Formulation of Rigorous Coupled Wave Analysis

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1/8/18 Instucto D. Ramon Rumpf (915) 747 6958 cumpf@utep.

1 1/8/18 Instucto D. Ramon Rumpf (915) EE 5337 Computational Electomagnetics Lectue #1 Fomulation of Rigoous Couple Wave Analsis Lectue 1 These notes ma contain copighte mateial obtaine une fai use ules. Distibution of these mateials is stictl pohibite Slie 1 Outline Backgoun Semi analtical fom of Mawell s equations in Fouie space Mati fom of Mawell s equations Mati wave equation Solution to the mati wave equation Multilae famewok: scatteing matices Calculate tansmission an eflection TMMPWEM RCWA Lectue 1 Slie 1

Galo Altenate Names fo the Metho Rigoous couple wave analsis Fouie

2 1/8/18 Backgoun Lectue 1 Slie 3 Rigoous Couple Wave Analsis Develope in 198 s D. M. G. Jim Mohaam D. Thomas K. Galo Altenate Names fo the Metho Rigoous couple wave analsis Fouie moal metho Tansfe mati metho with a plane wave basis D. M. G. Jim Mohaam D. Thomas K. Galo Lectue 1 Slie 4

3 1/8/18 Geomet of RCWA P Lectue 1 Slie 5 The D Unit Cell fo Single Lae homogeneous, All laes must be unifom in the iection. In ou coe, we just nee to escibe the coss section of the unit cell. Lectue 1 Slie 6 3

4 1/8/18 Sign Convention We will aopt the following sign convention fo a wave tavelling in the + iection. e jk Lectue 1 Slie 7 Semi Analtical Fom of Mawell s Equations in Fouie Space Lectue 1 Slie 8 4

5 1/8/18 5 Lectue 1 Slie 9 Stating Point fo RCWA H H k E H H k E H H k E E E k H E E k H E E k H We stat with Mawell s equations in the following fom Recall that we nomalie the magnetic fiel accoing to H j H Lectue 1 Slie 1 Unifom Meia We ae going to consie Mawell s equation insie a meium that is unifom in the iection. The meium ma still be inhomogeneous in the plane, but it must be unifom in the iection. homogeneous

6 1/8/18 Fouie Tansfom in an Onl Unlike PWEM, RCWA onl Fouie tansfoms along an. The paamete emains analtical an unchange. The Fouie epansion of the mateials in the plane ae, a e m n m, n m n j m n j, 1 amn, e, b e m n m, n m n j m n j, 1 bmn, e It follows that the Fouie epansion of the fiels ae j kmkn E,, Sm, n; e m n j kmkn E,, Sm, n; e m n j kmkn E,, Sm, n; e m n jkmkn H,, Um, n; e m n jkmkn H,, Um, n; e m n jkmkn H,, Um, n; e m n Lectue 1 Slie 11 Wave Vecto Components The tansvese components of the wave vectos ae equal thoughout all laes of the evice. m kmk,inc m,,, 1,,1,,, n knk,inc n,,, 1,,1,,, We nee the longituinal components of the wave vectos fo: (1) calculating iffaction efficiencies () calculating the eigen moes of a homogeneous lae analticall * * *, k m n k k m k n Fo genealie smmet, the tansvese components ae epane along the ecipocal lattice vectos T 1 an T. k m, n k mt nt T T,inc 1 These ae calculate fom the ispesion elation in the meium of inteest. The conjugate opeations enfoce ou negative sign convention. Lectue 1 Slie 1 6

