EE 5337 Computational Electromagnetics (CEM) Method of Lines

Transcription

1 11/30/017 Instucto D. Ramon Rumpf (915) Computational lctomagntics (CM) Lctu #4 Mto of Lins Lctu 4 Ts nots ma contain copigt matial obtain un fai us uls. Distibution of ts matials is stictl poibit Sli 1 Outlin Fomulation of 3D mto of lins Fomulation of D mto of lins Multila ics using scatting matics Compaison to RCWA Lctu 4 Sli 1

2 11/30/017 Dfinition of Mto of Lins T mto of lins as lop b matmaticians to sol patial iffntial quations (PDs). All but on inpnnt aiabls a iscti. Tis las to a lag st of coupl oina iffntial quations (ODs). T sstm of ODs a sol analticall. In lctomagntics, t inpnnt aiabls a usuall,, an. W tpicall iscti an an la analtical. An mto can b us to iscti t inpnnt aiabls. Tis inclus Foui tansfom, finit iffncs, finit lmnts, tc. In lctomagntics, t mto of lins implis tat finit iffncs a us to iscti an. RCWA is t mto of lins, but uss a Foui tansfom insta of finit iffncs to iscti an. Lctu 4 Sli 3 Sign Contion Tis fomulation of t mto of lins uss t folloing sign conntion fo as talling in t + iction. jk Lctu 4 Sli 4

3 11/30/017 Fomulation of 3D Mto of Lins Lctu 4 Sli 5 T D Unit Cll, Tis stuctu is unifom in t iction. Lctu 4 Sli 6 3

4 11/30/017 4 Lctu 4 Sli 7 Stating Point fo MOL W stat it Mall s quations in t folloing fom Rcall tat nomali t magntic fil an gi accoing to 0 0 j k k k Lctu 4 Sli 8 Unifom Mia W a going to consi Mall s quations insi a mia tat is unifom in t iction. T mia ma b inomognous in t plan, but it must b unifom in t iction.

5 11/30/017 Smi Analtical Mati Fom of Mall s quations W can go staigt to mati fom using t concpt of mati iati opatos. W kp t iction analtical. D μ D μ D D μ D ε D ε D D ε Lctu 4 Sli 9 liminat Longituinal Fil Componnts W sol t ti an sit quation fo an an substitut ts back into t maining fou quations. D μ D μ D D μ μ D D 1 D ε D μ D ε D μ D ε D D ε D D ε D ε D D ε ε D D 1 D μ D ε D μ D ε D μ D D μ D Lctu 4 Sli 10 5

6 11/30/017 Block Mati Fom W it t maining fou quations in block mati fom as μ D ε D D ε D D ε D μ D ε D μ DεD DεD Dε D μ Dε D D μ D ε D μ D ε D μ D D μ D Dμ D ε Dμ D 1 1 ε DμD DμD Lctu 4 Sli 11 Stana PQ Fom W can no it ou to quations in t stana P an Q fom. P P Dε D μ Dε D 1 1 μ D ε D D ε D 1 1 Q Dμ D ε Dμ D Q 1 1 ε Dμ D Dμ D 1 1 Lctu 4 Sli 1 6

7 11/30/017 Mati Wa quation W iffntiat t P quation it spct to an substitut t Q quation into t sult to i t mati a quation. P P PQ 0 PQ 0 Ω Q Ω 0 PQ Lctu 4 Sli 13 Solution to t Wa quation T solution to t a quation is ittn t sam as in RCWA an TMM. W c W c λ λ W an λ a t ign-ctos an ign-alus of Ω T oall solution is tn ψ λ W W 0 c λ V V 0 c V QWλ 1 Bas on pious lctus, ou kno t st of tis sto! Lctu 4 Sli 14 7

8 11/30/017 Intptation of t Solution λ ψ W c () Oall solution ic is t sum of all t mos at plan. c Column cto containing t amplitu cofficint of ac of t mos. Tis quantitis o muc ng is in ac mo. W Squa mati o s column ctos scib t mos tat can ist in t matial. Ts a ssntiall pictus of t mos ic quantif t lati amplitus of,,, an. Diagonal mati scibing o t mos popagat. Tis inclus accumulation of pas as ll as caing (loss) o going (gain) amplitu. Lctu 4 Sli 15 Solution in a omognous La In a omognous la, t P an Q matics a 1 D D ID D P IDD DD 1 DD IDD Q IDD DD Lctu 4 Sli 16 8

9 11/30/017 Visualiation of tis Solution Mos 1 c 1 1 c 3 c c c c 1 1 c 3 c c c 5 5 Lctu 4 Sli 17 Fomulation of D Mto of Lins Lctu 4 Sli 18 9

10 11/30/ Lctu 4 Sli 19 Stating Point D ε D ε D D ε D μ D μ D D μ W ill stat it Mall s quations in al spac an in mati fom as i ali tis lctu. Lctu 4 Sli 0 Ruction to To Dimnsions ε D ε D ε μ D μ D μ Fo iagonall anisotopic ics tat a unifom in t iction an n t is no a popagation in t iction, a D D 0 Mall s quations uc to

11 11/30/017 To Inpnnt Mos Mall s quations a coupl into to inpnnt mos. μ D μ D μ ε D ε D ε Mo D ε μ D μ Mo D μ ε D ε Lctu 4 Sli 1 Appl Dilctic Smooting To impo congnc, can incopoat ilctic smooting as follos: Mo D ε μ D μ 1 1 Mo D μ ε D ε 1 1 Lctu 4 Sli 11

