Metals and Alternative Grid Schemes

Transcription

1 EE 5303 Elecromagneic Analysis Using Finie Difference Time Domain Lecure #18 Meals and Alernaive Grid Schemes Lecure 18 These noes may conain copyrighed maerial obained under fair use rules. Disribuion of hese maerials is sricly prohibied Slide 1 Lecure Ouline Review of Seleced Topics PML Placemen Two dimensional TF/SF Compuing reflecance and ransmiance Incorporaing meals ino FDTD Alernaive grids Lecure 18 Slide 2 1

2 Review of PML Placemen Lecure 18 Slide 3 PML Placemen I is bes pracice o place he PML ouside any evanescen field. This is bes assessed by visualiing he fields during a simulaion. PML PML PML PML Lecure 18 Slide 4 2

3 Review of Two Dimensional Toal Field/Scaered Field Lecure 18 Slide 5 The Toal Field/Scaered Field Framework 2D FDTD Grid oal field scaered field Problem Poins! 1 src src Lecure 18 Slide 6 3

4 Correcion o Finie Difference Equaions a he Problem Cells (1 of 2) On he scaered field side of he TF/SF inerface, he finie difference equaion conains a erm from he oal field side. Due o he saggered naure of he Yee grid, his only occurs in he updae equaion for a magneic field. In fac, his only occurs in he compuaion of he curl of E used in he H field updae equaions. i, src src 1 E src 1 E E This is an equaion in he C scaered field, bu E src y We mus subrac he source from look like a scaered field quaniy. C E src 1 sc r i src E E E is a oal field quaniy. src E src, 1 src y o make i E E C y src src 1 E src 1 1 y E src src sandard curl equaion This is a correcion erm ha can be implemened afer calculaing he curl o inec a source. Lecure 18 Slide 7 Correcion o Finie Difference Equaions a he Problem Cells (2 of 2) On he oal field side of he TF/SF inerface, he finie difference equaion conains a erm from he scaered field side. Due o he saggered naure of he Yee grid, his only occurs in he updae equaion for an D field. In fac, his only occurs in he compuaion of he curl of H used in he D field updae equaions. C H src 2 src i1, src src src 1 y y H H H H o make i look like a oal We mus add he source o H field quaniy. C src 1 y This is an equaion in he scaered field, bu H is a oal field quaniy. src i 1, src i, src src src 1 s H, y src y H i , 1 i rc H H H H y src 1 C src i1, src src src 1 H, y H i y src H H H y y H src 1 src 2 sandard updae equaion This is a correcion erm ha can be implemened afer calculaing he curl o inec a source. Lecure 18 Slide 8 4

5 Calculaion of he Source Funcions (E Mode) We calculae he elecric field as E src i, src g The inde i indicaes he TF/SF correcion is incorporaed across he enire row of he grid. We calculae he magneic field as src 1 src r n H y g 2 r 2c0 2 Ampliude due o Mawell s equaions Delay hrough one half of a grid cell Half ime sep difference Lecure 18 Slide 9 TF/SF Block Diagram for E Mode Done? yes Finished! Main loop no Compue Curl of E Inec TF/SF Source ino curl of E Updae H Inegraions Updae H Field Compue Curl of H Inec TF/SF Source ino curl of H Updae D Inegraions Updae D Updae E Visualie Fields Lecure 18 Slide 10 5

6 Review of Compuing Reflecance and Transmiance Lecure 18 Slide 11 Comple Wave Vecors Purely Real k Purely Imaginary k Comple k Uniform ampliude Oscillaions move energy Considered o be a propagaing wave Decaying ampliude No oscillaions, no flow of energy Considered o be evanescen Decaying ampliude Oscillaions move energy Considered o be a propagaing wave (no evanescen) Lecure 18 Slide 12 6

7 10/19/2016 Evanescen Fields in 2D Simulaions n1 n2 n1 n2 n1 n2 No criical angle 1 C 1 C Lecure 18 Slide 13 Fields in Periodic Srucures Waves in periodic srucures ake on he same periodiciy as heir hos. k Lecure 18 Slide 14 7

