Plane Wave Expansion Method (PWEM)

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/15/18 Instucto D. Rymond Rumpf (915) 747 6958 cumpf@utep.

1 /15/18 Instucto D. Rymond Rumpf (915) EE 5337 Computtionl Electomgnetics Lectue #19 Plne Wve Expnsion Method (PWEM) Lectue 19 These notes my contin copyighted mteil obtined unde fi use ules. Distibution of these mteils is stictly pohibited Slide 1 Outline Fomultion of the Bsic Eigen Vlue Poblem 3D D Implementtion Clcultion of Bnd Digms Clcultion of Isofequency Contous Exmple Computing the bnd digm fo n EBG mteil with sque symmety Lectue 19 Slide 1

2 /15/18 Fomultion of the Bsic 3D Eigen Vlue Poblem Lectue 19 Slide 3 Block Mtix Fom We stt with the Fouie spce Mxwell s equtions in mtix fom. Ku Ku jk s y z z y x Ku Ku jk s z x x z y Ku Ku jk s x y y x z These cn be witten in block mtix fom s Ks Ks jk u y z z y x Ks Ks jk u z x x z y Ks Ks jk u x y y x z Kz K y ux sx z x y jk K K u sy y x z K K u s z Kz K y sx ux z x y jk K K s uy y x z K K s u z Lectue 19 Slide 4

3 /15/18 Compct Nottion We cn wite ou block mtix equtions even moe compctly s. Kz K y ux sx z x y jk K K u sy y x z K K u s z Kz K y sx ux z x y jk K K s uy y x z K K s u z K u jk s K s jk u Kz K y K Kz Kx Ky Kx ux sx u y s sy u z s u z Lectue 19 Slide 5 Eliminte the Mgnetic Field We stt with the following equtions. K s jk u K u jk s Solve fo the mgnetic field u 1 1 u jk K s Substitute expession fo u into this eqution. 1 1 K jk jk K s s Simplify 1 K K s k s Lectue 19 Slide 6 3

/15/18 The 3D Eigen Vlue Poblem The 3D eigen

K s k s Kz K y K Kz Kx Ky Kx 1 1 1 1 This hs

[V,D] = eig(a,b); A K K k 1 B x s sx s sy s z

It is possible to educe this to block mtix

. It is moe common to see this expessed in tems

A k x k V x x 1 Eigen vlues e some popety of

4 /15/18 The 3D Eigen Vlue Poblem The 3D eigen vlue poblem in tems of the electic field is 1 K K s k s Kz K y K Kz Kx Ky Kx This hs the fom of genelized eigen vlue poblem Ax Bx [V,D] = eig(a,b); A K K k 1 B x s sx s sy s z Notes: 1. It is possible to educe this to block mtix eqution. See PWEM Exts lectue.. It is moe common to see this expessed in tems of the mgnetic field becuse it is n odiny eigen vlue poblem. Lectue 19 Slide 7 Visulizing the Dt x y z Ax K A k x k V x x 1 Eigen vlues e some popety of the modes. Hee, it is fequency. 1 k k 3 k Eigen vectos e pictues of the modes. N k Inputs to PWEM Intemedite Dt V Lectue 19 Slide 8 Outputs of PWEM 4

5 /15/18 Consequences of k Being the Eigen Vlue The quntity k c k is elly just fequency scled by the speed of light. In ou fomultion, k is the eigen vlue so it is the unknown quntity. We do not know k when constucting the eigen vlue poblem. Since we do not know the fequency, it is not possible to build in fequency dependent mteil popeties (i.e. dispesion) without modifying the bsic PWEM lgoithm. The bsic PWEM cnnot incopote mteil dispesion. It must be modified to ccount fo this. See PWEM Exts. Lectue 19 Slide 9 Fomultion of Efficient D Plne Wve Expnsion Method Lectue 19 Slide 1 5

cystl slb s slb wveguides. Clculte the effective efctive index of ech coss section.

6 /15/18 Two Dimensionl Devices Fo D poblems, the device is unifom nd infinite in the z diection nd wve popgtion is esticted to the x y plne. z y x infinite nd unifom Lectue 19 Slide 11 Repesenting Slb Photonic Cystls 1D Slb Wveguide Anlysis neff,1 neff, Step 1 Anlyze veticl coss sections of photonic cystl slb s slb wveguides. Clculte the effective efctive index of ech coss section. Step Build D epesenttion of the photonic cystl slb using just the effective efctive indices fom Step 1. 3D Slb Photonic Cystl Be ceful to conside the poliztion. The lignment of the electic field must be consistent. D Repesenttion n eff,1 n eff, Lectue 19 Slide 1 6

/15/18 Reduction to Two Dimensions Fo the D devices descibed on the pevious slide whee the wves e esticted to the plne of the device, the wve hs no vecto components in the z diection.