7 1/8/18 Substitute Epansions into Mawell s Equations j kmkn E,, Sm, n; e m n, b e m n m, n m n j j kmkn E,, Sm, n; e m n jkmkn H,, Um, n; e m n E E k H m n j jkm kn jkm kn jkm kn Smn, ; e Smn, ; e k bm, ne Umn, ; e m n m n m n m n mq n, ; j j k m k n S m n jkmkn jkq k jkmsm, n; e e k bm q, n e U q, ; e m n m n m n q mq n, ; j j k m k n S m n j kmkn j kqk jk msm, n; e e k bmq, ne U q, ; e m n q S m, n; The eivative is oina jkmsm, n; k bmq, nuq, ; because is the onl q inepenent vaiable left. Lectue 1 Slie 13 Semi Analtical Fom of Mawell s Equations in Fouie Space If we o this fo all of Mawell s equations, we get Real Space Semi Analtical Fouie Space H H k E H H k E H H k E U m, n; jk n U m, n; k a S q, ; U mq, n q m, n; m q n q, ;,, ; jk m U m n k a S q, ;, ;,, ; jk mu mn jk nu mn k a S q m q n q E E kh E E kh E E kh S m, n; jk n S m, n; k b U q, ; S mq, n q m, n;, ;,, ; jk m S m n k b U q m q n q, ;, ;,, ; jk m S m n jk n S m n k b U q m q n q Note: U(m,n;) an S(m,n;) ae functions of.,, a, an b ae not. Lectue 1 Slie 14 7

8 1/8/18 Mati Fom of Mawell s Equations Lectue 1 Slie 15 Nomalie the Fouie Space Equations Define nomalie wave vectos. k k k k k k k k Nomalie the cooinate. k k Lectue 1 Slie 16 U m, n; jk n U m, n; a S q, ; U mq, n q m, n;, ;,, ; jk m U m n a S q m q n q, ;, ;,, ; jk m U m n jk n U m n a S q m q n q S m, n; jk n S m, n; b U q, ; S mq, n q m, n;, ;,, ; jk m S m n b U q m q n q, ;, ;,, ; jk m S m n jk n S m n b U q m q n q 8

9 1/8/18 Mati Fom of Mawell s Equations (1 of ) Stat with the fist equation. U m, n; jk n U m, n; a S q, ; mq, n q This equation is witten once fo eve combination of m an n. This lage set of equations can be witten in mati fom as jku u s U1,1 U 1,1 S1,1 U1, U 1, S1, u u s Toeplit convolution UM, N U M, N SM, N mati Note: onl tul Toeplit smmet fo 1D gatings. Lectue 1 Slie 17 k 1,1 k 1, K k M, N Mati Fom of Mawell s Equations ( of ) U m, n; jk num, n; M M N am N Sq, ;, q n q U m, n; jk mum, n; M N a Sq, ;, m q n q M N N jk mum, n; jk num, n; am Sq, ;, q n M q M N jku u s u jk u s Ku Ku j s M M N N, m q n q S m, n; jk n S m, n; k b U q, ; M N S m, n; jk m S m n k b U q, ;,, ; m q n q M N N, ;, ;,, ; jk m S m n jk n S m n k b U q m q n qm N M jks s u s jk su Ks Ks j u Lectue 1 Slie 18 9

10 1/8/18 Mati Wave Equation Lectue 1 Slie 19 Solve fo Longituinal Fiel Components We wish to eliminate the longituinal fiel components s an u. We stat b solving the thi an sith equation fo these tems. jku u s u jk u s Ku Ku j s 1 s j K u K u jks s u s jk su Ks Ks j u 1 u j K s K s Lectue 1 Slie 1

11 1/8/18 Eliminate Longituinal Fiel Components We substitute s an u back into the emaining fou equations. jku u s u jk u s 1 s j K u K u 1 K K s K s u s 1 u K K s K s s jks s u s jk s u 1 u j K s K s 1 K K u K u s u 1 s K K u K u u Lectue 1 Slie 1 Reaange the Tems Net, we epan the equations an eaange the tems. 1 K K s K s u s 1 u K K s K s s u K K s K K s u K K s K K s K K u K u s u 1 s K K u K u u s K K u K K u s K K u K K u Lectue 1 Slie 11

12 1/8/18 Block Mati Fom Just as we i fo the tansfe mati metho using scatteing matices, we wite ou mati equations in block mati fom. u K K s K K s u K K s K K s u s Q u s K K K K Q K K K K s K K u K K u s K K u K K u s u P s u K K K K P K K K K Lectue 1 Slie 3 P an Q in Homogeneous Laes When a lae is homogeneous, the P an Q matices euce to 1 KK I K P K I K K 1 KK I K Q K I K K P Notice that these matices o not contain computationall intensive convolution matices. Theefoe, the ae ve fast an efficient to calculate fo this special case. Lectue 1 Slie 4 1

