Finite Difference Time Domain

Transcription

1 11/9/ lecromagneic Analsis Using Finie Difference Time Domain Lecure #3 Finie Difference Time Domain Lecure 3 These noes ma conain coprighed maerial obained under fair use rules. Disribuion of hese maerials is sricl prohibied Slide 1 Lecure Ouline Two ommon Grid Sraegies Simplifing he updae equaions for crossed graing devices Boundar condiions Plane wave source using he oal field/scaered field echnique alculaion of ransmiance and reflecance MATLAB s slice() command for 3D visualiaion Lecure 3 Slide 1

2 11/9/016 Two ommon Grid Sraegies Lecure 3 Slide 3 Grid Scheme for Finie Sie Devices To simulae scaering from finie sied devices, a PML is needed a all boundaries. lo lo hi lo hi hi Lecure 3 Slide 4

3 11/9/016 Grid Scheme for rossed Graing Devices For crossed graing devices, a PML is onl needed a he ais boundaries. Reflecion Plane TF/SF Planes Spacer Region Uni cell of real device model valid in here Spacer Region Transmission Plane FDTD grid real device Lecure 3 Slide 5 Simplified Updae quaions for rossed Graing Devices Lecure 3 Slide 6 3

4 11/9/016 Simplif, simplif, simplif! A 3D grid conains man more poins han an equivalen D grid. For his reason, we will devoe more aenion o simplifing and sreamlining he updae equaions. D Grid cells = 8,000 cells 3D Grid = 30,000 cells We now have 40 more grid cells more field componens 80 more numbers o process Lecure 3 Slide 7 Imporan Iems o Simplif and Sreamline We do no need ais or ais PMLs. The should be eliminaed from he updae equaions. Inegraion erms are onl needed inside he PML. These should be spli ino low and high inegraion erms. Lecure 3 Slide 8 4

5 11/9/016 Simplif Updae quaion for m 0 m The following erms are eliminaed from he updae coefficiens: m 1 1 m c0 1 c0 1 m 3 m 4,, m0 m i j 0 0 m 0 0 For, we no longer need an inegraion erms! I T 0 I T T 0 i, j1, 1 The hird and fourh erms are eliminaed from he updae equaion. i, j, i j 1 3 m4 m m m I,, I Lecure 3 Slide 9 Final Form of he Updae quaion for The updae coefficiens are compued before he main FDTD loop. m m m c0 0 1,, i j 0 m 0 0 m0 The curl is compued inside he main FDTD loop, bu before he updae equaion. i, j1, 1 The updae equaion is compued inside he main FDTD loop immediael afer he curl erm is calculaed.,,,,,, 1 i j i j i j m m Lecure 3 Slide 10 5

6 11/9/016 Final Form of he Updae quaion for The updae coefficiens are compued before he main FDTD loop. m m m c0 0 1,, i j 0 m 0 0 m0 The curl erm is compued inside he main FDTD loop, bu before he updae equaion. 1 i1, j, The updae equaion is compued inside he main FDTD loop immediael afer he curl erm is calculaed. 1 m m Lecure 3 Slide 11 Final Form of he Updae quaion for The updae coefficiens are compued before he main FDTD loop. m c0 c0 m 3 0 The inegraion erm and curl are compued inside he main FDTD loop, bu before he updae equaion. i1, j, i, j1,,, i j T 0 I The updae equaion is compued inside he main FDTD loop immediael afer he inegraion erm and curl erm are updaed. i j i j m 3 i, j,,,,, m I Lecure 3 Slide 1 6

7 11/9/016 Final Form of he Updae quaion for D The updae coefficiens are compued before he main FDTD loop. m m m D D c0 D0 D1 D 0 m D0 0 md0 The curl erm is compued inside he main FDTD loop, bu before he updae equaion. i, j1, 1,, i j The updae equaion is compued inside he main FDTD loop immediael afer he curl erm is calculaed. D,, i j,, 1 D D i j,, i j m D m Lecure 3 Slide 13 Final Form of he Updae quaion for D The updae coefficiens are compued before he main FDTD loop. m m m D D c0 D0 D1 D 0 m 0 0 D md0 The inegraion erms are compued inside he main FDTD loop, bu before he updae equaion. 1 i1, j, The updae equaion is compued inside he main FDTD loop immediael afer he inegraion erms are updaed. D i j D1 D,, m D m Lecure 3 Slide 14 7

