Transfer Matrix Method Using Scattering Matrices

Transcription

1 Instructor Dr. Ramond Rumpf (915) EE 5337 Computational Electromagnetics Lecture #5 Transfer Matri Method Using cattering Matrices Lecture 5b These notes ma contain coprighted material obtained under fair use rules. Distribution of these materials is strictl prohibited lide 1 Outline Review Calculating reflected and transmitted power implifications for 1D transfer matri method Notes on implementation Parameter weeps Lecture 5b lide 1

2 Review Lecture 5b lide 3 Two Paths to Combined olution 4 4 Matri kk kˆ z z z z z z z z j k k jk k zz zz zz zz zz zz zz zz E z z z z k jk j k k z kk z z z E E zz zz zz zz zz zz zz zz E z H jk H H j k k kk z z k zz z z z z H zz zz zz zz zz zz zz zz k k z z k zz z z z z jk j k k zz zz zz zz zz zz zz zz Anisotropic ort Eigen Modes WE W E W W H W H e λz e λ z E E H H λ z e Mawell s Equations Field olution λ z E k r H WE W e E c ψ z z H VH V λ H e c k r E λz W We c ψ z λz Matrices V V e c Isotropic or 1 kk r r k P diagonall r k r r k k anisotropic No sorting! 1 kk r r k Q r k r r k k Lecture 5b PQ Method lide 4

3 Definition of A cattering Matri c 1 c 1 c c reflection transmission This is consistent with network theor and eperimental convention. Lecture 5b lide 5 cattering Matri for a ingle Laer The scattering matri i of the i th laer is still defined as: c 1 i c 1 c c i i i i i But the equations to calculate the elements reduce to 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 A XB A XB XB A X A B A XB A XB X A B A B i i i i Laers are smmetric so the scattering matri elements have redundanc. cattering matri equations are simplified. Fewer calculations. Less memor storage. i A W W V V 1 1 i i g i B W W V V 1 1 i i g i X ikl i i e λ g g Lecture 5b lide 6 3

4 Reflection/Transmission ide cattering Matrices The reflection side scattering matri is ref 1 ref ref ref 1 Aref A B ref 1.5 ref ref ref ref A B A B ref 1 ref ref B A trn 1 trn trn B A trn 1.5 trn trn trn trn A B A B trn 1 Atrn trn 1 trn trn A B A W W V V 1 1 ref g ref g ref B W W V V The transmission side scattering matri is 1 1 ref g ref g ref A W W V V 1 1 trn g trn g trn B W W V V 1 1 trn g trn g trn s ref r,i r,i s trn r,g r,g lim L lim L r,g r,g r,ii r,ii Lecture 5b lide 7 ummar of Using cattering Matrices ref 1 3 L1 L L3 device N LN trn Device in gap medium oba ref 1 N device gl l trn Lecture 5b lide 8 4

5 Redheffer tar Product Two scattering matrices ma be combined into a single scattering matri using Redheffer s star product. AB A B A A A A A B B B B B The combined scattering matri is then AB AB AB AB AB 1 AB A A B A B A I 1 AB A B A B I 1 AB B A B A I 1 AB B B A B A B I R. Redheffer, Difference equations and functional equations in transmission-line theor, Modern Mathematics for the Engineer, Vol., pp , McGraw-Hill, New York, Lecture 5b lide 9 Calculating Transmitted and Reflected Power Lecture 5b lide 1 5

6 Recall How to Calculate ource Parameters Incident Wave Vector urface Normal Unit Vectors in Direction of TE & TM sin cos aˆ k kn sin sin nˆ aˆ z aˆ nˆ k TE cos 1 nˆ k Unit vectors along,, and z aes. aˆ a a ˆ ˆz Right handed coordinate sstem aˆ TM aˆ aˆ TE TE k k Composite Polarization Vector P p aˆ p aˆ TE TE Can be an direction in the plane TM TM In CEM, we usuall make P 1 Lecture 5b lide olution Using cattering Matrices The eternal fields (i.e. ident wave, reflected wave, transmitted wave) are related through the global transfer matri. cref global c ctrn c E 1, W ref E, We get E, and E, from the polarization vector P. E, P Note that E E, P z, is not needed. E z, P z This matri equation can be solved to calculate the mode coefficients of the reflected and transmitted fields. global global ref global global trn c c c global ref global trn c c c c right c not tpicall used Lecture 5b lide 6

