EE 5337 Computational Electromagnetics

Transcription

1 Instucto D. Ramond Rumpf (95) EE 5337 Computational Electomagnetics Lectue # WEM Etas Lectue These notes ma contain copighted mateial obtained unde fai use ules. Distibution of these mateials is stictl pohibited Slide Outline Using 3D WEM fo D and D Analsis, and using D WEM fo D Analsis Visualiing the fields Fomulation of efficient 3D WEM Fomulation of efficient D WEM WEM with anisotopic mateials Incopoating dispesion Reduced Bloch mode epansion (RBME) technique Genealied smmet Lectue Slide

2 Using 3D WEM fo D and D Analsis, and Using D WEM fo D Analsis Lectue Slide 3 Using 3D WEM fo 3D, D & D Analsis 3D Analsis with 3D WEM D Analsis with 3D WEM D Analsis with 3D WEM along Q along R along along Q along R along along Q along R along Conventional 3D WEM The device is unifom along the ais so R can be set to. The device is unifom along the and aes so and Q can both be set to. Lectue Slide 4

3 Using D WEM fo D & D Analsis D Analsis with D WEM D Analsis with D WEM along Q along along Q along Conventional D WEM The device is unifom along the ais so Q can be set to. Lectue Slide 5 Visualiing the Fields Lectue Slide 6 3

4 Eigen Vectos Contain Field Infomation Fo D WEM, we solved the following eigen value poblem. k V K K K K s s Recall the field epansion into a plane wave basis jk p q eigen vectos λ eigen values,,, p ˆ q E S p q e k p q ˆ p q s The comple amplitudes of the plane waves ae stoed in the elements of the eigen vectos. Lectue Slide 7 Reaange Tems The field epansion is,,, p ˆ q E S p q e k p q ˆ jk p q p q j We substitute in the epession fo k p, q and facto out. p q j ˆ ˆ E e S p q e j, p q This is just an invese D FFT. We can now wite the electic field epession as j E e FFT Sp, q e Lectue Slide 8 4

5 Visualiing the Data S p, q S p, q on full gid q q p p j e Sp q FFT, E Lectue Slide 9 ocedue ( of ) Step Etact eigen vecto and eshape Step Inset Fouie coefficients into lage gid. s = V(:,m); s = eshape(s,,q); nc = ceil(n/); n = nc - floo(/); n = nc + floo(/); nc = ceil(n/); n = nc - floo(q/); n = nc + floo(q/); sf = eos(n,n); sf(n:n,n:n) = s; Lectue Slide 5

6 ocedue ( of ) Step 3 Calculate invese FFT of S p, q. S p, q on full gid A, on full gid Step 4 Calculate phase tem. q p a = ifft(ifftshift(sf)); a = a/ma(abs(a(:))); phase = ep(-i*(beta()*x + beta()*y)); Step 5 Calculate oveall field..* = E = phase.*a; Lectue Slide Fomulation of Efficient 3D lane Wave Epansion Method Lectue Slide 6

7 Notation Unbolded lettes ae scala quantities. Bold capital lettes ae matices, usuall squae matices. Bold lowe case lettes ae column vectos. a.678 a a a a a a A a a a M N N M M MN Bold lowe case lettes with a vecto sign ae column vectos of column vectos. e e e e a a a Lectue Slide 3 Vecto Mati Opeations Definitions A diagonal mati containing components of A A diagonal mati containing components of A A diagonal mati containing components of A A A i i A i AiN A A A A Vecto Mati Opeations A A A A A A A A A A A A A A A Unless A A B B A, A, A, B, B, and B ae all diagonal. Unless A A B B A, A, A, B, B, and B ae all diagonal. In ou fomulation, onl [ ] and [ ] ae not diagonal matices. Lectue Slide 4 7

8 Epand the Field Into Two Othogonal olaiations We wite the field as the sum of two othogonal polaiations. s s ˆ s ˆ Two polaiation vectos ae chosen fo each spatial hamonic in the plane wave epansion. These must be othogonal to the diection of the hamonic. If these ae chosen coectl, the following equations will be satisfied. ˆ ˆ k ˆ ˆ k K K k ˆ ˆ ˆ ˆ k K K k ˆ ˆ ˆ ˆ k K K Lectue Slide 5 Recipe fo Calculating Othogonal olaiation Vectos Given the wave vecto of the i th spatial hamonic k k aˆ k aˆ k aˆ i, i, i, i We constuct anothe vecto that is in a diffeent diection than k. i v 4k aˆ k aˆ 3k aˆ i, i, i, i We can now calculate two vectos that ae pependicula to the wave vecto. kivi pˆ, i kivi An choice of pˆ, i and pˆ, i can be used as long as pˆ ˆ, i p, i k i. k ˆ i p ehaps thee eists a bette choice than descibed hee., i pˆ, i k pˆ i, i Lectue Slide 6 8