8 11/9/016 Final Form of he Updae quaion for D The updae coefficiens are compued before he main FDTD loop.,, c m c m 0 D i j D 0 D3 0 The inegraion erm and curl erm are compued inside he main FDTD loop, bu before he updae equaion. i1, j, i, j1,,,,,,, i j i j i j T T 0 I The updae equaion is compued inside he main FDTD loop immediael afer he inegraion and curl erms are updaed. i j 3,, D D m D m D I Lecure 3 Slide 15 Final Updae quaions for,, and The updae coefficiens are compued before he main FDTD loop m m m 1,, 1 i j 1 The updae equaions are compued inside he main FDTD loop. m m m 1 D D 1 1 D Lecure 3 Slide 16 8

9 11/9/016 Boundar ondiions Lecure 3 Slide 17 Periodic Boundar ondiions All of he spaial derivaives appear in he curl calculaions. Therefore, all boundar condiions are implemened in he curl calculaions as well. We will implemen periodic boundar condiions all he wa around he grid. Lecure 3 Slide 18 9

10 11/9/016 url of The curl equaions are i, j1, 1 i, j, 1 i, j, i1, j, i, j, i1, j, i, j, i, j1, i, j, We see ha problems occur when calculaing hese a he high, high, and high boundaries. Lecure 3 Slide 19 url of The curl equaions are i, j1, 1 i, j, i, j, 1 i, j, i1, j, i1, j, i, j, i, j1, We see ha problems occur when calculaing hese a he low, low, and low boundaries. Lecure 3 Slide 0 10

11 11/9/016 omponen of url of wih PB There are four cases where he curl mus be compued eplicil. i, j1, 1 for j N and N i,1, in,, in,, 1 in,, for j N and N i, j1, N N 1 N for j N and N i,1, N i, N, N i, N,1 i, N, N for j N and N Lecure 3 Slide 1 MATLAB ode for omponen of url of wih PB % ompue for n = 1 : N for n = 1 : N-1 for n = 1 : N-1 (n,n,n) = ((n,n+1,n) - (n,n,n))/d... - ((n,n,n+1) - (n,n,n))/d; end (n,n,n) = ((n,n+1,n) - (n,n,n))/d... - ((n,n,1) - (n,n,n))/d; end for n = 1 : N-1 (n,n,n) = ((n,1,n) - (n,n,n))/d... - ((n,n,n+1) - (n,n,n))/d; end (n,n,n) = ((n,1,n) - (n,n,n))/d... - ((n,n,1) - (n,n,n))/d; end Noe: I imporan for speed and efficienc ha he boundar condiions be incorporaed wihou using if/hen saemens. This should be done eplicil. Lecure 3 Slide 11

12 11/9/016 omponen of url of wih PB There are four cases where he curl mus be compued eplicil. i, j1, 1 i,1, i, N,,1,,1, 1 i i, 1 i, j1,1 i j,1 N for j 1 and 1 for j 1 and 1 for j 1 and 1 i,1,1 i, N,1 i,1,1 i,1, N for j 1 and 1 Lecure 3 Slide 3 MATLAB ode for omponen of url of wih PB % ompue for n = 1 : N (n,1,1) = ((n,1,1) - (n,n,1))/d... - ((n,1,1) - (n,1,n))/d; for n = : N (n,1,n) = ((n,1,n) - (n,n,n))/d... - ((n,1,n) - (n,1,n-1))/d; end for n = : N (n,n,1) = ((n,n,1) - (n,n-1,1))/d... - ((n,n,1) - (n,n,n))/d; for n = : N (n,n,n) = ((n,n,n) - (n,n-1,n))/d... - ((n,n,n) - (n,n,n-1))/d; end end end Noe: I imporan for speed and efficienc ha he boundar condiions be incorporaed wihou using if/hen saemens. This should be done eplicil. Lecure 3 Slide 4 1

13 11/9/016 Plane Wave Source Using he Toal Field/Scaered Field Mehod Lecure 3 Slide 5 TF/SF Framewor Reflecion Plane TF/SF Planes Spacer Region Uni cell of real device Spacer Region Scaered Field Toal Field scaered field oal field Transmission Plane Lecure 3 Slide 6 13

14 11/9/016 orrecions Scaered Field Reflecion Plane TF/SF Planes ver cell along he TF/SF inerface requires a correcion erm. Spacer Region Uni cell of real device Toal Field Spacer Region Transmission Plane Lecure 3 Slide 7 orrecions on Scaered Field Side (1 of ) I is he curl erms ha require correcions on he scaered field side. ere, he source (i.e. correcions) are incorporaed hrough he and field componens. i, j1, i, j, 1 1 i1, j, i1, j, i, j, i, j1, i, j, i, j, Lecure 3 Slide 8 14

