光子晶体原理与计算

1、光子晶體原理與計算光子晶體原理與計算 (I) Bloch 定理定理 ,光子能帶光子能帶, 平面波展開法平面波展開法, 光子能流光子能流, 與多重散射法與多重散射法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 (Frequen

2、cy 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 “And

3、erson 局域化” (Anderson Localization).Eli Yablonovitch Sajeev John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,

4、 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”.光子晶體光子晶體 (Photonic Crystals)（a）一維光子晶體）一維光子晶體 （b）二維光子晶體）二維光子晶體 （c）三維光子晶體）三維光子晶體Photonic Crystals光子晶體能帶光子晶體能帶( (頻帶頻帶) , ) , 光子帶隙光子帶隙(2D System)Photonic band structure, Photonic band gap光子晶體光子晶體 (

5、帶隙帶隙) 的應用的應用: 共振腔共振腔與與波導波導缺陷模缺陷模 (Defect Mode) )波導模波導模 (Guiding Mode) )光子帶隙的其它應用光子帶隙的其它應用光子晶體光纖光子晶體光纖 (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.Photonic crystals as optical componentsP. Halevi et.al.A

6、ppl. Phys. Lett.75, 2725 (1999) 光子晶體傳導帶的應用光子晶體傳導帶的應用(I) 長波極限長波極限 (Long-wavelength Limit):非均向性透鏡非均向性透鏡 (Anisotropic Lens)10a光子晶體傳導帶的應用光子晶體傳導帶的應用(II) 超越長波極限超越長波極限 (Beyond the Long-wavelength Limit)負折射透鏡與次波長成像負折射透鏡與次波長成像 (Negative Refraction Lens and Subwavelength Imaging)光子晶體能帶理論光子晶體能帶理論, 計算方法計算方法, 與物

7、理詮釋與物理詮釋週期函數的處理: Fourier 級數與倒晶格 (Reciprocal Lattice)Bloch 定理 (Blochs 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Periodic function, Fourier Series and Reciprocal Lattice: ()( )(), , : perio

8、d f xaf xf xRRna a1D periodic functionTry ( )()exp()Gf xC GiGxWe have exp()1exp( 2)iGain22,( )expnnnGnbbf xCinbxaa001expexp( )expaamnnimbxinbx dxaCf xinbx dxa01Or ()( )expaC Gf xiGx dxa11112112 : ()()( )() , and : fundamental translation vectors ffffnn2D periodic functionrararrRRaaaa12(Primitive) Ce

9、ll Area: |cA aa12Try ( )()exp()exp()exp()1fCiiiGrGG rG aG a1122Define 2; ,1,2ijijnni jGbbab112212In a cell: ; 01, 01raa1 212121 122Thus ( )(,)exp 2n nnnffCi nn r211221212|()()|Using the fact: |cddd rddAaaaa1 2211121 12212001We have ()( )exp()Or (,)exp2c celln nCfid rACfi nndd GrG r31Fourier Componen

10、t: ()( )exp()c cellCfid rVGrG r123Cell Volumn: |() |cV aaa112233Reciprocal Lattice: , 2ijijnnnGbbbab233112123RL Bases: 2,2,2cccVVVaaaaaabbbFourier Expansion: ( )()exp()fCiGrGG r112233: ( )(), ffnnn3D periodic functionrrRRaaa1 1 11231122331230 0 0()(,)exp2Cfi nnnddd G112233123Point in a Unit Cell: ,

11、0,1 raaa晶格基底與倒晶格基底晶格基底與倒晶格基底 (二維二維) Lattice Bases vs. Reciprocal Lattice Bases (2D)Square Lattice 123232aaaxyaxy12213213aabxybxy12aaaxay1222aabxbyTriangular Lattice 4/ 3a二元系統與結構因子二元系統與結構因子Binary System and Structure Factor (2D)if region ( )if region ababrrr( )()exp()iGrGG r222221()( )exp()11exp()exp

12、()()exp()exp()c cellabccababbccaab cellid rAid rid rAAid rid rAA GrG rG rG rG rG r() if ()(1) if abSffGG0GG02region 1Structure factor: ()exp()Filling Fraction: caacSid rAAfAGG rExample 1: Square Lattice, Circular Rods/Holes20000200211211: ()exp(cos )22()( )22()2rcrGrccccSiGrdr drAr JGr drxJx dxAG AJ

