隨著科技的日新月異，新式光電元件的功能及重要性也與日俱增。近年來，超常材料就扮演相當重要的角色，其光學特性也應用於許多新型的光學元件中，如完美透鏡，主要原因是其具有很高的成像解析度以克服繞射極限的問題。擁有負折射率的材料在最近是相當熱門的，將人造複合材料或介電材料作週期性排列後，發現可產生負折射的現象，稱此結構為超常材料或光子晶體。由於光子晶體結構在其傳導區具有獨特的色散特性，因此也會發生特殊的波傳性質，而利用光子晶體來設計完美透鏡以聚焦電磁波的概念已被論證其可行性。
本文主要的研究為提出一新式的光子晶體來達到次波長聚焦之目的。此新式結構為在單位晶胞內包含不同半徑之介電圓柱，並作週期性排列所組成。我們利用有限元素法與時域有限差分法來模擬光學聚焦之特性，並用平面波展開法來求解馬克斯威爾方程式。值得注意地，相較於一般三角晶格之介電柱光子晶體，我們提出的複合型光子晶體能尋找到等效折射率為-1的聚焦特性。接著，在光子晶體平板介面設計一抗反射層，來增加光波的穿透率。而數值模擬也證明，在邊界適當的加入一排抗反射層，能使聚焦解析度有大幅的提升。此種光子晶體平面透鏡可應用於光學成像系統的設計，例如:光微影製程技術或近場光學顯微鏡。

Metamaterials currently play an important role in providing new functionalities and enhancements to the future electronic devices and components. The major trend today is to create the fundamentally new properties required by new technologies. Metamaterials make perfect lenses that image classical electromagnetic fields with significantly higher resolution than the diffraction limit. Materials with negative refractive index attracted lots of interest recently because they can be realized with metamaterials or photonic crystals (PCs), which are usually periodic structures. Specifically, the diffraction of PCs leads to the effective negative refraction such that a so-called perfect lens is feasible and able to focusing the electromagnetic wave beyond the diffraction limit.

This research focuses on the optical properties and the sub-wavelength focusing of a novel photonic crystal (PC). We propose an innovated PC slab and investigate its optical properties and the focusing numerically by both finite-difference-time-domain and finite element methods. The PC is composed of an array of two-dimensional hexagonal with dielectric cylinders inside each unit cell. As the different radius of the cylinder varies, the plane wave expansion method is employed for solving Maxwell’s equations. Interestingly, the complex PC, whose radius of each cylinder is different, can generate focusing with an effective refractive index of 1. The design as the antireflection structures to enhance the transmission efficiency of light at the interfaces between the air and the PC slab was proposed in this research. Numerical simulations show that the focusing resolution can be improved greatly by appropriately adding the surface of the slab. Such a mechanism of negative refraction PCs could open up a new application in optical imaging systems, such as optical lithography or near-field optical microscopy.

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