13 1/8/18 Mati Wave Equation Fom hee, we can eive a wave equation just as we i fo TMM. s u P s u Eq. (1) Fist, iffeentiate Eq. (1) with espect to. u s Q u s Eq. () Secon, substitute Eq. () into Eq. (3) to eliminate the magnetic fiels. P s s u u s s PQ s s Eq. (3) Thi, the final mati wave equation is s s Ω Ω PQ s s We aive at ou stana PQ fom! Lectue 1 Slie 5 Solution to the Mati Wave Equation Lectue 1 Slie 6 13

14 1/8/18 Analtical Solution in the Diection The mati wave equation is s s Ω s s This is eall a lage set of oina iffeential equations that can each be solve analticall. This set of solutions is s Ω e Ω s e s s The tems s an s ae the initial values fo this iffeential equation. The ± supescipts inicate whethe the petain to fowa popagating waves (+) o backwa popagating waves ( ). Lectue 1 Slie 7 Computation of e ± Recall fom Lectue 4 f 1 AW f λw A Abita squae mati (full ank) W Eigen-vecto mati calculate fom A λ Diagonal eigen-value mati calculate fom A We can use this elation to compute the mati eponentials. e We W e We W Ω λ 1 Ω λ 1 W Eigen-vecto mati of Ω λ Eigen-value mati of Ω e λ e 1 e e N Lectue 1 Slie 8 14

15 1/8/18 Revise Solution We stat with the following solution. s Ω e Ω s e s s Substituting Eq. () into Eq. (1) iels s λ 1 1 e λ W W s We W s s c Ω 1 e Wep λ W Eq. (1) Eq. () Ω 1 e Wep λ W c The tems s an s ae initial values that have et to be calculate. Theefoe W 1 can be combine with these tems to pouce column vectos of popotionalit constants c + an c -. s We c We c s λ λ c c W s W s 1 1 Lectue 1 Slie 9 Solution fo the Magnetic Fiels (1 of ) We can similal wite a solution fo the magnetic fiels. u Ve c Ve c u λ λ We nee to calculate V fom the eigen value solution of. To put this equation in tems of the electic fiel, we iffeentiate with espect to. u λ e λ Vλ c Vλe c u The negative sign is neee so both tems will be positive afte iffeentiation. Lectue 1 Slie 3 15

16 1/8/18 Solution fo the Magnetic Fiels ( of ) Recall, u s Q u s s We c We c s λ λ u Vλe c Vλe c u λ λ Eq. (1) Eq. () Eq. (3) Substitute Eq. () into Eq. (1). u QWe c QWe c u λ λ Compae this epession to Eq. (3). Vλ QW 1 V QWλ Lectue 1 Slie 31 Oveall Fiel Solution The fiel solutions fo both the electic an magnetic fiels wee s We c We c s λ λ u Ve c Ve c u λ λ Combining these into a single mati equation iels ψ s u λ s W We c λ u V V e c whee V QWλ 1 Lectue 1 Slie 3 16

17 1/8/18 Intepetation of the Solution () Oveall solution which is the sum of all the moes at plane. ψ We λ c c Column vecto containing the amplitue coefficient of each of the moes. This quantities how much eneg is in each moe. W Squae mati who s column vectos escibe the moes that can eist in the mateial. These ae essentiall pictues of the moes which quantif the elative amplitues of E, E, H, an H. e Diagonal mati escibing how the moes popagate. This inclues accumulation of phase as well as ecaing (loss) o gowing (gain) amplitue. Lectue 1 Slie 33 Visualiation of this Solution Moes 1 c 1 v 1 w c v w 3 c 3 v 3 w 4 c 4 v 4 w 5 c 5 v 5 w e 1 e 3 e 4 e 5 e 1 e e 3 e 4 e 5 e 1 c 1 v 1 w c v w 3 c 3 v 3 w 4 c 4 v 4 w 5 c 5 v 5 w Lectue 1 Slie 34 17

