Scattering Matrices for Semi Analytical Methods

Transcription

1 Instructor Dr. Raymond Rumpf (95) EE 5337 Computatonal Electromagnetcs Lecture #5a catterng Matrces for em Analytcal Methods Lecture 5a These notes may contan copyrghted materal obtaned under far use rules. Dstrbuton of these materals s strctly prohbted lde Outlne catterng matrx for a sngle layer Multlayer structures Longtudnally perodc structures Dsperson analyss Alternatves to scatterng matrces Lecture 5a lde

2 catterng Matrx for a ngle Layer R. C. Rumpf, Improved Formulaton of catterng Matrces for em Analytcal Methods That s Consstent wth Conventon, PIER B, ol. 35, pp. 4 6,. Lecture 5a lde 3 Motvaton for catterng Matrces catterng matrces offer several mportant features and benefts: Uncondtonally stable method. Parameters have physcal meanng. Parameters correspond to those measured n the lab. Can be used to extract dsperson. ery memory effcent. Can be used to explot longtudnal perodcty. Mature and proven approach. Much greater wealth of lterature avalable. However, excellent alternatves to matrces do exst! Lecture 5a lde 4

3 Geometry of a ngle Layer Indcates a pont that les on an nterface, but assocated wth a partcular sde. Medum Layer Medum z kl z kl c c c c c c z L z feld wthn th layer c mode coeffcents nsde c mode coeffcents outsde th Lecture 5a lde 5 th layer layer Defnton of A catterng Matrx c c c c reflecton transmsson Ths s consstent wth network theory and expermental conventon. Lecture 5a lde 6 3

4 Feld Relatons Feld nsde the th layer: E z H y, z x, λz Ey, z e z W W c z H x, z λ e c Boundary condtons at the frst nterface: W Wc W W c c c Boundary condtons at the second nterface: kl λkl W W e c W W c λkl e c c Note: k has been ncorporated to normalze L. Lecture 5a lde 7 Dervaton of the catterng Matrx olve both boundary condton equatons for the ntermedate mode coeffcents c and c. c W W W Wc c c Both of these equatons have the term λ kl c e W W W W c λkl c e c Wj Wj Aj Bj j j j j j Bj Aj j j j W W A W W B W W We set substtute ths result nto the frst two equatons and then set them equal to elmnate the ntermedate mode coeffcents. λkl A B c e A Bc kl λ B A c e B Ac We wrte ths as two matrx equatons and rearrange the terms untl they have the form of a scatterng matrx. c?? c?? c c ee HW Lecture 5a lde 8 4

5 The catterng Matrx The scatterng matrx of the th layer s defned as: c c c c After some algebra, the components of the scatterng matrx are computed accordng to A XB A XB XB A X A B A XB A XB X A B A B A XB A XB X A B A B A XB A XB XB A X A B A W W j j j B W W j j j X kl e λ s the layer number. j s ether or dependng on whch external medum s beng referenced. Lecture 5a lde 9 catterng Matrces n the Lterature For some reason, the computatonal electromagnetcs communty has: () devated from conventon, and () formulated neffcent scatterng matrces. c c c c transmsson reflecton c c Here s c c s not reflecton. Instead. t s backward transmsson! Here s s not transmsson. Instead, t s a reflecton parameter! catterng matrces can not be nterchanged. R. C. Rumpf, "Improved formulaton of scatterng catterng matrces are not symmetrc so they take twce matrces for sem analytcal methods that s consstent wth conventon," PIER B, ol. 35, 4 6,. the memory to store and are more tme consumng to calculate. Lecture 5a lde 5

6 Lmtaton of Conventonal Matrx Formulaton Note that the elements of a scatterng matrx are a functon of materals outsde of the layer. Ths makes t dffcult to nterchange scatterng matrces arbtrarly. For example, there are only three unque layers n the multlayer structure below, yet separate computatons of scatterng matrces are needed. Three unque layers layer stack Lecture 5a lde oluton To get around ths, we wll surround each layer wth external regons of zero thckness. Ths lets us connect the scatterng matrces n any order because they all calculate felds that exst outsde of the layers n the same medum. Ths wll have no effect electromagnetcally as long as we make the external regons have zero thckness between layers. Gap Medum Layer Gap Medum L Lecture 5a lde 6