8 The Plane Wave Specrum (1 of 2), E y m S m e k m ky m y We rearranged erms and saw ha a periodic field can also be hough of as an infinie sum of plane waves a differen angles. This is he plane wave specrum of a field. Lecure 18 Slide 15 The Plane Wave Specrum (2 of 2) The plane wave specrum can be calculaed as follows kmr, E y m S m e ˆ k m k ˆ m ky m y 2 m kmk,inc y 2 2 k m k n k m Each wave mus be separaely phase mached ino he medium wih refracive inde n k inc k 5 k 4 k 3 k 2 k 1 k 0 k 1 k 2 k 3 k 4 k 5 n 1 n 2 k y is imaginary. k y is real. k y is imaginary. Lecure 18 Slide 16 8

9 Power Flow From Graings 1. Power flow is in he direcion of he Poyning vecor. 1 2 * 1 Re E H Re k k E I is only he componen ha flows power ino and ou of he graing. 2 1 E k Re 2 k 3. Diffracion efficiency is defined as he fracion of power diffraced ino a paricular mode. m DE m,inc 4. The diffracion efficiencies of he spaial harmonics are 2 2 Sref m k rn,ref m S m k,rn m r,ref DEref m Re DE 2 rn m Re 2 S k,inc S k,inc r,rn inc 5. Conservaion of power requires m 1 maerials have loss DEref m DE rn m 1 maerials have no loss m 1 maerials have gain inc Lecure 18 Slide 17 Calculaing Transmiance and Reflecance FDTD Simulaion Fourier Transform f Seady Sae Fields FFT km, kym Spaial Harmonics 2 Sref m k,ref m DEref m Re 2 S,inc inc k 2 Srn m k,rn m r,ref DE rn m Re 2 S k,inc r,rn inc Diffracion Efficiencies R f DE ref m, f N T f DE rn m, f N Reflecance and Transmiance frequency 1 frequency 2 frequency 3 frequency 4 frequency NFREQ frequency 1 frequency 2 frequency 3 frequency 4 frequency NFREQ Lecure 18 Slide 18 9

10 Procedure for FDTD 1. Simulae he device using FDTD and calculae he seady sae field a he reflecion and ransmission record planes. Eref, f, Ern, f and Esrc f 2. Calculae he inciden wave vecor: ky,inc k0ninc frequency 3. Calculae periodic epansion of he ransverse wave vecor km2 m L mfloorn 2,, 1, 0,1, floor N 2 4. Calculae longiudinal wave vecor componens in he refleced and ransmied regions k m kn k m k m kn k 2 m y,ref 0 ref y,rn 0 rn 5. Normalie he seady sae fields o he source Eˆ, f E, f E f Eˆ, f E, f E f ref ref src rn rn src 6. Calculae he comple ampliudes of he spaial harmonics S m, f FFT Eˆ, f S m, f FFT Eˆ, f ref ref rn rn 7. Calculae he diffracion efficiencies of he spaial harmonics 2 ky,ref m 2 ky,rn m ref DE ref m, f Sref m, f Re DE rn m, f Srn m, f Re ky,inc ky,inc rn 8. Calculae reflecance, ransmiance, and conservaion of energy. DE ref, DE rn, R f m f T f m f C f R f T f m m Lecure 18 Slide 19 Incorporaing Meals ino FDTD Lecure 18 Slide 20 10

11 Wha are Meals? Meals are maerials wih very high conduciviy and usually very negaive dielecric consan. The elecric field approaches ero inside a meal. Increasing Conduciviy Maerial Conduciviy (S/m) Glass Carbon Mercury 10 6 Lead Tin Iron Nickel Aluminum Gold Copper Silver Lecure 18 Slide 21 Mehods for Incorporaing Meals Easier Implemenaion More Accurae Simulaion Ereme Dielecric Consan Easies because no modificaion o he code is necessary, bu i does no accoun for loss. Perfec Elecric Conducor Requires minimal modificaion o he code, bu does no accoun for loss. Requires greaer modificaion o he formulaion of he updae equaions. I can accoun for loss, bu canno accoun for frequency dependence. Loren Drude Model Requires a much more complicaed formulaion and implemenaion, bu i can accoun for loss and frequency dependence. Lecure 18 Slide 22 11

12 Mehod #1: Ereme Dielecric Consan Recall he updae equaions for he elecric field: E D E D E D y y yy In meals, he elecric field approaches ero. We can force his o happen by choosing dielecric consans ha are very large (i.e or higher). As, 0 E yy E y E As, 0 As, 0 Lecure 18 Slide 23 Mehod #2: Perfec Elecric Conducor Recall he updae equaions for he elecric fields: E m D E m D E m D E y Ey y E We can force he fields o ero by seing he updae coefficiens o 0 everywhere here is a meal and 1 everywhere ha here is no. Define he placemen of meals using hree arrays:, k, k, k y PEC PEC PEC We modify he updae coefficiens as follows: m m m PEC m, k, k, k E1 E1 PEC m, k, k, k Ey1 y Ey1 PEC m, k, k, k E1 E1 Lecure 18 Slide 24 12