12 11/30/017 liminat Longituinal Componnts W sol fo t longituinal componnts μ D 1 1 ε D an substitut ts pssions into t maining quations. Mo ε D μ D 1 1 μ 1 Mo μ D ε D 1 1 ε 1 Lctu 4 Sli 3 Stana P an Q Fom W it ou mati quations in t stana PQ fom. P Q Mo P Mo Q P μ Q ε D μ D P ε Q μ D ε D an no ou kno t st of t sto. Lctu 4 Sli 4 1

13 11/30/017 Multila Dics Using Scatting Matics R. C. Rumpf, "Impo fomulation of scatting matics fo smi analtical mtos tat is consistnt it conntion," PIRS B, Vol. 35, 41 61, 011. Lctu 4 Sli 5 ign Sstm in ac La Bouna conitions qui tat all las a t sam K an K matics. 0 BCs Dε D μ Dε D Dμ D ε Dμ D , 1, 1, 1, 1, 1, P1 Q , 1, 1, μ D ε D D ε D ε1, Dμ1, D Dμ1, D Ω1 PQ 1 1 W1, λ1 V1 c1, c1 Z 1 BCs Dε D μ Dε D Dμ D ε Dμ D ,,,,,, P Q ,,, μ D ε D D ε D ε, Dμ, D Dμ, D Ω PQ W, λ V c, c Z BCs Z 3 D ε D μ D ε D D μ D ε D μ D , 3, 3, 3, 3, 3, P3 Q , 3, 3, μ D ε D D ε D ε3, Dμ3, D Dμ3, D Ω3 PQ 3 3 W, λ V3 c3, c3 BCs Lctu 4 Sli 6 13

14 11/30/017 Fil Rlations & Bouna Conitions Fil insi t i t la: ψ i λi Wi Wi 0 c i λi Vi Vi 0 ci Bouna conitions at t fist intfac: ψ ψ 0 1 W1 W1c 1 Wi Wic i V1 V1c 1 Vi Vi ci i Bouna conitions at t scon intfac: ψ klψ i 0 i λikl 0 i Wi Wi 0 c i W Wc λikl 0 i Vi Vi 0 c i V V c Not: k 0 as bn incopoat to nomali L i. Lctu 4 Sli 7 Aopt t Smmtic S Mati Appoac T scatting mati S i of t i t la is still fin as: c 1 i c 1 S c c S i S S i i 11 S1 i i 1 S But t lmnts a calculat as i i i i i i i i i i i i i i i i i i i i i i i i i S A XB A XB XB A X A B S A XB A XB X A B A B S S i i 1 S1 i i S11 Las a smmtic so t scatting mati lmnts a unanc. Scatting mati quations a simplifi. F calculations. Lss mmo stoag. S i A W W V V 1 1 i i 0 i 0 B W W V V 1 1 i i 0 i 0 X i ikl 0 i λ Lctu 4 Sli 8 14

15 11/30/017 Global Scatting Mati Scatting mati fo all las. 0 BCs ic 1 3 S S S S Z 1 1 S BCs S Connction to outsi gions global f ic tn S S S S Z Z 3 BCs 3 S Rcall tis pocu fom Lctu 5. BCs Lctu 4 Sli 9 Rflction/Tansmission Si Scatting Matics T flction si scatting mati is f 1 11 f f f 1 1 Af S A B S f f f f f S A B A B f 1 f f S B A T tansmission si scatting mati is tn 1 11 tn tn S B A tn tn tn tn tn S A B A B S tn 1 1 Atn tn 1 tn tn S A B A W W V V 1 1 f 0 f 0 f B W W V V 1 1 f 0 f 0 f A W W V V 1 1 tn 0 tn 0 tn B W W V V 1 1 tn 0 tn 0 tn s f,i 0,I s tn 0 0 lim L0 lim L0 0,II,II Lctu 4 Sli 30 15

16 11/30/017 Calculating t Tansmitt an Rflct Fils T lctic fil souc is calculat as c W 1 inc inc f T inc p p jk jk,inc,inc Lctu 4 Sli 31 k k,inc,inc Gin t global scatting mati, t cofficints fo t flct an tansmitt fils a c S c c S c f 11 inc tn 1 inc an a column ctos containing t cooinats of t points in t coss sction. Rcall calculating t polaiation componnts p an p in TMM. p 1 T tanss componnts of t flct an tansmitt fils a tn f Wfcf Wf S11cinc tn t Wtnctn Wtn S1cinc t Calculat t Amplitus of t Spatial amonics Rmo t pas tilt,, p,, p,, p,, p A jk k,f,inc,inc A jk k,f,inc,inc A t jk k,tn,inc,inc A,tn t jk,inc k,inc Calculat amplitus of t spatial amonics FFT A, t t D,f FFT A, D,f FFT A, D,tn FFT A, D,tn Lctu 4 Sli 3 16

17 11/30/017 Calculating t Longituinal Componnts T longituinal fil componnts a calculat fom t tanss componnts using t ignc quation (s Lctu 4). K K K 1,f t K K t K t 1,tn Not, t K matics a not nomali. Lctu 4 Sli 33 Calculating t Diffaction fficincis T iffaction fficincis R an T a calculat as t t t t R K R R k R K T R k,f inc,tn inc,inc,inc,tn t,inc Rmmb tat ts quations assum t souc as gin unit amplitu. Not, t K matics a not nomali. Lctu 4 Sli 34 17

18 11/30/017 Compaison to RCWA Lctu 4 Sli 35 Dilctic Dic Compaison nf 1.0 t 134 nm 314 nm n.1 ntn 1.5 n.0 L 314 nm Lctu 4 Sli 36 18

19 11/30/017 Compaison of Mol Rsults RCWA 99 spatial amonics.4 minut simulation tim Not noug alngt points. Numical glitcs MOL 1919 gi points (/10) 3. ou simulation tim Lctu 4 Sli 37 19