13、GrrGrJ GrG AAGrJGrfGr Structure FactorG10:exp(cos )( ),( )( )ninnndixi Jx exJ xxJxdxIdentities22rfa2222121224rGrnnf nnaExample 2: Triangular Lattice, Circular Rods/Holes1: ( )2JGrSfGrStructure FactorG1213,22aaaxaxy122122,33aabxyby2122222|sin( /3)23rfraraaa122221122112222112221223322324383Grnnraannrn

14、arnn nnafnn nnxyyExample 3: Simple Cubic Lattice, Spheres1201203: 2( )exp(cos ) cos4sin()sin()cos()3()rcrcS GiGrdrdrVGrrdrVGrGrGrGrfGr Structure Factor21/3222(6)xyzGrfnnn334 3spherecVfVra123,aaaaxayaz123222,aaabxbybz布洛赫定理與布里淵區布洛赫定理與布里淵區Blochs Theorem and Brillouin zoneBlochs Theorem (Electron System

15、s):22: ( )( ) ( )( ), ( )() 2VEVVmEigenvalue problemrrrrrrR: ( )( )exp(), ( )() ( )expuiuuuikkkkkGSolutionrrk rrrRkFirst Brillouin Zone GG rFirst Brillouin Zone3Try ( )()exp()d KCirKK rHere ( )()exp()VViGrGG r223Thus we have ()( ) () exp()02KE CVCid KmGKGKGK r22Since ( )( ) ( )( )2VEmrrrr33( )()exp

16、()( )()expVCid KVCid KGGGKKGrGKGK r22Or ()() ()02KE CVCmGKGKG3. .3. .3. .3. .3. .( )()exp ()()exp ()exp()()exp()exp()( )( )B ZB ZB ZB ZB Zd K Cid kCid kiCid kiud k GGGGkkrkGkGrkGkGrk rkGG rk rrrDefine ( )()exp()( )exp()( )uCiiukGkkKkGrkGG rrk rrProving Blochs Theorem22()We have ()() ()()2CVCE CmkGkG

17、kGGGkGkG22()Difine ( )=(), ( )()2MVCCmGGGGGkGkGGkkGSo we have the eigenvalue equation( ) ( )= ( )( )MCECGGGGGkkkkMore explicitly:( ) ( )=( )( ), : Band Index; : Bloch Wave Vector( ) : Energy (Hyper)SurfacesnnnnMCECnEGGGGGkkkkkkBlochs Theorem and Photonic Band Structure2-1-12( )1( )( ), ( )()exp()( )

18、nickkGkHrHrrGG rr( )()exp ()ikkGHrhGkGr212( )(), = +, = +nc GGGkGG PPhhP k GPk G 2221 2 11( )(), nhhhcGGGGGkGGPePehe1(Faradays Law)ictc BEH1(Amperes Law)ictc DHETwo-Dimensional Inhomogeneous Wave Systems2221ct ( ), ( )ccrr2222222222221, Acoustic wave in fluid1, -polarized shear wave: =1, E-polarized

19、 EM wave: .1, H-polarized EM watppctuuzuctEEEctHHctuzEzve: .HHz0cc2211( ), ( )eettccrrClassical WavesUnified TreatmentBand Structure Calculation (2D, Scalar Wave)222( , )1( , )Wave Equation: 0( )( )( )ttct rrrrrHarmonic wave: ( , )( )exp()ti t rr22( )( )Thus ( )( )( )c rrrrr112Media Parameters: ( )(

20、)exp(), ( )( )( ) ()exp()iciGGrGG rrrrGG r12or ( )( )( )( ) (*) kkkrrrr(1) if 0()() () if 0ababffSGGGG(1) if0()() () if0ababffSGGGGSubstitute ( )()exp into Eq. (*)i kkGrGkGr2We get ()() ()()()()kkkGGGGkGkGGGGG2Generalized Eigenvalue Problem: kkkkkM vN vwhere()()() ()()()()() kGGkGGkGMGGkGkGNGGvG2121