7 Calculation of Transmitted and Reflected Fields The procedure described thus far calculated c ref and c trn. The transmitted and reflected fields are then ref E global global E 1 ref Wrefcref Wref c Wref Wref E E trn E global global E 1 trn Wtrnctrn Wtrn c Wtrn Wref E E Lecture 5b lide 13 Calculation of the Longitudinal Components We are still missing the longitudinal field component E z on the reflection and transmission sides. These are calculated using Mawell s divergence equation. E Note: E jk r jk r jk r E, e E, e E, ze E z jk r jk r jk r jke, e jke, e jkze, ze ke, ke, ke z, z ke z, z ke, ke, ref ref ke, ke ke, ref ke E, z Ez ref k k z z trn trn ke trn ke Ez trn k Lecture 5b lide 14 z reduces to when is homogeneous. 7

8 Calculation of Power Flow Reflectance is defined as the fraction of power reflected from a device. Eref R E E E Ez E Transmittance is defined as the fraction of power transmitted through a device. trn Etrn Re kz Note: We will derive these r,trn T formulas in Lecture 7. E Re k z r, It is alwas good practice to check for conservation of power. 1 materials have loss Note: Recall RT 1 materials have no loss and no gain A RT 1 1 materials have gain Lecture 5b lide 15 Reflectance and Transmittance on a Decibel cale Decibel cale P P log db 1 1log db 1 A P How to calculate decibels from an amplitude quantit A. How to calculate decibels from a power quantit P. P A P db 1log1 A log1 A Reflectance and Transmittance Reflectance and transmittance are power quantities, so R T 1log db 1 1log db 1 R T Lecture 5b lide 16 8

9 implifications for 1D Transfer Matri Method Lecture 5b lide 17 Analtical Epressions for W and The dispersion relation with a normalized wave vector is k k k r r z Using this relation, we can simplif the matri equation for. 1 kk r r k kk r r k k z Ω PQ k z I r r k r r k k k r r k k k z 1 I identit matri 1 A lot of algebra e is a diagonal matri, we can conclude that 1 WI 1 jk λ λ Ω z jk zi jk z For isotropic materials and diagonall anisotropic materials, we don t actuall have to solve the eigen value problem to obtain the eigen modes! Lecture 5b lide 18 e λz e jk z z e jk z z 9

10 implifications for TMM in LHI Media In LHI media, 1 1 Wi I 1 and Ωi jk z, ii I identit matri 1 Now we do not actuall have to calculate because λ Ω i i Given all of this, the eigen vectors for the magnetic fields can be calculated as V Q Wλ QΩ 1 1 i i i i i i When calculating scattering matrices, the intermediate matrices A i and B i reduce to A W W V V IV V i i g i g i g B W W V V IV V i i g i g i g The fields and mode coefficients are now related through P P E E c W W c c W c c ref trn 1 ref ref ref trn trn P P E E Lecture 5b lide 19 implified Eternal Matrices in LHI Media The reflection side scattering matri is ref 1 ref ref ref 1 Aref A B ref 1.5 ref ref ref ref A B A B ref 1 ref ref B A trn 1 trn trn B A trn 1.5 trn trn trn trn A B A B trn 1 Atrn trn 1 trn trn A B A IV V 1 ref g ref B IV V The transmission side scattering matri is 1 ref g ref A IV V 1 trn g trn B IV V 1 trn g trn s ref r,i r,i s trn r,g r,g lim L lim L r,g r,g r,ii r,ii Lecture 5b lide 1

11 Notes on Implementation Lecture 5b lide Outline tep Define problem tep 1 Dashboard tep Describe device laers tep 3 Compute wave vector components tep 4 Compute gap medium parameters tep 5 Initialize global scattering matri tep 6 Main loop through laers tep 7 Compute reflection side scattering matri tep 8 Compute transmission side scattering matri tep 9 Update global scattering matri tep 1 Compute source tep Compute reflected and transmitted fields tep Compute reflectance and transmittance tep 13 Verif conservation of power human does this computer does the rest tep 6: Iterate through laers Compute P and Q Compute eigen modes Compute laer scattering matri Update global scattering matri Lecture 5b lide

12 toring the Problem How is a device described and stored for TMM? We don t use a grid for this method! tore the permittivit for each laer in a 1D arra. tore the permeabilit for each laer in a 1D arra. tore the thickness of each laer in a 1D arra. ER = [.5, 3.5,. ]; UR = [ 1., 1., 1. ]; L = [.5,.75,.89 ]; Input arras for three laers We will also need the eternal materials, and source parameters. er1, er, ur1, ur, theta, phi, pte, ptm, and lam Lecture 5b lide 3 toring cattering Matrices We often talk about the scattering matri as a single matri. However, we ver rarel all use the scattering matri this wa. We usuall use the individual terms,,, and separatel. o, scattering matrices are actuall best stored as the four separate components of the scattering matri.,,, and Lecture 5b lide 4