9 Wave Equation with olaiation Epansion We substitute the polaiation epansion into ou mati wave equation. K K s k s K ˆ ˆ ˆ ˆ K s s k s s Lectue Slide 7 ojecting a Vecto Mati Equation Onto Two Othogonal olaiations We can poject one vecto mati onto anothe using ab oj a onto b b b ab oj a onto b a b b b b a Using this equation, we can poject both sides of a vectomati equation onto ou two othogonal polaiations to get two sepaate equations. A B ˆ ˆ A ˆ B ˆ A B ˆ ˆ A ˆ B ˆ A B One 33 mati equation Two mati equations Note: ˆ I ˆ Lectue Slide 8 9

10 oject Ou Vecto Mati Equation Onto Two Othogonal olaiations We appl the esults fom the last slide to ou vecto mati equation. K K s s s s ˆ ˆ ˆ ˆ k oject onto ˆ ˆ K K s s s s ˆ ˆ ˆ ˆ ˆ k oject onto ˆ ˆ K K s s s s ˆ ˆ ˆ ˆ ˆ k We now have two mati equations. Lectue Slide 9 Block Mati Fom We can put ou two pevious equations into block mati fom as follows. ˆ ˆ ˆ ˆ ˆ ˆ K K s s s s ˆ ˆ ˆ ˆ ˆ ˆ K K s s s s k k ˆ ˆ ˆ ˆ K K K K s ˆ k ˆ ˆ ˆ ˆ K K K K s ˆ ˆ ˆ s ˆ ˆ ˆ ˆ s This is a genealied eigen value poblem. It is smalle than the 33 genealied eigen value poblem that we deived peviousl. Lectue Slide

11 Final Fom of Genealied Eigen Value oblems Reduced Eigen Value oblem fo Electic Fields ˆ ˆ ˆ ˆ s s k K K K K A B A s s ˆ ˆ ˆ ˆ K K K K ˆ ˆ ˆ ˆ B ˆ ˆ ˆ ˆ Reduced Eigen Value oblem fo Magnetic Fields ˆ ˆ ˆ ˆ u u k K K K K A B A u u ˆ ˆ ˆ ˆ K K K K ˆ ˆ ˆ ˆ B ˆ ˆ ˆ ˆ Note that both of these eigen value poblems ae valid fo geneal anisotopic media. Lectue Slide Genealied Eigen Value oblem with NO MAGNETIC RESONSE Genealied Eigen Value oblem fo Electic Fields ˆ ˆ ˆ ˆ s s K K A k B A ˆ ˆ ˆ ˆ s s K K ˆ ˆ ˆ ˆ B ˆ ˆ ˆ ˆ Odina Eigen Value oblem fo Magnetic Fields ˆ ˆ ˆ ˆ u u k K K K K A A u u ˆ ˆ ˆ ˆ K K K K This is the most common fomulation because it is an odina eigen value poblem. Note that both of these eigen value poblems ae valid fo geneal anisotopic media. Lectue Slide

12 Recoveing s fom s and s The genealied eigen value poblem fo the electic fields is s s A kb s s Let the eigen vecto and eigen value matices be s s W eigen-vecto mati A kb s s λ eigen-value mati If we etact a single eigen mode, w i and i, we have si wi si Recall that s s ˆ ˆ s. Theefoe, s ˆ ˆ ˆ ˆ si s ˆ ˆ i s is i wi s si s These tems ae now independent of the chosen polaiation vectos. Lectue Slide 3 Fomulation of Efficient D lane Wave Epansion Method Lectue Slide 4

13 Reduction to One Dimension Fo the D devices whee popagation is onl in the diection, the wave has no wave vecto components in the and diections. K and K Mawell s equations educe to Ku jk Ku jk s s jk s Ks jk Ks jk u u jk u Immediatel, we see that s u The field will have no components in the longitudinal diection. Lectue Slide 5 Two Modes Mawell s equations have again decoupled into two distinct sets of equations. Ku K u jk jk jk s s s Ks K s jk jk jk u u u Ku E Mode jk Ks jk E Mode s Ku jk u Ks jk s u Lectue Slide 6 3

