11/1/16 EE 533 Eletromgneti nlsis Using Finite Differene Time Domin Leture # Wveguide Ciruit nlsis Using FDTD Leture These notes m ontin oprighted mteril obtined under fir use rules. Distribution of these mterils is stritl prohibited Slide 1 Leture Outline Slb Wveguides Frequen Domin nlsis of Slb Wveguides Wveguide Soures in FDTD Refletion nd Trnsmission in Wveguides Leture Slide 1
11/1/16 Slb Wveguides Leture Slide 3 D pproimtion of Optil Integrted Ciruits It is possible to ver urtel simulte n optil integrted iruit in two dimensions using the effetive inde method. n 1,eff n,eff Effetive indies re best omputed b modeling the vertil ross setion s slb wveguide. simple verge inde n lso produe good results. n 1,eff n,eff Leture Slide 4
11/1/16 The Critil ngle nd Totl Internl Refletion When n eletromgneti wve is inident on mteril with lower refrtive inde, it is totll refleted when the ngle of inidene is greter thn the ritil ngle. n 1 n 1 n sin 1 n1 in in Emple Wht is the ritil ngle for fused sili (glss). The refrtive inde t optil frequenies is round 1.5. n n 1 1. sin 41.81 1.5 Leture Slide 5 The Slb Wveguide If we sndwih slb of mteril between two mterils with lower refrtive inde, we form slb wveguide. TIR n n 1 TIR Conditions n n n 1 nd n 3 n 3 Leture Slide 6 3
11/1/16 R Tring nlsis m kn eff kn sin The round trip phse of r must be n integer multiple of. euse of this, onl ertin ngles re llowed to propgte in the wveguide. Leture Slide 7 Frequen Domin nlsis of Slb Wveguides Leture Slide 8 4
11/1/16 Mwell s Equtions In Leture 11, we normlied the eletri fields nd rrived t Mwell s equtions in the following form: 1 D H t r H E t D r E For this slb wveguide nlsis in the frequen domin, we eliminte the D field. r H j E r E j H H H j E H H j E H H j E E E j H E E j H E E j H Leture Slide 9 Wveguide Modes ssume Solution Modes in wveguide hve the following form: j E,, e, j H,, e,,, j e The derivtive E, e j e j, H j j, e e, j j, e j E j j, e j H Conlusion j Leture Slide 1 5
11/1/16 6 Leture Slide 11 Redution of Dimensions Slb Wveguides re Uniform long nd j Mwell s Equtions Redue to j j j j j j j j j j j j j j j j j j j j Leture Slide 1 Two Distint Modes E Mode H Mode Mwell s equtions hve deoupled into two sets of three equtions. j j j j j j j j j j
11/1/16 Normlie the Grid We normlie the grid oordinte s follows k We lso reognie tht kn n eff eff Under these onditions, Mwell s equtions for the two modes beome E Mode jneff j n eff j H Mode jneff j n eff j Leture Slide 13 Mtri Representtion of Fields on Grid 1 D Sstems E 1 E E E E 4 5 3 E1 E E E 3 E4 E 5 D Sstems E1 E E3 E4 E5 E6 E7 E8 E9 E1 E11 E1 E13 E14 E15 E16 E1 E E 3 E4 E 5 E6 E 7 E8 E E 9 E1 E11 E 1 E13 E 14 E15 E 16 Leture Slide 14 7
11/1/16 Mwell s Equtions in Mtri Form The equtions for the E mode n be written in mtri form s jneff j n eff j Db jn b jε h eff n eff μ b D jμ b e μ ε ε μ 1 ε ε N 1 μ μ N 1 1 1 1 1 e D k 1 1 1 1 1 1 h 1 D k 1 1 1 1 Leture Slide 15 1 N 1 i i bi in Mtri Wve Eqution We strt with Mwell s equtions in mtri form. Db jn b jε h eff n eff μ b D jμ b e We solve the seond two equtions for the mgneti field quntities. b n μ b 1 eff jμ D 1 e We will use this eqution gin. We substitute these into the first eqution to get the mtri wve eqution. h Db jn b jε eff h j jn n j 1 e 1 eff eff D μ D μ ε jdμ D jn μ jε This is generlied eigen vlue problem h 1 e 1 eff h 1 e 1 h 1 e Dμ D ε neffμ Dμ D ε 1 h 1 e 1 Dμ D ε neffμ μ neff Leture Slide 16 8
11/1/16 Solving the Eigen Vlue Problem We n use MTL s built in eig() funtion to solve this eigen vlue problem. [V,D] = eig(,); The solution n be interpreted s 1 1 1 1 1 1 1 1 1 3 3 3 V 1 1 N 1 N 1 N 1 1 1 N N N 1 n eff neff D N n eff The eigen vetors desribe the mplitude profile of the modes. e jkneff The eigen vlues desribe the umultion of phse. Leture Slide 17 Conept of the Eigen Vetor Mtri The olumns of the eigen vetor mtri re the modes of the wveguide. V Leture Slide 18 9
