Band Structure Analysis of Three-Dimensional Photonic Crystal by the Finite-Difference Frequency-Domain Method

摘要 本論文係採用以余氏網格為基礎所建立之三維有限差分頻域法來分析三維光子晶體元件的能帶結構。利用均勻網格及週期性的邊界條件，我們可以很容易地從馬克斯威爾旋度方程式，推導出三維光子晶體能帶結構的特徵矩陣方程式，並求出能帶結構圖。為了處理光子晶體結構中介質介面不連續的問題，我們利用折射率加權平均法以增進數值計算的精確性並加速數值計算的收斂性。 本研究中，我們先探討兩種三維結構的簡單立方光子晶體，包括球型結構與鷹架型結構。利用三維有限差分頻域法來獲得這兩種光子晶體的能帶結構，並將該數值結果與其他數值方法所獲得的結果相互比較與探討。另外，折射率加權平均法對數值結果的影響也會加以討論。我們也分析與探討有限厚度的三維光子晶體平板，成功地獲得此種光子晶體平板的能帶結構，並在能帶結構中以光錐表示非導波模態。為了減少計算的時間與計算器記憶體的使用量，我們在數值模擬中引入此三維光子晶體的鏡射對稱。利用這個概念，我們成功地獲得偶模態與奇模態的縱剖面模態輪廓圖，並與鏡射對稱條件相符合。此外，我們也分別計算了被支撐於空氣、介質材料與週期性結構的三維光子晶體平板，從其能帶結構圖中可以發現光子晶體能隙出現於光子晶體折射率與支撐背景折射率差異較大的時候。我們的三維有限差分頻域法的計算可以有效地獲得三維光子晶體結構圖，並可繼續應用於分析更多實際的結構。An Yee-mesh-based three-dimensional (3-D) finite-difference frequency-domain (FDFD) method is proposed for the band structure analysis of 3-D photonic crystal (PC) devices. With uniform meshes and periodic boundary conditions in the numerical implementation, it is easy to derive eigenvalue matrix equations from Maxwell's curl equations to obtain the band structures of the 3-D PCs. For dealing with the dielectric discontinuity in the PC structures, the index-averaging scheme is employed to improve the numerical accuracy and accelerate the numerical convergency. In this work, two kinds of 3-D simple-cubic PCs are first considered, including spheres structures and scaffold structures. The band structures of these two PCs are obtained by using our 3-D FDFD method. Comparsion of our results with those from other numerical methods is performed and discussed. The influence of the index-averaging scheme on the 3-D PCs analysis is also investigated. We also consider 3-D PC slabs with finite thickness. Their band structures can be successfully obtained with non-guided modes referred as light cone in the band diagrams. To reduce the computing time and memory, the mirror symmetry of 3-D PC slabs is introduced in the numerical simulations. The mode profiles of even and odd modes can be obtained and coincide with other numerical investigations. We also calculate the band structures of 3-D PC slabs suspended in air, dielectric materials, and periodic backgrounds, respectively. It is found that the phtonic bandgaps appear for larger index difference between the PC slabs and the background region. Our 3-D FDFD algorithm is demonstrated to be a useful numerical tool to find out the band diagrams of 3-D PC structures for more practical applications.Contents 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Formulation of the Finite-Di erence Frequency-Domain Method 7 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Bloch Function for 3-D Photonic Crystals and Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Formulae for 3-D Anisotropic Band Structures . . . . . . . . . . . . . 9 2.3.1 Central Di erence Scheme . . . . . . . . . . . . . . . . . . . . 9 2.3.2 The FDFD Method . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Improvements for the Dielectric Medium Interfaces . . . . . . . . . . 16 3 Band Structures of 3-D Simple Cubic Photonic Crystal Structures 23 3.1 3-D Simple Cubic Photonic Crystal: Sphere Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.1 Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 3-D Simple Cubic Photonic Crystal: Sca old Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Band Structures of 3-D Photonic Crystal Slabs 51 4.1 Computational Method for Band Structures of 3-D Photonic Crystal Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Band Structures for PC Slabs Suspended in Air . . . . . . . . . . . . 54 4.3 Band Structures for PC Slabs with Solid Background . . . . . . . . . 56 4.4 Band Structures for PC Slabs with Periodic Background . . . . . . . 57 5 Conclusion 8