15 11/9/016 orrecions on Scaered Field Side ( of ) i, j1, 1 1 i, j, 1 i, j, i1, j, i, j, Sandard curl equaions % TF/SF orrecion (:,:,n_-1) = (:,:,n_-1) + _(T)/d; (:,:,n_-1) = (:,:,n_-1) - _(T)/d; orrecions for TS/SF Lecure 3 Slide 9 orrecions on Toal Field Side (1 of ) I is he curl erms ha require correcions on he oal field side. ere, he source (i.e. correcions) are incorporaed hrough he and field componens. i, j1, 1 1 i1, j, i1, j, i, j, i, j1, Lecure 3 Slide 30 15

16 11/9/016 orrecions on Toal Field Side ( of ) % TF/SF orrecion (:,:,n_) = (:,:,n_) + _(T)/d; (:,:,n_) = (:,:,n_) - _(T)/d; orrecions for TS/SF Sandard curl equaions i, j1, 1 i, j, 1 i, j, i, j, 1 i, j, i1, j, Lecure 3 Slide 31 Source Funcions We Need for TF/SF Reflecion Plane TF/SF Planes Spacer Region Uni cell of real device Spacer Region Scaered Field Toal Field 1 1 Transmission Plane Lecure 3 Slide 3 16

17 11/9/016 Polariaion Vecor The elecric field can be polaried in an direcion in he plane. Le he inciden elecric field be epressed as Pcos 0ninc P cos n 0 inc P cos n 0 inc Noe: P 1 Noe ha he P is no needed in his formulaion. Wh isn his informaion needed? Lecure 3 Slide 33 The Magneic Field The magneic field is relaed o he elecric field hrough Mawell s curl equaion. inc c 0 Subsiuing he plane wave form of he elecric field leads o an epression for he magneic field source. P n P n r,inc r,inc cos cos 0 inc 0 inc r,inc r,inc 1 r,inc P cos 0 inc 1 n r,inc 1 r,inc P cos 0 inc 1 n r,inc Lecure 3 Slide 34 17

18 11/9/016 Generaliaion for a Gaussian Source A Gaussian source is implemened as P g P g 1 r,inc P g r,inc 1 r,inc P g r,inc ninc c 0 Lecure 3 Slide 35 Tpical View of 3D FDTD wih TF/SF Gaussian Pulse Lecure 3 Slide 36 18

19 11/9/016 alculaion of Transmiance and Reflecance Reflecance recorded here Transmiance recorded here Lecure 3 Slide 37 Bloc Diagram of Power alculaion alculae Sead Sae and During FDTD Simulaion alculae Transverse Wave Vecor pansion Frequenc alculae Longiudinal Wave Vecors Inerpolae Fields a Origin Normalie o Source alculae Ampliudes of Spaial armonics alculae S alculae Diffracion fficiencies alculae Overall Reflecance and Transmiance Lecure 3 Slide 38 19

20 11/9/016 alculae of Sead Sae Fields During Simulaion,ref,ref Reflecion Plane TF/SF Planes Spacer Region Uni cell of real device Spacer Region Noe: We do no calculae. We will do his anoher wa afer FDTD.,rn,rn Transmission Plane Lecure 3 Slide 39 Inerpolae Field omponens The fields are saggered because he are on a Yee grid. To calculae ransmiance and reflecance in 3D, we need o inerpolae he fields a a common poin in each Yee cell in order o perform calculaions wih vecor quaniies. ere, we will inerpolae hem a he origin of he Yee cell. % Inerpolae Fields a Origin r = eros(n,n); r(1,:) = (ref(n,:,nfreq) + ref(1,:,nfreq))/; r(:n,:) = (ref(1:n-1,:,nfreq) + ref(:n,:,nfreq))/;,ref,ref,rn,rn ref i1, j, ref origin, ref,ref,ref ref i, j1, ref origin, ref,ref,ref rn i1, j, rn origin, rn,rn,rn rn i, j1, rn origin, rn,rn,rn Lecure 3 Slide 40 0

21 11/9/016 Transverse Wave Vecor pansion (1 of ) rossed graing devices diffracion along wo dimensions, and. To quanif diffracion for crossed graing srucures, we mus calculae an epansion for boh and. m m,inc m,,, 1,0,1,, n n,inc n,,, 1, 0,1,, m n m n ˆ, ˆ % TRANSVRS WAV VTOR XPANSION M = [-floor(n/):floor(n/)]'; N = [-floor(n/):floor(n/)]'; = - *pi*m/s; = - *pi*n/s; [,] = meshgrid(,); Lecure 3 Slide 41 Transverse Wave Vecor pansion ( of ) The vecor epansions can be visualied his wa. m n m n m n ˆ, ˆ Lecure 3 Slide 4 1