7 /15/18 Reduction to Two Dimensions Fo the D devices descibed on the pevious slide whee the wves e esticted to the plne of the device, the wve hs no vecto components in the z diection. K z Mxwell s equtions educe to Ku y z Ku z y jk sx Ku z x K xuz jk sy Ks y z Ks z y jk u x Ks z x K xsz jk uy Ku x y Ku y x jk sz Ks x y Ks y x jk u z Mxwell s equtions hve septed into two distinct modes. E Mode H Mode Ku x y Ku y x jk sz Ks x y Ks y x jk uz Ks y z jk ux Ku y z jk sx Ks jk u Ku jk s x z y x z y Lectue 19 Slide 13 D Eigen Vlue Poblems We cn now deive two eigen vlue poblems fo two dimensionl devices. E Mode Ku Ku jk s x y y x z j Ks u u Ks 1 y z jk x x y z k j Ks u u Ks H Mode 1 x z jk y y x z k 1 1 k K K K K s x x y y z s z Ks Ks jk u x y y x z j Ku s s Ku 1 y z jk x x y z k j Ku s s Ku 1 x z jk y y x z k 1 1 k Lectue 19 Slide 14 K K K K u x x y y z Note: Fo non mgnetic mteils, this mode clcultes slowe. u z 7

/15/18 Solution in Homogeneous Unit Cell Fo homogeneous unit cells, the A nd B mtices of the genelized eigen vlue poblem e

Av Bv 1 V I AK x Ky 1 B I λ B A When solved numeiclly, the columns of the eigen modes e scmbled.

sptil hmonics (i.e. plne wves in ou expnsion) fo tht mode.

8 /15/18 Solution in Homogeneous Unit Cell Fo homogeneous unit cells, the A nd B mtices of the genelized eigen vlue poblem e digonl. In this cse, the modes hve closed fom solution. Av Bv 1 V I AK x Ky 1 B I λ B A When solved numeiclly, the columns of the eigen modes e scmbled. V numeicl V closed fom Lectue 19 Slide 15 Intepeting the Eigen Vectos An eigen vecto contins the complex mplitudes of ll the sptil hmonics (i.e. plne wves in ou expnsion) fo tht mode. The numbes e complex to eflect both mgnitude nd phse of the sptil hmonics. v1 v v 3 v4 v 5 v6 v 7 v8 v 9 v1 v11 v 1 v v13 v 14 v15 v 16 v17 v18 v 19 v v 1 v v 3 v4 v 5 v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 1 v 11 v 1 v 13 v 14 v 15 v 16 v 17 v 18 v 19 v v 1 v v 3 v 4 v 5 Lectue 19 Slide 16 8

/15/18 Implementtion Lectue 19 Slide 17 PWEM fom Use s Pespective Given the unit cell of n infinitely peiodic lttice nd the Bloch wve vecto (i.e. wve diection nd sptil peiod), the PWEM clcultes ll the modes with these conditions.

9 /15/18 Implementtion Lectue 19 Slide 17 PWEM fom Use s Pespective Given the unit cell of n infinitely peiodic lttice nd the Bloch wve vecto (i.e. wve diection nd sptil peiod), the PWEM clcultes ll the modes with these conditions. The eigen vlues contin the fequencies nd the eigen vectos contin the fields. & Lectue 19 Slide 18 9

10 /15/18 Choosing the Numbe of Sptil Hmonics The only tue wy to detemine the coect numbe of sptil hmonics is to test fo convegence. Thee e howeve, some ules of thumb you cn follow to mke good guess. 1 M min Hee e some exmples Fo ech diection. M = 1 N = 7 M = 1 N = 15 M = 7 N = 1 M = 15 N = 1 M = 15 N = 15 M = 7 N = 7 M = 11 N = 7 M = 15 N = 7 M = 1 N = 5 M = 7 N = 7 M = 1 N = 1 M = 1 N = 1 M = 31 N = 31 M = 1 N = 13 Lectue 19 Slide 19 Nomlizing the Fequency We deived ou eigen vlue poblem so tht eigen vlue. 1 k K K s s z z We think of constnt. k c k (fequency) is the s fequency becuse it is fequency divided by It is vey useful to scle the eigen vlue to /. n k c Given /, we cn esily scle ou design to opete t ny fequency tht we wnt. Lectue 19 Slide k 1