18 1/8/18 Solution in Homogeneous Laes Recall that in homogeneous laes we have 1 KK I K P Q P K I K K The solution to the eigen value poblem is Ω PQ Eigen-Vectos: Eigen-Values: I W I λ K jk λ K jk * * * K IK K The eigen moes fo the magnetic fiels ae simpl V Qλ 1 Lectue 1 Slie 35 Multilae Famewok: Scatteing Matices R. C. Rumpf, "Impove fomulation of scatteing matices fo semi analtical methos that is consistent with convention," PIERS B, Vol. 35, 41 61, 11. Lectue 1 Slie 36 18

Z 1 BCs BCs 1 1 K,1 K μ,1 K,1 K P1 1 1,1,1 K K μ K,1 K 1 1 K,1 K,1 K,1 K Q 1 1 1,1,1 K K K,1 K Ω1 PQ 1 1 W1, λ1 V1 c1, c1 Z BCs 1 1 K, K

19 1/8/18 Geomet of a Multilae Device Z 1 Z 1 Z Z Z 3 Sample of an infinitel peioic lattice Unit cell Z 3 Lectue 1 Slie 37 Eigen Sstem in Each Lae Bouna conitions equie that all laes have the same K an K matices. Z 1 BCs BCs 1 1 K,1 K μ,1 K,1 K P1 1 1,1,1 K K μ K,1 K 1 1 K,1 K,1 K,1 K Q 1 1 1,1,1 K K K,1 K Ω1 PQ 1 1 W1, λ1 V1 c1, c1 Z BCs 1 1 K, K μ, K, K P 1 1,, K K μ K, K 1 1 K, K, K, K Q 1 1,, K K K, K Ω PQ W, λ V c, c Z K,3 K,3 K,3 K P3 1 1,3,3 K K K,3 K 1 1 K,3 K,3 K,3 K Q 3 1 1,3,3 K K K,3 K Ω3 PQ 3 3 W3, λ3 V3 c3, c3 BCs Lectue 1 Slie 38 19

1/8/18 Fiel Relations & Bouna Conitions Fiel insie the i th lae: ψ i si, λi s i, Wi Wie c i λi ui, Vi Vi e ci u i, Bouna conitions at the fist inteface: ψ ψ 1 W1 W1c 1 Wi Wic i V1 V1c 1 Vi Vi ci i

Lectue 1 Slie 39 Aopt the Smmetic S Mati Appoach The scatteing mati S i of the i th lae is still efine as: c 1 i c 1 S c c S i S S i i 11 S1 i i 1 S But the elements ae calculate as i 1 1 1 11 i i i

20 1/8/18 Fiel Relations & Bouna Conitions Fiel insie the i th lae: ψ i si, λi s i, Wi Wie c i λi ui, Vi Vi e ci u i, Bouna conitions at the fist inteface: ψ ψ 1 W1 W1c 1 Wi Wic i V1 V1c 1 Vi Vi ci i Bouna conitions at the secon inteface: ψ klψ i i λikl i Wi Wie c i W Wc λikl i Vi Vi e c i V V c Note: k has been incopoate to nomalie L i. Lectue 1 Slie 39 Aopt the Smmetic S Mati Appoach The scatteing mati S i of the i th lae is still efine as: c 1 i c 1 S c c S i S S i i 11 S1 i i 1 S But the elements ae calculate 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 Laes ae smmetic so the scatteing mati elements have eunanc. Scatteing mati equations ae simplifie. Fewe calculations. Less memo stoage. A W W V V 1 1 i i i B W W V V 1 1 i i i X S i i ikl i e λ X = epm(-lam*k*l(nla)); Lectue 1 Slie 4

