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光子晶體原理與計算 (I) Bloch 定理 , 光子能帶 , 平面波展開法 , 光子能流 , 與多重散射法

Pi-Gang Luan ( 欒丕綱 ) Wave Engineering Lab ( 波動工程實驗室 ) Institute of Optical Sciences National Central University ( 中央大學光電科學研究所 ). 光子晶體原理與計算 (I) Bloch 定理 , 光子能帶 , 平面波展開法 , 光子能流 , 與多重散射法. 光子晶體研究的由來.

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光子晶體原理與計算 (I) Bloch 定理 , 光子能帶 , 平面波展開法 , 光子能流 , 與多重散射法

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  1. Pi-Gang Luan (欒丕綱) Wave Engineering Lab(波動工程實驗室) Institute of Optical Sciences National Central University (中央大學光電科學研究所) 光子晶體原理與計算 (I) Bloch 定理 ,光子能帶, 平面波展開法, 光子能流, 與多重散射法

  2. 光子晶體研究的由來 • 問題:電子 (機率) 波在週期性晶格位能中傳輸時, 其能譜具有能隙 (Energy Bandgaps). 光波在具有週期性介電質分布的環境中傳輸時, 是否也會出現帶隙 (Frequency Bandgaps)? • 西元1987年, 兩位科學家Eli Yablonovitch 與 Sajeev John 幾乎同時提出上述概念 (Phys. Rev.Lett. 20, 2059 (1987) 與 Phys. Rev.Lett. 23, 2486 (1987 )). 他們將這種人工製造的介電質週期結構稱作 “光子晶體” (Photonic Crystals). • Eli Yablonnovitch 的目的是想藉著光子晶體的 Bandgaps 抑制自發輻射, 增進雷射的效率. Sajeev John 則是想藉著先由週期介電質產生 Bandgaps, 再適度弄亂此結構, 以實現光波的 “Anderson 局域化” (Anderson Localization).

  3. Sajeev John Eli Yablonovitch Famous People

  4. 光子晶體研究的現在與未來 • 在忽視物質對光能量的吸收的情形下, 光子能帶的形式與特性在工作波長與系統尺寸的相對比例維持不變的情形下是固定的. 因此, 只要等比例縮小系統尺寸與波長, “大” 光子晶體結構在微波頻段所表現出的光學特性將與 “小” 光子晶體結構在紅外光區或可見光區的特性一樣. • 早期的技術只能製造出工作於微波頻段的光子晶體結構, 近年利用半導體蝕刻與其它先進技術, 已可製造出用於可見光光頻區的光子晶體. • 利用整塊光子晶體週期結構的塊材, 或是在其中製造點缺陷與線缺陷, 並選擇適當的工作頻率, 可以做出 “光子反射器”, “濾波器”, “共振腔”, “波導”, “光纖”, 以及 “光子晶體透鏡” 等. 未來的目標是要利用光子晶體全面性地控制光流, 製造出 “給光子使用的半導體”.

  5. 光子晶體 (Photonic Crystals) （a）一維光子晶體 （b）二維光子晶體 （c）三維光子晶體

  6. Photonic Crystals

  7. 光子晶體能帶(頻帶) , 光子帶隙(2D System)Photonic band structure, Photonic band gap

  8. 光子晶體 (帶隙) 的應用: 共振腔與波導 缺陷模 (Defect Mode) 波導模 (Guiding Mode)

  9. 光子帶隙的其它應用 • 光子晶體光纖 (Photonic Crystal Fiber) 其截面為具有缺陷的光子晶體, 如上一張投影片的左上第一圖. 優點為: 1. 利用的是光子帶隙特性, 而非全反射, 故不必受全反射臨界角條件之限制, 原則上可任意轉彎. 2. 在其中的光是在空氣或真空中傳播, 減少被介電物質吸收的機會.

  10. 光子晶體傳導帶的應用(I) 長波極限 (Long-wavelength Limit): 非均向性透鏡 (Anisotropic Lens) Photonic crystals as optical components P. Halevi et.al. Appl. Phys. Lett. 75, 2725 (1999) See also Phys. Rev. Lett. 82, 719 (1999)

  11. 長波極限 :

  12. 光子晶體傳導帶的應用(II) 超越長波極限 (Beyond the Long-wavelength Limit) 負折射透鏡與次波長成像 (Negative Refraction Lens and Subwavelength Imaging)

  13. 光子晶體能帶理論, 計算方法, 與物理詮釋 • 週期函數的處理: Fourier 級數與倒晶格 (Reciprocal Lattice) • Bloch 定理 (Bloch’s Theorem) 與平面波展開法 (Plane Wave Expansion Method) • 光子能流 (Photonic energy flow), 能量速度 (Energy velocity) 與群速度 (Group velocity) • 等頻率線 (Constant frequency curves) 的應用

  14. Periodic function, Fourier Series and Reciprocal Lattice

  15. 晶格基底與倒晶格基底 (二維) Lattice Bases vs. Reciprocal Lattice Bases (2D) Square Lattice Triangular Lattice

  16. 二元系統與結構因子Binary System and Structure Factor (2D)

  17. Example 1: Square Lattice, Circular Rods/Holes

  18. Example 2: Triangular Lattice, Circular Rods/Holes

  19. Example 3: Simple Cubic Lattice, Spheres

  20. 布洛赫定理與布里淵區Bloch’s Theorem and Brillouin zone Bloch’s Theorem (Electron Systems): First Brillouin Zone

  21. Proving Bloch’s Theorem

  22. Band Structure (Electron System)

  23. Bloch’s Theorem and Photonic Band Structure

  24. Two-Dimensional Inhomogeneous Wave Systems Classical Waves Unified Treatment

  25. Band Structure Calculation (2D, Scalar Wave)

  26. 約化布里淵區與約化本徵頻率Reduced Brillouin Zone and Dimensionless Frequency

  27. 布洛赫模態 (Bloch Modes)

  28. 光子帶隙 (Photonic Band Gaps) 與穿透率 (Transmission) Left: Photonic Band , Right: Transmission (20layers)

  29. See, for example, Kazuaki Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001). 光子能流 (Photon Energy Flow) 與群速度 (Group Velocity)

  30. 頻率等高線 Frequency Contours Square Lattice

  31. 多重散射理論 (Multiple Scattering Theory) • 1. An array of dielectric cylinders in a uniform medium. • 2. In response to the incident wave from the wave source and the scattered waves from other scatterers, each scatterer will scatter waves repeatedly.Scattered waves can thus be expressed in terms of a modal series of partial waves. • 3. Regarding these scattered waves as the incident wave to other scatterers,a set of coupled equations can be formulated and computed rigorously. • 4.The total waveat any spatial pointis the sum of the direct wavefrom the sourceand the scattered wavesfrom all scatterers. • 5. The details about MST can be found inJ. Appl. Phys. 94, 2173 (2003) (B. Gupta and Z. Ye (葉真)). • 6. For brevity,we only consider the E-polarized wave.

  32. 波源 (wave source) 與散射體 (scatterers)

  33. 波場計算 : 點波源的情形

  34. 入射波 (Incident Waves), 散射波 (Scattered Waves)

  35. 加法定理與座標轉換Addition Theorem and Coordinate Transformation

  36. We also have the relations

  37. 解出總波場 (Solving the Total Waves)

  38. Scattered wave Incident wave 平面波波源 (Plane Wave source)

  39. 一個計算實例 (次波長成像)

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