/15/18 Block Digm of D Anlysis Stt 's Finish Define Poblem, Build

Mtices ERC = convmt(er,p,q); URC = convmt(ur,p,q); E Mode: H Mode:

k, k y, pq x, pq Build Eigen Vlue Poblem 1 1 k k 1 1 K K K K s s x

Poblem Ax k Bx Compute List of Bloch Wve Vectos Scle k 's n k

11 /15/18 Block Digm of D Anlysis Stt 's Finish Define Poblem, Build Unit Cell on High Resolution Gid x, y nd x, y Compute Convolution Mtices ERC = convmt(er,p,q); URC = convmt(ur,p,q); E Mode: H Mode: Clculte Wve Vecto Expnsion k x p xpq, x k yq ypq, y Ky, Kx meshgid k, k y, pq x, pq Build Eigen Vlue Poblem 1 1 k k 1 1 K K K K s s x x y y z z K K K K u u x x y y z z Solve Genelized Eigen Vlue Poblem Ax k Bx Compute List of Bloch Wve Vectos Scle k 's n k Recod Dt Lectue 19 Slide 1 Clcultion of Photonic Bnd Digms Lectue 19 Slide 11

/15/18 Bnd Digms (1 of ) Bnd digms e compct, but incomplete, mens of chcteizing the

Along the hoizontl xis is list of Bloch wve vectos (diection nd peiod of the Bloch wve).

Lectue 19 Slide 3 Bnd Digms ( of ) To constuct bnd digm, we mke smll steps ound the peimete of

When we plot ll these eigen vlues s function of, the points line up to fom continuous bnds.

12 /15/18 Bnd Digms (1 of ) Bnd digms e compct, but incomplete, mens of chcteizing the electomgnetic popeties of peiodic stuctue. Along the hoizontl xis is list of Bloch wve vectos (diection nd peiod of the Bloch wve). Veticlly bove ech Bloch vecto e ll of the fequencies which hve mode with tht Bloch wve vecto. Lectue 19 Slide 3 Bnd Digms ( of ) To constuct bnd digm, we mke smll steps ound the peimete of the ieducible Billouin zone (IBZ) nd compute the eigen vlues t ech step. When we plot ll these eigen vlues s function of, the points line up to fom continuous bnds. CAUTION: When solved numeiclly, the ode of the modes will be diffeent fo diffeent vlues of. Fo this eson, it is difficult to plot the bnds s continuous lines when they coss. Lectue 19 Slide 4 1

/15/18 Animtion of the Constuction of Bnd Digm Lectue 19 Slide 5 List of Bloch Wve Vectos Next, we genete n y of Bloch wve vectos tht mch ound the peimete of the IBZ.

13 /15/18 Animtion of the Constuction of Bnd Digm Lectue 19 Slide 5 List of Bloch Wve Vectos Next, we genete n y of Bloch wve vectos tht mch ound the peimete of the IBZ. M T X T 1 X Use moe points in pts of this y tht cove longe distnces. Do not epet djcent points. Lectue 19 Slide 6 M 13

14 /15/18 Animtion of Bnd Clcultion & Bloch Wves Lectue 19 Slide 7 Clcultion of Isofequency Contous Lectue 19 Slide 8 14

15 /15/18 The Bnd Digm is Missing Infomtion y x Diect lttice: We hve n y of i holes in dielectic with n=3.. Recipocl lttice: We constuct the bnd digm by mching ound the peimete of the ieducible Billouin zone. y x M X The bnd extemes lmost lwys occu t the key points of symmety. But we e missing infomtion fom inside the Billuoin zone. Lectue 19 Slide 9 The Complete Bnd Digm The Full Billouin Zone k y c k x Thee is n infinite set of eigen fequencies ssocited with ech point in the Billouin zone. These fom sheets s shown t ight. y x Lectue 19 Slide 3 15

16 /15/18 IFCs Fom Second Ode Bnd Index ellipsoids e isofequency contous in k spce. Lectue 7 Slide 31 IFCs Fom Fist Ode Bnd c c y x y x Lectue 19 Slide 3 16

contou() b = [ Bx(nbx,nby) ; By(nbx,nby) ]; k = pwemd(.