22 11/9/016 Longiudinal Wave Vecor pansion (1 of ) The longiudinal componens of he wave vecors are compued as, m n n m n,ref 0 ref, m n n m n,rn 0 rn % ompue Longiudinal Wave Vecor omponens 0 = *pi/lam; inc = 0*nref; R = sqr((0*nref)^ -.^ -.^); T = sqr((0*nrn)^ -.^ -.^); The cener few modes will have real s. These correspond o propagaing waves. The ohers will have imaginar s and correspond o evanescen waves ha do no ranspor energ. m, n m, n Lecure 3 Slide 43 Longiudinal Wave Vecor pansion ( of ) The overall wave vecor epansion can be visualied his wa m, n,ref m, n m n ref, m n rn, Lecure 3 Slide 44

23 11/9/016 Normalie o Source When a Gaussian source is used, he energ decreases a high frequencies. If nohing is done, lower energ observed a he high frequencies will be confused wih low reflecance or ransmiance. The soluion is o normalie he sead sae fields o he source.,ref,ref,rn,rn f f f f,ref,,ref,,rn,,rn, f f f f f f f f % Normalie o Source r = r / SR(nfreq); r = r / SR(nfreq); = / SR(nfreq); = / SR(nfreq); Lecure 3 Slide 45 alculae Spaial armonics We calculae he and ampliude coefficiens of he spaial harmonics b Fourier ransforming he sead sae and. % alculae Ampliude omponens of Spaial armonics Sr = ffshif(ff(r))/(n*n); Sr = ffshif(ff(r))/(n*n); S = ffshif(ff())/(n*n); S = ffshif(ff())/(n*n); Refleced spaial harmonics Transmied spaial harmonics Lecure 3 Slide 46 3

24 11/9/016 alculae S I is no necessar o calculae he sead sae field. We can calculae S from S and S using he divergence equaion. r m, n 0 m, n 0 jm j n jm, n Sm, ne e e 0 j,,,,,, m j n j m n j m j n j m n j m j n j m n S m n e e e S m n e e e S m n e e e 0 msm, nnsm, nm, nsm, n0 S m, n,, m, n m S m n n S m n % alculae Longiudinal omponens Sr = -(.*Sr +.*Sr)./R; S = -(.*S +.*S)./T; Lecure 3 Slide 47 alculae Diffracion fficiencies The diffracion efficiencies are calculaed from he wave vecor componens and he ampliudes of he spaial harmonics. D ref f; m, n S f; m, n Re D rn f; m, n S f; m, n Re,ref ref,inc f,rn r,ref rn,inc f r,rn f; m, n f; m, n % alculae Ampliude of Spaial armonics Sref = abs(sr).^ + abs(sr).^ + abs(sr).^; Srn = abs(s).^ + abs(s).^ + abs(s).^; % alculae Diffracion fficiencies ref = real(r/inc).* Sref; rn = real(t*urref/inc/urrn).* Srn; Lecure 3 Slide 48 4

25 11/9/016 alculae Reflecance and Transmiance The overall reflecance is he sum of all he diffracion efficiencies of he refleced modes. Similarl, he overall ransmiance is he sum of all he diffracion efficiencies of he ransmied modes. D ref f; m, n R f n m D rn ;, T f f m n n m % ompue Reflecance and Transmiance RF(nfreq) = sum(ref(:)); TRN(nfreq) = sum(rn(:)); I is alwas good pracice o chec for conservaion of energ. When no gain or loss is incorporaed ino he model, he sum of he reflecance and ransmiance should be 100%. 100% R f T f % ALULAT ONSRVATION OF NRGY ON(nfreq) = RF(nfreq) + TRN(nfreq); Lecure 3 Slide 49 MATLAB s slice() ommand for 3D Visualiaion Lecure 3 Slide 50 5

26 11/9/016 MATLAB s slice() ommand slice(x,y,z,v,s,s,s) X, Y, and Z are 3D arras generaed using meshgrid(). V is he 3D arra of daa o visualie. s is a 1D arra of numbers of where o place slices ha are perpendicular o he ais. s is a 1D arra of numbers of where o place slices ha are perpendicular o he ais. s is a 1D arra of numbers of where o place slices ha are perpendicular o he ais. Lecure 3 Slide 51 Visualiing a 3D Daa Arra % Draw Field slice(y,x,-z,,0,0,0); ais equal igh off; colorbar; view(-75,0); Lecure 3 Slide 5 6