14 Final Eigen Value oblems We can substitute one equation into the othe to deive two eigenvalue poblems. E Mode Kμ Ks k u Kε Ku jk o k u s jk s K s K u E Mode Kμ Ks k s u jk Kε Ku o k s u K s K u jk Lectue Slide 7 WEM with Full Anisotopic Mateials Lectue Slide 8 4

15 Anisotopic 3D WEM The mati equations deived fo 3D WEM ae valid fo full anisotopic mateials. In the anisotopic case, howeve, the mateial tensos become block matices composed of convolution matices fo each tenso element individuall. Lectue Slide 9 Anisotopic D WEM In geneal, all field components emain coupled in anisotopic media. Since Mawell s equations will not simplif in this case, thee is almost no numeical advantage to developing a D WEM fo anisotopic media. Instead, use ou anisotopic 3D WEM fo D poblems b using onl on spatial hamonic in the unifom diection. infinite and unifom ais # hamonics along Q # hamonics along R # hamonics along In this case, ou would set R =. Lectue Slide 3 5

16 Incopoating Dispesion Lectue Slide 3 Techniques fo Incopoating Dispesion. Iteative WEM Can onl handle one fequenc at a time.. Do not use k as the eigen value Lectue Slide 3 6

17 Altenate D WEM Fomulation ( of 5) Recall ou stating point fo E and H modes E Mode Ku Ku jk s Ks Ks jk jk u u H Mode Ks Ks jk u Ku Ku We will choose as ou eigen value instead of k. jk jk s s jk jk I G u I G u s E Mode I G s u IG s jk u jk jk I G s I G s u H Mode I G u s IG u jk s G G G G components of wave vecto epansion components of wave vecto epansion Lectue Slide 33 Altenate D WEM Fomulation ( of 5) Net we nomalie ou wave vecto tems b dividing b k. K k K K k K G k G G k G k k Ou govening equations can now be witten as E Mode IG u IG u j s j j I G s u I G s u H Mode j j j I G s I G s u I G u s I G u s Lectue Slide 34 7

18 Altenate D WEM Fomulation (3 of 5) Solving fo s and u, we get u I G s We eliminate these tems b substituting these epessions into ou govening equations. E Mode j IG IG s IG u js IG s j u H Mode j E Mode j H Mode s I G u j j j I G I G u I G s u I G u s Lectue Slide 35 Altenate D WEM Fomulation (4 of 5) We eaange these equations to bing to the ight hand side. E Mode H Mode Gu j IG IG s u j u G s s Gs j IG IG u s j s G u u Lectue Slide 36 8

19 Altenate D WEM Fomulation (5 of 5) Finall, we wite ou equations in block mati fom. E Mode G j IG IG u u j s s G H Mode G j IG IG s s j u u G Lectue Slide 37 Altenate 3D WEM Fomulation ( of ) We choose to be the eigen value. j j Ku IG u s IG u K u s Ku Ku j s Solve fo longitudinal components. Eliminate longitudinal components. j j Ks IG s u Lectue Slide 38 IG s K s u Ks Ks j u jk K s K K s G u u j s j K u K u u j K s K s j K K s jk K s G u u j j j j j Gs K Ku K K u s Gs K K u K Ku s 9

20 Altenate 3D WEM Fomulation ( of ) In block mati fom, we have j j j j j G K K K K s s G j K K K K s s j K K K K G u u u u j K K K K G Lectue Slide 39 Altenate D WEM Fomulation Fo D, ou eigen value poblem is E Mode I G u j s j Ku jk Gs j u s j s G u u G j s s j u u G Ks jk u I G s u s E Mode j j Ks Ku jk s jk Gs j u s j s G u u G j s s j u u G I G u s IG s u u Lectue Slide 4

21 Reduced Bloch Mode Epansion (RBME) Technique Mahmoud I. Hussein, Reduced Bloch mode epansion fo peiodic media band stuctue calculations, oc. R. Soc. A 465, pp , 9. Lectue Slide 4 The oblem Suppose we wish to calculate a photonic band diagam o calculate the full bands thoughout the entie Billouin one. Fo D simulations, tpical mati sie is 4 4. Fo 3D simulations, tpical mati sie is,,. These lage matices must be solved fo each Bloch wave vecto of inteest. This is tpicall s of vectos. a c Run time is long and computation is bogged down. a a a Lectue 4 a

