov radiation, etc. Many people are devoted to simulate and fabricate the left-handed materials and photonic crystals to study the abnormal properties. However, some arguments are still left unsolved.
In the literature, the light propagation in the photonic crystals (PCs) cannot be stated by the ray-tracing method due to the complex scattering in the photonic crsytals. The refraction direction of the light propagating in the PCs and materials interface is generally predicted by using the equal frequency surface (EFS). One can calculate the band structure of the PC structure to obtain the EFS. Group velocity direction (Vg) is obtained by the
gradient of the EFS.
We found the fact that there is difference between the light propagation direction and the group velocity direction. In this thesis, we propose to use the energy velocity direction to define light propagation direction in PCs.
This will be presented in Chapter 3-2-3. We will introduce the FDTD and PWE simulation methods in Chapter 2. The third section can be split into 3
parts. The first part shows the view of point of the partial band gap to state the light propagation in PCs. The second part shows the EFS method. The relation between the incident angles and the refraction angles of Vg is
obtained. The result of the ray-tracing and the FDTD methods can be proved to be inconsistent. In the third part, we calculate the relation between the incident and the refraction angles by calculating the energy velocity (Ve)
direction of the light propagating in photonic crystals. We use this relation into the ray-tracing to analyze the imaging phenomena of the point source by the PC slab. The results show an excellent agreement. The results imply that
the ray-tracing can be used to analyze the imaging phenomena of the point source by the PC slab.

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