光子晶體是由週期性排列所組成的結構，假若經由適度的設計則具備控制光波傳遞的效果。也因為二十年前Yablonovitch與John對光子晶體性質的發現，光子晶體吸引了學者做廣泛的研究。由於可藉由變數來設計光子晶體能隙的範圍，不計其數的文獻致力於光子晶體中的缺陷應用。近幾年，將研究延伸至傳導區獨特的新現象，例如: 超稜鏡、自我對準效應、負折射以及透鏡聚焦。
而負折射的另一項重大的應用就是由Notomi首先提出的“開放式共振腔”。本文將針對所提出的新式晶格排列由其基本性質到應用做深入地探討，利用平面波展開法、有限元素法、及時域有限差分法來檢視Archimedean-4的性質；與正方晶格、三角晶格相較可縮小物件的尺寸、減少材料、降低成本；與Archimedean-1相較，符合負折射應用的頻域也較廣。關於負折射方面應用上: Archimedean-4亦可產生自我準直效應；完美透鏡聚焦上藉由頻率的改變可使聚焦位置改變進而產生平面波。而運用光子晶體形成具有負折射開放式共振腔的設計上，有許多參數必須加以考量，比方:晶格的種類、晶格週期、柱體或空氣洞的直徑、材料以及開放式共振腔的幾何外貌。結果可藉由第一TE能帶的 產生共振現象，並發現藉由加入不同折射率之材料可使共振頻率呈線性偏移。此新式晶格排列除了能應用於自我準直、完美透鏡之聚焦以及共振腔生醫感測，或許能為積體光學光迴路帶來新的應用。

Photonic crystals (PhCs) are synthetic periodic structures that, when suitably designed, have the ability to control the propagation of light. It has attracted extensive research over the last two decades following Yablonovitch and John. Because the photonic band gap can be purposely designed, the vast majority of research in this field has been devoted to applications of the bandgap and defects in PhCs. Recently, this research has been extended to new transmission phenomena, such as the superprism, self-collimation, negative refraction, and slab lens.
Another important application of negative refraction is the “open cavity”, originally proposed by Notomi for PhCs. In the present studies, we investigated the energy band properties of Archimedean-4 lattices with the plane wave expansion method, the finite element method, and the finite-difference time-domain method. Compared with square, triangular lattices, Archimedean-4 can shrink into smaller scale and reduce the cost. The Archimedean-4 has wider frequency domain to fit the condition of negative refraction than the Archimedean-1 lattice. In the application of negative refraction, Archimedean-4 can form self-collimation effect. The position of focus form by the Perfect lens can change by tunning frequencies of the source. Perfect lens even can form a plan wave. To form an open cavity using PhCs with negative refraction, there are many parameters to optimize, such as the lattice type, lattice period, the diameter of the hole or rod, materials, and the geometrical configurations. It is shown that resonance can occur at the first TE band with . A small change of the refractive index of the measurand (filling only the three air wedges and not the holes in the PhCs) will cause a linear shift in the resonant wavelength. The novel lattices not only provide for several applications such as self-collimation, perfect lens, and biosensor but also have novel applications- in the photonic integrated circuit.

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