22 Step : Calculate the Full Solution at Onl the Ke oints of Smmet Γ M Notes Ensue that no edundant points ae used o RBME will fail. An asmmetic distibution of ke points will lead to asmmet in the data calculated fom them. You can impove accuac b adding moe ke points, but this will be slowe and less efficient. X 3 N V Γ v Γ v Γ v Γ v Γ 3 N V X v X v X v X v X V M 3 N v M v M v M v M Done? Constuct full WEM Eigen Value oblem Solve full WEM Eigen Value oblem Loop though ke points of smmet Append Tansfomation Mati U Lectue 43 Step : Combine Eigen Vecto Matices Using Lowest Ode Modes 3 N V Γ vγ vγ v Γ vγ 3 N VX vx vx v X vx VM 3 N v M vm v M vm Take the M lowest ode modes fom each eigen vecto mati and constuct a new eigen vecto mati U. M M M U v Γ vγ v X vx v M vm U does not have to be a squae mati. Lectue 44

23 Step 3: efom Gam Schmidt Othonomaliation on the Mati U M M M U v Γ vγ v X vx v M vm M M M U v Γ v Γ v X v X v M v M Lectue 45 Gam Schmidt Othonomaliation U v v v 3 v M vm vm v Step : v v v vvv Step : v v vvv v v v v v v v Step 3: v 3 v v v v v v v This algoithm essentiall just emoves the components of the pevious vectos and then nomalies the amplitude. Lectue 46 3

24 Step 4: Constuct Eigen Value oblem fo the Net Constuct the standad WEM eigen value poblem fo the net, but do not solve it et. A λb E Mode H Mode K K K K A K K K K B E Mode H Mode Lectue 47 Step 5: Reduce Mati Sie b Epanding into Bloch Modes as the Basis efom a linea tansfomation that epesses the field in tems of the new basis fomed fom the educed set of Bloch modes. This damaticall educes the sie of the matices while etaining ecellent accuac because those Bloch modes moe efficientl descibe the fields in the stuctue. A B H U AU H U BU H A U Lectue 48 A U 4

25 Step 6: Solve the Reduced Eigen Value oblem The educed eigen value poblem is solved accoding to A λb V, λ The eigen values ae intepeted the same as in the standad WEM. If the ae needed, the eigen vectos must be tansfomed back into a plane wave basis to be compatible with the standad WEM. V UVU H Lectue 49 Step 7: Repeat ocedue fo Each Bloch Wave Vecto in IBZ Done? Constuct full WEM Eigen Value oblem Tansfom to U Solved Reduced Eigen Value oblem Recod Loop though list of s Lectue 5 5

26 Block Diagam of RBME Dashboad Build Unit Cell Build Convolution Matices Constuct IBZ and Ke oints Geneate List of s Done? Constuct WEM Eigen Value oblem Solve WEM Eigen Value oblem Loop though ke points of smmet Append Tansfomation Mati U Gam Schmidt Done? Constuct WEM Eigen Value oblem Tansfom to U Solved Reduced Eigen Value oblem Recod Loop though list of s Daw Band Diagam Lectue 5 Genealied Smmet Lectue Slide 5 6

27 Mste Bands When Modeling Heagonal Aas with Rectangula Smmet Unit Cell 3 Recipocal Lattice Vectos a c Γ M K Γ Unit Cell 3 Recipocal Lattice Vectos a c Mste nonphsical bands Γ M K Γ Lectue Slide 53 Coect phsical bands Cause of Mste Bands Lattice smmet imposes phase conditions between adjacent unit cells epessed b the Bloch theoem. The highe the smmet, the tighte the phase conditions that ae imposed. Thee ae fewe phase conditions with a ectangula unit cell so thee ae additional allowed states. Lectue Slide 54 7

28 The Fi The fi is to constuct the convolution matices fo the geneal smmet case. The standad FFT can no longe be used and this pocess is moe numeicall intensive. Fo D peiodic functions, the epansion is jptqt jptqt f, apq, e apq, fe, da A p q A Fo 3D peiodic functions, this is jptqtt3 jptqtt3 f apq,, e apq,, f e dv V p q V These must be numeicall evaluated. Lectue Slide 55 8