光子晶體為週期性排列的人工結構，此結構會產生光子能隙，能使得某些頻段的光無法在結構中傳遞，因此可以利用光子晶體結構的特性來進行一些光學元件上的設計與應用。也因為數十年前Yablonovitch與John對光子晶體性質的發現，吸引了許多學者對光子晶體開始感興趣，積極的朝這方面做廣泛的研究。由於可藉著調變參數來設計光子晶體能隙的範圍，因此許多文獻致力於光子晶體的缺陷應用，近幾年將研究延伸至傳導區獨特的現象，例如超稜鏡、自我準直效應、負折射及透鏡聚焦。本論文我們研究討論一個二維扇形漸變式光子晶體，其結構以一個正方晶格空氣柱排列之漸變式光子晶體透鏡為基礎，空氣柱半徑大小由中心至外圍兩側在垂直方向逐漸增大，而在每一橫列中的空氣柱半徑大小皆為相同。我們調變幾何形狀、曲率半徑以及週期排列數以對此結構最佳化，並探討其結構對於焦距、焦深、聚焦點大小及聚焦強度的影響，接著再將此扇形漸變式光子晶體應用在光子晶體波導的耦合，並和其它光子晶體波導耦合器進行比較，包括變式光子晶體透鏡波導以及弧形漸變式光子晶體波導。
本文中我們主要以平面波展開法和時域有限差分法做為分析的工具，來計算所設計的光子晶體之各種特性。所有的元件被設計並製作在一個矽基板上。從模擬和量測結果得知，我們所設計的扇形漸變式光子晶體耦合器可以在較短的耦合長度限制下提供較高的耦合效率。而我們也討論了其它光子晶體耦合器的優缺點。漸變式光子晶體透鏡的焦距及耦合長度較短，但是其缺點為耦合效率較低；弧形漸變式光子晶體焦深為三種結構中最長，但其焦距及耦合長度較長，導致此結構在製程上空間需求較大。由此可知扇形漸變式光子晶體比漸變式光子晶體透鏡及弧型漸變式光子晶體更適合應用在耦合器，其耦合效率可以達到56.2%，相較於漸變式光子晶體透鏡耦合器之耦合效率28.4%，提升了將近一倍。

The photonic crystal is a periodic artificial structure. This structure produces photonic bandgaps, make the light of certain frequency bands can not pass in it. So we can use the characteristic of photonic crystal structure to have application and design some optical components. Because of Yablonovitch and John found the characteristic of photonic crystal structure few decades ago, many researchers began to be interested in the photonic crystal and studied extensively in this direction as possible as they can. Because the photonic band gap can be designed purposely, the vast majority of research in this field has been devoted to applications of the bandgap and defects in photonic crystal. Recently, the research has been extended to new transmission phenomena such as the superprism, self-collimation, negative refraction and slab lens. In this paper, we research a two-dimensional sectorial graded photonic crystal base on a graded photonic crystal lens with square lattice of air columns. The radii of the air columns are identical in horizontal arrangement and are modified from the central to the edge in vertical arrangement. We change the geometric shape, radius of curvature and periodic arrangement to optimize the structure, and discuss the parameters of the focal length, the depth of focus, the size of focal point and the focusing intensity. Then we apply the sectorial graded photonic crystal to the photonic crystal waveguide coupling, and compare the coupling efficiency of it with other couplers including the graded photonic crystal lens waveguide and the curved graded photonic crystal waveguide.
We calculate and analyze the properties of the photonic crystal we design by using plane wave expansion method and finite-difference time-domain method here. All of the couplers are designed and fabricated on a silicon-on-insulator substrate. The simulation and experimental results suggest that the sectorial graded photonic crystal coupler could offer a higher coupling efficiency in a short coupling length. Besides, the innate properties of each coupler are discussed. The graded photonic crystal lens has the short focal length and coupling length, but its disadvantage is that the coupling efficiency is low. The curved graded photonic crystal has the longest depth of focus than the others, but its disadvantages are that the focal length and coupling length are long resulting in demand for space during the manufacturing process. Therefore, the sectorial graded photonic crystal is the most suitable for application in the coupler than the graded photonic crystal lens and the curved graded photonic crystal. Its coupling efficiency is 56.2% almost double of the graded photonic crystal lens’s coupling efficiency 28.4%.

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