7 sualzaton of the Technque We calculate the scatterng matrces for just the unque layers. Three unque layers Then we just manpulate these same three scatterng matrces to buld the global scatterng matrx. Gaps between the layers are made to have zero thckness so they have no effect electromagnetcally. Faster! mpler! Less memory needed! Lecture 5a lde 3 Revsed Geometry of a ngle Layer same medum Gap Medum Layer Gap Medum r,g r,g z z kl kl r,g r,g c c c c c c L Lecture 5a lde 4 7

8 Calculatng Revsed catterng Matrces The scatterng matrx of the th layer s stll defned as: c c c c But the equatons to calculate the elements reduce to A XB A XB XB A X A B A XB A XB X A B A B Layers are symmetrc so the scatterng matrx elements have redundancy. catterng matrx equatons are smplfed. Fewer calculatons. Less memory storage. A W W g B W W g X kl e λ g g Lecture 5a lde 5 Layers n TMM are Actually Four Port Networks We have wrtten the scatterng matrces as block matrces. For TMM, ths actually expands to a 44 element scatterng matrx. s s s s3 s4 s s s3 s 4 s3 s3 s33 s 34 s4 s4 s43 s44 s s s 3 4 s s s3 s 4 s s s s s4 s 4 s43 s 44 Each mode provdes an I/O mechansm and there are two modes on each sde n each drecton. 4 3 Lecture 5a lde 6 8

9 catterng Matrces of Lossless Meda If a scatterng matrx s composed of materals that have no loss and no gan, the scatterng matrx must conserve power. That s, all ncdent power must ether reflect or transmt. Ths mples that the scatterng matrx s untary. If the scatterng matrx s untary, t must obey the followng rules: H H H I Note: If the regons external to the layer are dfferent from each other, the scatterng matrces wll not be untary. Ths s because the feld ampltudes wll be dfferent even though the feld carres the same amount of power. Lecture 5a lde 7 Hnts About tablty n These Formulatons Dagonal elements and tend to be the largest numbers. Dvde by these nstead of any off dagonal elements for best numercal stablty. X descrbes propagaton through an entre layer. Don t dvde by X or your code can become unstable. Lecture 5a lde 8 9

10 Multlayer tructures Lecture 5a lde 9 oluton Usng catterng Matrces The scatterng matrx method conssts of workng through the devce one layer at a tme and calculatng an overall scatterng matrx devce Redheffer star product. NOT matrx multplcaton! Lecture 5a lde

11 Dervaton of the Redheffer tar Product We start wth the equatons for the two adjacent scatterng matrces. A A B B c c c A A B B c c3 c3 c c We expand these nto four matrx equatons. A A B B c c c Eq. c c c3 Eq. 3 A A B B c c c Eq. c 3 c c3 Eq. 4 We substtute Eq. () nto Eq. (3) to get an equaton wth only c. We substtute Eq. (3) nto Eq. () to get an equaton wth only c. B A I B A B c c c Eq. 5 A B A A B 3 I c c c3 Eq. 6 We elmnate c and c by substtutng these equatons nto Eq. () and (4). We then rearrange terms nto the form of a scatterng matrx. c?? c?? c 3 c3 Overall, ths s just algebra. We start wth 4 equatons and 6 unknowns and reduce t to equatons wth 4 unknowns. Lecture 5a lde Redheffer tar Product Two scatterng matrces may be combned nto a sngle scatterng matrx usng Redheffer s star product. AB A B A A A A A B B B B B The combned scatterng matrx s then AB AB AB AB AB AB A A B A B A I AB A B A B I AB B A B A I AB B B A B A B I R. Redheffer, Dfference equatons and functonal equatons n transmsson-lne theory, Modern Mathematcs for the Engneer, ol., pp , McGraw-Hll, New York, 96. Lecture 5a lde

12 Puttng t All Together ( of ) Frst, we calculate the devce scatterng matrx by teratng through each layer of the devce and combnng the scatterng matrces usng the Redheffer star product. L L 3 L 3 N L N devce 3 N Lecture 5a lde 3 Puttng t All Together ( of ) econd, we must connect the devce scatterng matrx to the external regons to get the global scatterng matrx. We use connecton scatterng matrces to do ths. ref 3 L L L3 N LN trn global ref 3 N trn Lecture 5a lde 4