21、Transform(21)()(1)(21)()(1)iNnNnNjNnNnN11221122nnnnGbbGbb1212Choose, , NnNNnNNnNNnN()() , ()() , ()()ijijikGGkkGGkkGkMMNNvv12222We can also solve the Eigenvalue Problem: The dimension of ,: (21)(21)There are (21) plane waves or eigenvaluesM NNNNkkkkkN M vv約化布里淵區與約化本徵頻率約化布里淵區與約化本徵頻率Reduced Brillouin

22、Zone and Dimensionless Frequency/2/aca ca布洛赫模態布洛赫模態 (Bloch Modes)(Bloch Modes)光子帶隙光子帶隙 (Photonic Band Gaps) 與穿透率與穿透率 (Transmission)Left: Photonic Band , Right: Transmission (20layers) 光子能流光子能流 (Photon Energy Flow) 與群速度與群速度 (Group Velocity)*1Time averaged Poynting vector: ( )Re( )( )2S rE rH r221Time

23、 averaged energy density: ( )( )|( )|( )|( )|4UrrE rrH rHarmonic fields: ( , )( ), ( , )( )i ti tteteE rE rH rH rBloch Eigenmodes: ( )( ), ( )( )iieek rk rkkkkErerHrhrPhase velocity: |Group velocity: ( )Energy velocity: ( )pgegUkvkkvS rvvrSee, for example, Kazuaki Sakoda, Optical Properties of Photo

24、nic Crystals (Springer-Verlag, 2001).頻率等高線頻率等高線Frequency ContoursSquare LatticegvConstant Frequency Curve多重散射理論多重散射理論 (Multiple Scattering Theory). 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 scatter

25、ers, 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 wave a

26、t any spatial point is the sum of the direct wave from the source and the scattered waves from all scatterers. 5. The details about MST can be found in J. Appl. Phys. 94, 2173 (2003) (B. Gupta and Z. Ye (葉真).6. For brevity, we only consider the E-polarized wave.波源波源 (wave source) 與散射體與散射體 (scatterer

27、s)波場計算波場計算 : 點波源的情形點波源的情形22(2)221Helmholtz Equation: ()( )4( ), ()( )0.extintkk rrr1Boundary Condition:11extintextintnn 1111Media Parameters:( , ), (,)Wave Numbers: / , /Cylinder Radius: , Cylinder Center: ( cos, sin)Position Vector: ( cos, sin)(|cos,|sin)Location oiiiiiiiiiiicckc kcarrrrrrrrr rr rr

28、rrrrrrrf the Point Source: 0r222222D Laplacian: /xy 22(2)(1)0Source wave satisfies: () ( )4( )Solution: ( )() kGGi Hkr rrr(1)(1)( )( )(z) , :Bessel function, : Neuman function:First kind Hankel functionnnnnnnHzJziYJYH入射波入射波 (Incident Waves), 散射波散射波 (Scattered Waves)(1)Sacttered wave from the th sact

29、terer:,|jinjsjnnjnji A Hker rr rrr 1,The incident wave towards the th scatterer:,Niincsjjj iiGrrr rThe incident wave has no singularity:( |)iiniiincnniB Jker rrr22(2)Exterior wave: ()( )4( )extiiiextincscatk rr()(1),( |)( |)( |)( |)ijiiisji n liljnn lijlilnili jllilini jnnini A HkeJ keCJ keCJke rrr

30、rr rr rr rrrrrrrrr = (1)(1):|jijiinnji n liln lijlilHkeHkeJker rrrr rAddition Theoremrrrrrr(),(1)( |)iji l ni jjnll nijlCi A Hkerrrr加法定理與座標轉換加法定理與座標轉換Addition Theorem and Coordinate Transformation(1)0(1)( )()( |)( |)( |)iiiininnininininninGi HkriHkeJkeS Jkerr rr rrrrrrr(1)( |)iininniSi Hkerr ,1,()(1

31、)1,( |)(*)i jijNiinnnjj iNi l nijnll nijjj i lBSCSi A Hke rrrrWe also have the relations(1)Exterior wave:( )( |)( |)iiniiextnninninB Jki A Hker rrrrrr(1)(1) 1iiiiiinnnnnniiiiiinnnnnnB Jkai A HkaD Jka hB Jkai A HkaD Jka hgh1Interior wave has no singularity:( )(|)iiniiintnninD Jker rrrr111/,/ghk kcc1Bo