13 Initializing the Global cattering Matri Before we iterate through all the laers, we must initialize the global scattering matri as the scattering matri of nothing. What are the ideal properties of nothing? 1. Transmits 1% of power with no phase change. global global I. Does not reflect. global global We therefore initialize our global scattering matri as global I I This is NOT an identit matri! Look at the position of the s and I s. Lecture 5b lide 5 Calculating the Parameters of the Gap Media Our analtical solution for a homogeneous gap medium is k k k 1 kk r,gr,g k Qg Wg I r,g k r,gr,g k k λ jk I z,g r,g r,g g z,g 1 g QgWgλg We are free to choose an r,g and V r,g that we wish. We also wish to avoid the case of k z,g =. For convenience, we choose r,g 1. and r,g 1 k k We then have Q kk 1 k g 1 k kk W I g V jq g g W not even used in TMM. Lecture 5b lide 6 13

14 Calculating X i = ep( i k L i ) Recall the correct answer: X i e z i ΩikL i e jk zkli WRONG jk k L e It is orrect to use the function ep() because this calculates a point b point eponential, not a matri eponential. X = ep(omega*k*l); Approach #1: epm() X = i i Approach #: diag() X = epm(omega*k*l); X = i i X = diag(ep(diag(omega)*k*l)); X = i i Lecture 5b lide 7 Efficient Calculation of Laer Matrices There are redundant calculations in the equations for the scattering matri elements. i i Ai XB i iai XB i i XB i iai XiAi Bi i i Ai XBA i i i XiBi i i i i Bi X A B A These are more efficientl calculated as 1 Ai IVi Vg 1 Bi IVi Vg λikl i Xi e 1 DAi XB i iai XB i i i i 1 1 D XB A X A B i i i i i i i i 1 1 i i i i i D X A B A B Lecture 5b lide 8 14

15 Efficient tar Product After observing the equations to implement the Redheffer star product, we see there are some common terms. Calculating these multiple times is inefficient so we calculate them onl once using intermediate parameters. 1 AB A A B A B A I 1 AB A B A B I AB A B 1 AB B A B A I 1 AB B B A B A B I A B A D I B A B F I 1 1 AB A B A D AB B D AB A F AB B A B F Lecture 5b lide 9 Using the tar Product as an Update Ver often we update our global scattering matri using a star product. When we use this equation as an update, we MUT pa close attention to the order that we implement the equations so that we don t accidentall overwrite a value that we need. global i global global global i reverse order i global i D I 1 global i global F I global global i global global i F global global D global i global i F D 1 standard order global i global D I i global i F I 1 global global i global global i D global global F global i global i D F 1 Lecture 5b lide 3 15

16 Block Diagram of TMM Using Matrices Initialize Global cattering Matri global I I Calculate Gap Medium Parameters kk 1 k Qg V g jq 1k k k g Calculate Transverse Wave Vectors k n sin cos k n sin sin es Calculate Parameters for Laer i k zi, i ik k 1 kk i i k Qi i k i i k k 1 Ω jk I V QΩ i z, i i i i no Done? Calculate cattering Matri for Laer i 1 λikl i Ai IVi Vg Xi e 1 Bi IVi Vg 1 D Ai XB i iai XB i i D XB A X A B D X A B A B i i 1 1 i i i i i i i i 1 1 i i i i i Loop through all laers Update Global cattering Matri i global global 1 D i I global global 1 i F I i global i D global global global global i D F global global i F i global global tart Finish Connect to Eternal Regions ref global global trn global global Calculate ource P p ˆ ˆ TEaTE ptmatm P 1 P esrc P Calculate Transmitted and Reflected Fields ref E eref ref esrc E trn E etrn trn esrc E Calculate Longitudinal Field Components ref ref ke ref ke Ez ref k z trn trn ke trn ke Ez trn k z Calculate Transmittance and Reflectance R E ref T E trn Re k Re k trn z r,trn z r, Lecture 5b lide 31 How to Handle Zero Number of Laers Follow the block diagram!! etup our loop this wa NLAY = length(l); for nla = 1 : NLAY... end For zero laers: ER = []; UR = []; L = []; If NLAY =, then the loop will not eecute and the global scattering matri will remain as it was initialized. global I I Lecture 5b lide 3 16

