本研究討論一個漸變式光子晶體，其結構為一個以四方晶格排列的空氣柱所組成。空氣柱的大小由中心至外圍在水平方向逐漸增大，而在每一直行中的空氣柱大小是相同的。經由有限時域差分法的計算，我們發現這個漸變式光子晶體可將電磁波聚焦，我們更進一步研究其幾何參數對於焦距、光點大小及聚焦強度的影響。由於此結構可將電磁波聚焦到約兩個晶格常數的範圍內，兩個晶格常數約是等於一個光子晶體波導的寬度，因此我們將此結構應用至光子晶體波導與傳統介電質波導的耦合。並和其它光子晶體波導耦合器進行比較。我們比較的光子晶體波導耦合器包括:光子晶體錐形波導、傳統錐形波導和拋物面鏡耦合器。所以的元件被設計並製作在一個 silicon-on-insulator 基板上。從模擬和量測結果得知，我們所設計的漸變式光子晶體波導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波導耦合器的優缺點。

In this work, we studied the graded photonic crystal consisting of spatial-varying air holes in a square array theoretically and experimentally. The radii of the air holes are identical in each column and modified from the central column to the edge column. We found that the graded photonic crystal could focus the electromagnetic waves. Through finite-difference time-domain method, we studied the relations between the geometry of the graded photonic crystal, focal length, spot size, and the intensity at focal point. The structure focuses the electromagnetic waves to a focal point measuring only two lattice constants which is close to the width of a single-line-defect photonic crystal waveguide. Therefore, the graded photonic crystal is applied to the issue of the coupling between photonic crystal waveguides and conventional dielectric waveguides. The coupling efficiency is compared with various couplers including the photonic crystal taper, the waveguide taper, and the parabolic mirror coupler. All of the couplers were designed and fabricated on a silicon-on-insulator substrate. The simulation and experimental results suggest that the graded photonic crystal coupler could offer a higher coupling efficiency in a shorter coupling length. Besides, the innate properties of each coupler are discussed.

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