21 1/8/18 Global Scatteing Mati Scatteing mati fo all laes. BCs evice 1 3 S S S S Z 1 1 S BCs S Connection to outsie egions global ef evice tn S S S S Z Z 3 BCs 3 S Recall this poceue fom Lectue 5. BCs Lectue 1 Slie 41 Reflection/Tansmission Sie Scatteing Matices The eflection sie scatteing mati is 1 1 ef 1 S11 AefB A W W V V ef 1 1 ef 1 B W W V V S1 Aef ef 1 S.5 A B A B 1 ef ef ef ef ef 1 ef ef S B A The tansmission sie scatteing mati is ef ef ef ef ef ef A = W\Wef + V\Vef; B = W\Wef - V\Vef; SR.S11 = -A\B; SR.S1 = *inv(a); SR.S1 =.5*(A - B/A*B); SR.S = B/A; s ef,i s tn tn S11 BtnA A tn tn W Wtn V Vtn,II 1 1 tn 1 S1.5Atn BtnAtnB Btn W Wtn V Vtn,II tn A = W\Wtn + V\Vtn; tn 1 S1 A B = W\Wtn - V\Vtn; tn ST.S11 = B/A; tn 1 S AtnB ST.S1 =.5*(A - B/A*B); tn ST.S1 = *inv(a); ST.S = -A\B; lim L Etenal egions ae homogeneous so we o not nee to constuct convolution matices. Lectue 1 Slie 4,I lim L 1

22 1/8/18 Calculating Tansmission an Reflection Lectue 1 Slie 43 Calculating the Tansmitte an Reflecte Fiels The electic fiel souce is calculate assuming unit amplitue polaiation vecto P. p inc δ, pq st inc p c W s δ, pq 1 inc ef T Given the global scatteing mati, the coefficients fo the eflecte an tansmitte fiels ae c S c c S c ef 11 inc tn 1 inc The tansvese components of the eflecte an tansmitte fiels ae then T s W c W S c t s W c W S c T ef T tn T ef ef ef 11 inc tn tn tn 1 inc p P p P 1 p elta function δ, pq 1 pq, position T This ae amplitue coefficients of the tansvese components of the spatial t tt hamonics, not eflectance o t tansmittance. Lectue 1 Slie 44

1/8/18 Calculating the Longituinal Components The longituinal fiel components ae calculate fom the tansvese components using the ivegence equation (see Lectue 5).

IK K * *,ef,ef,ef K IK K * *,tn,tn,tn * * Calculating the Diffaction Efficiencies The iffaction efficiencies R an T ae calculate as t t t t R, Re K R Re Re K T

23 1/8/18 Calculating the Longituinal Components The longituinal fiel components ae calculate fom the tansvese components using the ivegence equation (see Lectue 5). K K K 1,ef t K K t K t 1,ef Deivation E E E E jkmn, Smn, jkmn, Smn, jkmn, Smn, km, nsm, nkm, nsm, nkm, nsm, n Ks Ks Ks Ks Ks Ks s K K s K s 1 Lectue 1 Slie 45 K IK K * *,ef,ef,ef K IK K * *,tn,tn,tn * * Calculating the Diffaction Efficiencies The iffaction efficiencies R an T ae calculate as t t t t R, Re K R Re Re K T Re k,ef,inc inc k,inc,tn inc,tn,inc t Remembe that these equations assume a unit amplitue souce. Don t foget to eshape R an T back to D aas! tt, Lectue 1 Slie 46 3

24 1/8/18 Calculating Oveall Reflectance an Tansmittance The oveall eflectance R an tansmittance T ae calculate b summing all of the iffaction efficiencies. R R T T Reflectance an Tansmittance on a Decibel Scale R 1log R T 1log T B 1 B 1 Be caeful NOT to use log1! Lectue 1 Slie 47 Powe Consevation It is alwas goo pactice to check fo consevation of powe. A R T 1 When no loss o gain is incopoate into the simulation (i.e. A = ), consevation euces to R T 1 no loss o gain Lectue 1 Slie 48 4

Transfer Matrix Method

Transfer Matrix Method 9/6/17 Instucto D. Ramond Rumpf (915) 747 6958 cumpf@utep.edu 5337 Computational lectomagnetics Lectue #4 Tansfe Mati Method Lectue 4These notes ma contain copighted mateial obtained unde fai use ules.