17 Can TMM Fail? Yes! The TMM can fail to give an answer and behave numericall strange an time k z =. This happens at a critical angle when the transmitted wave is at or ver near its cutoff. We fied this problem in the gap medium, but this can also happen in an of the laers or in the transmission region. This happens in an laer where k k r r Lecture 5b lide 33 Parameter weeps Lecture 5b lide 34 17

18 What is a Parameter weep? o far, we have learned to simulate a single device at a single frequenc, or wavelength. im R = 81% T = 19% uppose we calculate this data as we continuousl change one or more parameters? This is called a parameter sweep. im Device Behavior Parameter Lecture 5b lide 35 Block Diagrams of Common Parameter weeps Conventional im (No weep) Wavelength or Frequenc weep Device Parameter weep Convergence weep for NRE Dashboard Dashboard Dashboard Dashboard Compute Params. Build Device Perform im how Results Compute Params. Build Device et or frequenc Compute Params. d et Parameter Build Device NRE et NRE Compute Params Build Device Good idea to visualize our results during simulation. You can abort earl if something is wrong. Perform im Record Results how Results Perform im Record Results how Results Perform im Record Results how Results Lecture 5b lide 36 18

19 Make a Generic TMM Function A great wa to simplif programming our parameter sweeps is to first make a generic function out of our TMM code. The basic TMM simulation will take as input arguments: ource: Device:,,, polarization, etc. UR, ER, L, etc. Given these input arguments, our TMM function will simulate the device and calculate reflectance, transmittance, fields, etc. It ma return REF, TRN, or whatever else ou wish. Lecture 5b lide 37 Eample Header for a Generic TMM Function These comments are displaed at the command prompt b tping >> help tmm1d It is alwas a good idea to lude a help section at the start of our codes. function DAT = tmm1d(dev,rc) % TMM1D One-Dimensional Transfer Matri Method % % DAT = tmm1d(dev,rc); % % INPUT ARGUMENT % ================ % DEV Device Parameters %.er1 relative permittivit in reflection region %.ur1 relative permeabilit in reflection region %.er relative permittivit in transmission region %.ur relative permeabilit in transmission region %.ER arra containing permittivit of each laer %.UR arra containing permeabilit of each laer %.L arra containing thickness of each laer % % RC ource Parameters %.lam free space wavelength %.theta elevation angle of idence (radians) %.phi azimuthal angle of idence (radians) %.ate amplitude of TE polarization %.atm amplitude of TM polarization % % OUTPUT ARGUMENT % ================ % DAT Output Data %.REF Reflectance %.TRN Transmittance Lecture 5b lide 38 19

20 What teps are Performed b TMM1D() tep Define problem tep 1 Dashboard tep Describe device laers tep 3 Compute wave vector components tep 4 Compute gap medium parameters tep 5 Initialize global scattering matri tep 6 Main loop through laers tep 7 Compute reflection side scattering matri tep 8 Compute transmission side scattering matri tep 9 Update global scattering matri tep 1 Compute source tep Compute reflected and transmitted fields tep Compute reflectance and transmittance tep 13 Verif conservation of power human does this computer does the rest tep 6: Iterate through laers Compute P and Q Compute eigen modes Compute laer scattering matri Update global scattering matri Lecture 5b lide 39 Wavelength or Frequenc Parameter weep B far, the most common parameter sweep is calculating the device behavior as a function of frequenc or wavelength. Dashboard for nlam = 1 : NLAM RC.lam = LAMBDA(nlam); DAT = tmm1d(dev,rc); REF(nlam) = DAT.REF; end UR = [ ]; ER = [ ]; L = [ ]; Build Device et Wavelength Perform im Record Results how Results Lecture 5b lide 4

21 Incorporating Material Dispersion in a Parameter weep ometimes the material properties change significantl as a function of frequenc, or wavelength. Dashboard This is called dispersion. Dispersion can be orporated into our parameter sweep b: (1) Calculate the material properties at the given wavelength or frequenc. () Rebuild the device each iteration with the material properties that were just calculated. Determine and Build Device from and et Wavelength Perform im Record Results how Results Lecture 5b lide 41 Bad Vs. Good Parameter weeps LABEL LINE THICKNE WHITE PACE CONERVATION? WHITE PACE WHITE PACE TRIANGLE LABEL # DIGIT FONT IZE CALE Lecture 5b lide 4