13 Mehod #3: Conduciviy Recall from Lecure 10 ha we can incorporae maerial conduciviy ino Mawell s equaions as follows: H E E H E We incorporae meals by reaining his conduciviy erm, deriving new updae equaions, and revising he FDTD algorihm. Noe, you will end up creaing hree new maerials arrays:, yy, Lecure 18 Slide 25 Mehod #4: Loren Drude Model Recall from Lecure 10 ha we can wrie he consiuive relaion as M 2 fm D 0 E P P pe 2 2 m1 0, m m We can implemen his in he ime domain as follows Jm mjm 0, mpm fm pe Jm Pm 2 2 Lecure 18 Slide 26 13

14 Placing Meals on a 2D Grid E Mode For he E mode, he elecric field is always angenial o meal inerfaces and few problems arise when modeling meallic srucures. H Mode For he H mode, he elecric field can be polaried perpendicular o meal inerfaces. This is problemaic and i is bes o place meals wih he ouermos fields being angenial o he inerfaces. Bad placemen of meals Good placemen of meals E H Mode H E y y Lecure 18 Slide 27 Alernaive Grids Lecure 18 Slide 28 14

15 Drawbacks of Uniform Grids Uniform grids are he easies o implemen, bu do no conform well o arbirary srucures and ehibi high anisoropic dispersion. Anisoropic Dispersion (see Lecure 10) Saircase Approimaion (see Lecure 18) Lecure 18 Slide 29 Heagonal Grids Heagonal grids are good for minimiing anisoropic dispersion suffered on Caresian grids. This is very useful when eracing phase informaion. Phase Velociy as a Funcion of Propagaion Angle Yee FDTD 10 He FDTD See Te, pp Lecure 18 Slide 30 15

16 Nonuniform Orhogonal Grids (1 of 2) Nonuniform orhogonal grids are sill relaively simple o implemen and provide some abiliy o refine he grid a localied regions. See Te, pp Lecure 18 Slide 31 Nonuniform Orhogonal Grids (2 of 2) Uniform Grid Simulaion cells 140,800 cells Nonuniform Grid Simulaion cells 77,824 cells Conclusion: Roughly 50% memory and ime savings. Lecure 18 Slide 32 16

17 Curvilinear Coordinaes Mawell s equaions can be ransformed from curvilinear coordinaes o Caresian coordinaes o conform o curved boundaries of a device. See Te, pp M. Fusco, FDTD Algorihm in Curvilinear Coordinaes, IEEE Trans. An. and Prop., vol. 38, no. 1, pp , Lecure 18 Slide 33 Srucured Nonorhogonal Grids This is a paricularly powerful approach for simulaing periodic srucures wih oblique symmeries. M. Fusco, FDTD Algorihm in Curvilinear Coordinaes, IEEE Trans. An. and Prop., vol. 38, no. 1, pp , Lecure 18 Slide 34 17

18 Irregular Nonorhogonal Unsrucured Grids Unsrucured grids are more edious o implemen, bu can conform o highly comple shapes while mainaining good cell aspec raios and global uniformiy. Comparison of convergence raes ln P. Harms, J. Lee, R. Mira, A Sudy of he Nonorhogonal FDTD Mehod Versus he Convenional FDTD Technique for Compuing Lecure 18 Resonan Frequencies of Cylindrical Caviies, IEEE Trans. Microwave Theory and Techniq., vol. 40, no. 4, pp , Slide 35 Bodies of Revoluion (Cylindrical Symmery) Three dimensional devices wih cylindrical symmery can be very efficienly modeled using cylindrical coordinaes. Devices wih cylindrical symmery have fields ha are periodic around heir ais. Therefore, he fields can be epanded ino a Fourier series in. even m0 E e m e m,,, cos, sin H h m h m,,, cos, sin m0 even odd odd Due o a singulariy a r=0, updae equaions for fields on he ais are derived differenly. See Te, Chaper 12 Lecure 18 Slide 36 18

19 Some Devices wih Cylindrical Symmery Ben Waveguides Cylindrical Waveguides Dipole Anennas Conical Horn Anenna Diffracive Lenses Focusing Anennas Lecure 18 Slide 37 19