17 /15/18 Clculting nd Visulizing the IFCs Full Bnd Dt wn Isofequency Contous fom Full Bnd Dt wn y y x x fo nby = 1 : NBY fo nbx = 1 : NBX pcolo() contou() b = [ Bx(nbx,nby) ; By(nbx,nby) ]; k = pwemd(...); wn(nbx,nby) = *k/(*pi); end end Lectue 19 Slide 33 Stndd View of IFCs y x Lectue 19 Slide 34 17

18 /15/18 Exmple Bnd digm fo D EBG with sque symmety Lectue 19 Slide 35 Define the Lttice Extended Lttice D Unit Cell Lectue 19 Slide 36 18

19 /15/18 Build Lttice on High Resolution Gid Unit Cell 9 s High Resolution Gid 51 1 s Lectue 19 Slide 37 Constuct Convolution Mtices (x,y) o UR o URC 1 s URC = convmt(ur,p,q) 9 s (x,y) o ER 1 s ERC = convmt(er,p,q) o ERC Lectue 19 Slide 38 19

20 /15/18 Compute the Recipocl Lttice Diect Lttice Recipocl Lttice t ŷ T yˆ t1 ˆx T 1 xˆ Lectue 19 Slide 39 Constuct the Billouin Zone T T 1 Lectue 19 Slide 4

/15/18 Identify the Ieducible Billouin Zone 1. Stt with the full Billouin zone.

IBZ Lectue 19 Slide 41 Identify the Key Points of Symmety T T 1 M X The key points of symmety e

M.5T.5T 1 Fomuls fo clculting the key points of symmety long with thei nming convention cn be

21 /15/18 Identify the Ieducible Billouin Zone 1. Stt with the full Billouin zone.. Lttice hs left/ight symmety. 3. Lttice hs up/down symmety. 4. Lttice hs 9 ottionl symmety. IBZ Lectue 19 Slide 41 Identify the Key Points of Symmety T T 1 M X The key points of symmety e clculted fom line combintion of the ecipocl lttice vectos. X.5T1 M.5T.5T 1 Fomuls fo clculting the key points of symmety long with thei nming convention cn be found in [M. Lx, Symmety Pinciples in Solid Stte nd Molecul Physics, (Dove, New Yok, 1974). See supplementl notes fo Lectue 6 Peiodic Stuctues. Lectue 19 Slide 4 1

/15/18 The Numbes Typiclly the lttice constnt is nomlized to the vlue of 1.. 1 The diect nd ecipocl lttice vectos e t 1 t 1 1 6.8 T1 T 6.8 The key points of symmety e 3.14 3.14 X M 3.

22 /15/18 The Numbes Typiclly the lttice constnt is nomlized to the vlue of The diect nd ecipocl lttice vectos e t 1 t T1 T 6.8 The key points of symmety e X M 3.14 Lectue 19 Slide 43 Genete List of s Next, we genete n y of Bloch wve vectos tht mch ound the peimete of the IBZ. M T X X M T 1 X Use moe points in pts of this y tht cove longe distnces. Do not epet djcent points. Lectue 19 Slide 44 M

23 /15/18 Fo Ech, Constuct KX nd KY Compute k x nd k y wve vecto expnsions long the x nd y xes espectively. p kxp x, i p P,, 1,,1,, P q kyq y, i qq,, 1,,1,, Q kx = bx - *pi*p/; ky = by - *pi*q/; Compute wve vecto meshgid() expnsion. kx p, q ky p, q [ky,kx] = meshgid(ky,kx); Fom digonl mtices KX nd KY. K x K y KX = dig(spse(kx(:))); KY = dig(spse(ky(:))); Lectue 19 Slide 45 Solve Eigen Vlue Poblem fo Ech The eigen vlues clculted fom this poblem e: k X Lectue 19 Slide 46 M 3

/15/18 Plot Eigen Vlues Vs. k Ay index of Bloch wve vecto Lectue 19 Slide 47 Genete Pofessionl Looking Plot The fequency xis hs been nomlized fo esy scling.

24 /15/18 Plot Eigen Vlues Vs. k Ay index of Bloch wve vecto Lectue 19 Slide 47 Genete Pofessionl Looking Plot The fequency xis hs been nomlized fo esy scling. Bnd Digm of Sque Lttice Given meningful title n c k how it is lbeled how it is used how it is clculted 1 X M Bloch wve vecto Hoizontl xis is lbeled with the key points of symmety. A pictue of the lttice unit cell, Notice the spcing between key Billouin zone, IBZ, nd key points points is consistent lengths of symmety should be shown. ound peimete of IBZ. Lectue 19 Slide 48 4

This immediately suggests an inverse-square law for a "piece" of current along the line.

This immediately suggests an inverse-square law for a piece of current along the line. Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line