11/1/16 MTL Code for Slb Wveguide nlsis funtion [E_sr,H_sr,neff,Z,ind] = emode(ur,ur,er,dp) % EZMODE Clulte the Fundmentl Mode of Slb Wveguide % for the E Mode % % [E_sr,H_sr,neff,Z,ind] = fmode(ur,ur,er,dp) % % dp is the normlied grid resolution % dp = k*d ε 1 ε ε ε N % DETERMINE NUMER OF POINTS ON GRID N = length(er); % CONSTRUCT DIGONL MTERIL MTRICES UR = dig(sprse(ur(:))); UR = dig(sprse(ur(:))); ER = dig(sprse(er(:))); % UILD DERIVTIVE OPERTORS DHX = spdigs(-ones(n,1)/dp,-1,sprse(n,n)); DHX = spdigs(ones(n,1)/dp,,dhx); DEX = spdigs(-ones(n,1)/dp,,sprse(n,n)); DEX = spdigs(ones(n,1)/dp,1,dex); % SOLVE EIGEN-VLUE PROLEM = full(er + DHX/UR*DEX); = full(inv(ur)); [Z,NEFF] = eig(,); NEFF = sqrt(dig(neff)); % FIND FUNDMENTL MODE [neff,ind] = m(rel(neff)); E_sr = Z(:,ind); 1 1 1 1 1 1 1 1 e h 1 D D k k 1 1 1 1 1 1 1 h 1 e DμD ε μ 1 neff We identif the fundmentl mode s the mode with the lrgest rel vlue. % COMPUTE H_sr H_sr = -neff*(ur\e_sr); See Slide 16 Leture Slide 19 Tpil Modes in Slb Wveguide (E Mode) n1 1. 3 n. n3 1. Leture Slide 1
11/1/16 Wveguide Soures in FDTD Leture Slide 1 Rell Totl Field/Sttered Field D FDTD Grid totl field sttered field Problem Points! j 1 j sr sr Leture Slide 11
11/1/16 Rell Injeting Plne Wve (E Mode) We lulte the eletri field s E sr j sr t g t We lulte the mgneti field s jsr 1 sr r H n t g t t t r mplitude due to Mwell s equtions Del through one hlf of grid ell Hlf time step differene Leture Slide 3 Modifition for Wveguide Soures Plne Wve Soure H E sr jsr t jsr 1 sr t t g t r gtt r Del neff t t Wveguide Soure E sr jsr t jsr 1 sr t t Re ep r t v j ft E Re ep H r t vh j f t t Rmp funtion Comple mode mplitudes from emode() Leture Slide 4 Hrmoni osilltion (pure frequen) Note: These soures re t single frequen f. 1
11/1/16 Etrting the Slb Wveguide(s) from FDTD window just outside the top PML is used for the soure nd to nle refleted wves. nother window just outside the side PML is used for nling trnsmitted wves. Leture Slide 5 nimtion of Wveguide Simultion Soure Profile Wves eiting trnsmission plne Wves sttered from wveguide Leture Slide 6 13
11/1/16 Refletion From nd Trnsmission Through Wveguides Leture Slide 7 Modif the Fourier Trnsform For wveguide iruits, we tpill use soure tht is t pure frequen. To lulte Fourier trnsform from sinusoidl soure, we run the simultion until sted stte hs been rehed nd then integrte over single period. This is not neessr, but is fster. We strt with the stndrd Fourier trnsform, but we onl hve to integrte over one period beuse the funtion will just keep repeting s long s it is t sted stte.. t 1 f j ft F f f f t e dt This is implemented in FDTD s t j ft m F f t f e f m 1 f Leture Slide 8 14
11/1/16 MTL Code for Revised Fourier Trnsform We must ensure tht one wve le is resolved with n integer number of time steps. % SNP TIME STEP SO WVE PERIOD IS N INTEGER NUMER OF STEPS period = 1/f; Nt = eil(period/dt); dt = period/nt; The Fourier trnsform is omputed during the lst wve le of the simultion. % Updte Fourier Trnsform if T>(STEPS-Nt) Eref = Eref + (K^(T-STEPS+Nt))*E(:,nref); Etrn = Etrn + (K^(T-STEPS+Nt))*E(ntrn,:); end fter the min loop, we finish the trnsform s % FINISH TRNSFORMS Eref = Eref * (*dt/period); Etrn = Etrn * (*dt/period); Note: pure sinusoid soure is used so there is no need to Fourier trnsform the soure or divide b its mplitude. Leture Slide 9 Field ross Wveguide During the FDTD simultion, we use the Fourier trnsform proedure to lulte the sted stte field ross the wveguide. 1 16.34 Leture Slide 3 15
11/1/16 Field In Terms of Eigen Modes The field ross the wveguide must be liner sum of the eigenmodes. v 1 v v3 v4 v5 1 3 4 5.8.4..1. 1 e 3 1v1v 3v34v4 5v5 V mode 1 4 5 mode mode 3 mode 4 mode 5 Leture Slide 31 Clulting the Energ in Eh Mode Using FDTD, we lulte the sted stte field e round the input nd output(s) of the wveguide iruit. We n then lulte the omple mplitudes of ll the modes. e V V e ref trn 1 ref ref ref ref ref V V e e 1 trn trn trn trn trn Now we n lulte the frtion of power in ll of the modes. p p ref trn 1 in 1 in ref trn Most of the time we onl re bout the frtion of power in the fundmentl mode. p ref ref trn ptrn in in Leture Slide 3 16
11/1/16 MTL Code for Power Clultion First we lulte the omple mplitudes of the eigen modes. % CLCULTE MODE MPLITUDES ref = EZR\Eref(NPML(1)+1:N-NPML()); trn = EZT\Etrn(NPML(3)+1:N-NPML(4))'; Eigen vetor mtries Grb fields outside of PML (simplifies mode lultion) Seond, we lulte refletion nd trnsmission. % CLCULTE TRNSMITTNCE ND REFLECTNCE OF FUNDMENTL MODE REF = bs(ref(ind_ref))^; TRN = bs(trn(ind_trn))^; Leture Slide 33 Emple Trnsmission Clultion 1 16 ssuming the wveguide ws soured with onl the fundmentl mode with unit mplitude sr 1..87. trn.6.15.5.34 R % T.87 1 76% Leture Slide 34 17
11/1/16 enhmrk Simultions The wveguide prmeters re n n 1.55 m ld ore 1.5..5 m n ld n ore n ld Leture Slide 35 18