在早期研究及探討光子晶體結構主要方向為其能隙區之運用，如當光波導或於結構中加入缺陷當共振腔等。而近幾年來發現在其傳導區擁有獨特的色散特性，尤其具有負折射的特性，使得該方面之研究更成為最近學術界極度熱門的話題。

本文將介紹如何以平面波展開法求得二維光子晶體等頻圖及解說二維時域有限插分法之運算原理，再使用這些方法分析光子晶體的異常折射現象。

當光子晶體結構的折射係數改變時，等頻圖會產生變化，折射角也因此改變。以此概念，我們若添加折射係數可調變之材料到光子晶體中，則應可設計出一可調折射角之光學元件。本文選用液晶材料當為可調變之材料，即利用液晶材料的非等向性和可藉由電場控制液晶導軸角度的特性，將液晶材料加入二維光子晶體結構中，來觀察與分析不同液晶導軸角度對於折射角的影響。我們針對兩無因次化頻率做分析，發現負折射角可做大角度的調變，及可把折射角度從負調整為正，論證了添加液晶之光子晶體具有折射角之可調性。最後，對於調至臨界角之後的全反射，則提出可做為光開關之用途。

The photonic crystals (PhCs) are structures with periodically modulated dielectric constants. Recently, it was proved that the diffraction effects of PhCs can produce the effective negative refraction or the negative index. Hence, the studying of PhCs is not limited in the band gap region. The anomalous refractive properties (especially negative refraction) of PhCs have become hot topics of scientific research over the past few years.
In this thesis, the plane wave expansion method is used to obtain the equifrequency surface. The anomalous refractive properties of photonic crystal are analyzed by using the equifrequency surface. Then we use the finite difference time domain method to simulate the light propagation in PhCs and compare the refractive angle with that predicted by the equifrequency surface.
From the fact that the EFS varied as the refractive index changed, we speculate that the propagation direction in photonic crystal structures could change. Therefore, the index-tunable materials can be used to control the refractive direction.
The properties of liquid crystals (LCs) can be changed easily by applying an external electric field. Hence, they are adaptable for tuning. The LCs are infiltrated into the photonic crystal structures and the relationship between the refractive angle and the angle of the director is studied. It is demonstrated that the refraction of PhCs can be tuned by LCs. The direction of the negative refraction is controlled by changing the direction of the LCs director. We select two frequencies to analyze the tunablility. A large tunable range of the negative refraction is achieved at one frequency. At the other frequency, the refractive angle can vary from negative to positive. Finally, we find that the refraction of a tunable PhC with nematic liquid crystals can be used to design an optical switch.

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