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系統識別號 U0026-1907201617594100
論文名稱(中文) 二維週期結構能隙之數值研究
論文名稱(英文) A Numerical Study on Band Gap of Two-Dimensional Periodic Structure
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 104
學期 2
出版年 105
研究生(中文) 吳怡潔
研究生(英文) Yi-Jie Wu
學號 l16011051
學位類別 碩士
語文別 中文
論文頁數 44頁
口試委員 指導教授-舒宇宸
口試委員-侯世章
口試委員-陳旻宏
中文關鍵字 光子晶體  光子頻率禁帶  馬克斯威爾方程式  能隙比例 
英文關鍵字 photonics crystals  photonic crystal bandgap  bandgap ratio 
學科別分類
中文摘要 光子晶體為不同介電係數的兩種或兩種以上的物質,所排列而成的週期性結構。此空間上的週期性,導致光在結構中會產生建設性和破壞性干涉,因而在某些特定的結構中,某些頻段的光無法穿透。這些被終止的頻率區間稱為「光子頻率禁帶」,也稱為能隙。本論文即在尋找具有最大能隙比例的光子晶體。
本論文的工作從馬克斯威爾方程式,推導出橫向磁波,橫向電波兩種極化的電磁波頻率的特徵方程式。並利用有限元素法將特徵方程式離散化,以計算光子晶體之能帶。藉由討論橫向磁波的頻率、和橫向電波的頻率,對不同介電係數的偏導數,進而推導出能隙比對介電係數的偏導數,藉此調整光子晶體的介電係數以達到最大的能隙比例。
在這篇論文中,主要完成了三件事。第一,我們找到了當目標方程式在無因次化介電係數和磁導率為1時的解析解。而且從數值結果說明了解析解和數值解的誤差隨著三角化網格大小為二階收斂。第二,由有限元素法離散化目標方程式,並藉此推導出能隙比例對介電係數的偏導數。第三,我們將一個給定的初始結構(初始能隙比例6.71%經過三次加密網格並透過梯度調整介電係數,其能隙比例可以增加到19.73%。
英文摘要 SUMMARY
This thesis focus on finding the photonic crystal periodic structures which maximize the bandgap. We use Maxwell's equations and Bloch boundary conditions to derive the eigenfunctions of transverse magnetic wave (TM-wave) and transverse electric wave (TE-wave), then apply finite element method to solve the eigenvalue problem to derive the gradient of bandgap ratio with respect permittivity. We change the permittivity of the uni-cell for increasing the bandgap ratio according to the gradient. In this work, we have mainly three conclusions. First, we find the exact solution of the eigenvalue problem with the relative permittivity and permeability are 1. From the numerical results, we obtain the second-order convergence for numerical solution. Second, we derive the gradient of bandgap ratio with respect permittivity with the help of the governing equations. Third, for a given configuration with bandgap ratio 6.71%, we refine the mesh and adjust the permittivity by the gradient ascent method, we obtain a better configuration with bandgap ratio 19.73%.
INTRODUCTION
Photonic crystals are periodic structures composed of two or more dielectric materials. Owing to the periodicity, electric-magnetic waves in some frequencies cannot propagate. The ranges of frequencies are called "photonic crystals bandgap"(PBG). The so-called photonic bandgap engineering has been an fast-developing field of study in modern micro-fabrication techniques. Those study is considered about how to maximize bandgap.
This thesis exactly focus on finding the periodic structures which maximize the bandgap. For discussing bandgap of photonic crystal, Maxwell's equation is used to describe wave behavior in crystal structure and derive TM and TE wave eigenfunction. In numerical simulation, finite element method is adpoted to discrete equation and calculate bandgap of crystal. We can derive the gradient of bandgap ratio with respect permittivity through correlation between frequency and permittivity. Base on this, we can determine how to change uni-cell structure and get larger bandgap ratio of photonic crystal.
MATERIALS AND METHODS
In this thesis, we discuss the two-dimensional photonic crystals. There are two independent polarized waves, transverse magnetic wave (TM-wave) and transverse electric wave (TE-wave). Because of the periodicity structure, we use Maxwell's equations and Bloch boundary conditions to derive the eigenfunctions of TM-wave frequencies and TE-wave frequencies.
Owing to the complicated determinant of coefficient matrix, we cannot obtain the explicit relation of frequency and wave-number directly. Thus we propose the numerical method to overcome this difficulty. We apply finite element method to solve the eigenvalue problem and substitute the frequency into the determinant of coefficient matrix. To verify the numerical simulation, we find the exact solution with permittivity and permeability are eqaul to 1, and the wavevector are (0,0). From the numerical results, we obtain the second-order convergences. In order to maximize the bandgap ratio, we wnat to know how to adjust the uni-cell to achieve our purpose.Through the discuss of the gradient of frequencies with respect permittivity, we derive the gradient of bandgap ratio with respect permittivity. With the results, we can change the permittivity of the uni-cell for increasing the bandgap ratio.
RESULTS AND DISCUSSION
In this numerical simulations first demonstration, air and silicon are used. Their relative permittivities are 1 and 13 respectively. We found a two-parameter global search, and adopt similar uni-cell structure in N=48 to demonstrate numerical simulation. The corresponding band gap ratio result is got in our demonstration, which can be verification of this numerical simulation.
We choose a uni-cell structure which has bandgap ratio 6.71% in N=6 as begining. By gradient ascent method, we successfully increase the bandgap ratio from 6.71% to 19.73%.
CONCLUSION
First, we find the exact solution of the eigenvalue problem with the relative permittivity and permeability are 1. From the numerical results, we obtain the second-order convergence for numerical solution. Second, we derive the gradient of bandgap ratio with respect permittivity with the help of the governing equations. Third, for a given configuration with bandgap ratio 6.71%, we refine the mesh and adjust the permittivity by the gradient ascent method, we obtain a better configuration with bandgap ratio 19.73%.
論文目次 1 前言 1
1.1 光子晶體(PhotonicCrystal)........................ 1
1.2 本研究工作................................. 3
2 理論基礎 4
2.1 馬克斯威爾方程式(Maxwell’sEquations) ................ 4
2.2 布洛赫定理 (Bloch Theorem) 和第一布里淵區 (First Brillouin Zone) . 6
2.3 橫向磁波 (Transverse magnetic wave) 和橫向電波 (Transverse electric
wave) .................................... 8
3 數值方法 10
3.1 有限元素法(FiniteElementMethod)................... 10
3.1.1 離散化TM波和TE波 ...................... 10
3.1.2 誤差分析 .............................. 14
3.2 光子能隙對介電係數的變化........................ 17
3.2.1 能隙(BandGap) .......................... 17
3.2.2 Gradient of Bandgap ratio with respect permittivity . . . . . . 18
3.3 演算法.................................... 21
4 數值結果 23
4.1 數值運算之驗證............................... 23
4.2 數值運算結果................................ 25
4.2.1 N=6初始結構 ........................... 25
4.2.2 N=12 ................................ 26
4.2.3 N=24 ................................ 28
4.2.4 N=48 ................................ 30
A 離散化矩陣推導 35
Reference 43
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