13 Reflecton/Transmsson de catterng Matrces The reflecton sde scatterng matrx s ref ref ref ref Aref A B ref.5 ref ref ref ref A B A B ref ref ref B A trn trn trn B A trn.5 trn trn trn trn A B A B trn Atrn trn trn trn A B A W W ref g ref g ref B W W The transmsson sde scatterng matrx s ref g ref g ref A W W trn g trn g trn B W W trn g trn g trn s ref r,i r,i s trn r,g r,g lm L lm L r,g r,g r,ii r,ii Lecture 5a lde 5 ummary of Usng catterng Matrces ref 3 L L L3 devce N LN trn Devce n gap medum oba ref N devce gl l trn Lecture 5a lde 6 3

14 Longtudnally Perodc Devces Lecture 5a lde 7 Longtudnally Perodc Devces uppose we just calculated the scatterng matrx for the unt cell of a longtudnally perodc devce. Unt Cell A B C A B C Unt Cell Unt Cell Unt Cell 3 Unt Cell 4 Unt Cell 4 Unt Cell 6 Unt Cell 7 Unt Cell 8 There exsts a very effcent way of calculatng the global scatterng matrx of a longtudnally perodc devce wthout calculatng and combnng all the ndvdual scatterng matrces. 8 A B C A B C A B C A B C A B C A B C A B C A B C 8 Both are neffcent!!! Lecture 5a lde 8 4

15 Cascadng and Doublng We can quckly buld an overall scatterng matrx that descrbes hundreds and thousands of unt cells. We start by calculatng the scatterng matrx for a sngle unt cell. A B C A B C Next, we keep connectng the scatterng matrx to tself to keep doublng the number of unt cells t descrbes and so on What f you have a non nteger power of number of layers? Lecture 5a lde 9 Block Dagram for Modfed Cascadng and Doublng Algorthm Inputs -matrx of one unt cell N Number of tmes to repeat unt cell Convert N to bnary Intalze Algorthm global I bn I Loop through all bnary dgts startng wth the least sgnfcant dgt. Done? dgt =? no Perform Doublng bn bn bn Update (N) global global bn Lecture 5a lde 3 no yes Output global 5

16 Example of Cascadng and Doublng Algorthm tep Calculate scatterng matrx for one unt cell A B C Inputs to algorthm: A B C scatterng matrx for a sngle unt cell N number of unt cells to combne tep Convert N to bnary tep Intalze bnary and global scatterng matrces bn 6 s 8 s 4 s s global global I global I global s Lecture 5a lde 3 Example of Cascadng and Doublng Algorthm tep 3 Loop through bnary dgts s dgt = Update (global) global global bn (global) now encompasses unt cells s dgt = Do not update (global) (global) stll encompasses unt cells Double (bn) bn bn bn (bn) now represents unt cells Double (bn) bn bn bn (bn) now represents 4 unt cells 4 s dgt = Update (global) global global bn (global) now encompasses 6 unt cells Double (bn) bn bn bn (bn) now represents 8 unt cells 8 s dgt = Do not update (global) (global) stll encompasses 6 unt cells Double (bn) bn bn bn (bn) now represents 6 unt cells 6 s dgt = Update (global) global global bn (global) now encompasses unt cells Double (bn) bn bn bn (bn) now represents 3 unt cells Oops! Ths algorthm performs one unnecessary doublng operaton. How can we fx ths? Lecture 5a lde 3 6

17 Dsperson Analyss Lecture 5a lde 33 Dsperson Analyss ( of ) An overall scatterng matrx s calculated that descrbes the unt cell. uc uc c c uc uc uc same as on prevous slde cn cn The terms are rearranged n almost the form of a transfer matrx. uc uc c N I c uc uc I cn c If the devce s nfntely perodc n the z drecton, then the followng perodc boundary condton must hold. c N jkz c z e cn c Here k z s the effectve propagaton constant of the mode. Lecture 5a lde 34 7

18 Dsperson Analyss ( of ) We substtute the perodc boundary condton nto our rearranged equaton to get uc uc Ic e jk z c z uc uc c I c Ths s a generalzed egen value problem. Ax Bx uc x uc c uc e jk z uc I c A B I Egen vectors Egen values Bloch modes k z s Lecture 5a lde 35 z [,D] = eg(a,b); Who Cares? Gven k z, we can. Calculate the effectve propertes of the unt cell. k z r,eff r,eff Ths s an over smplfcaton and beyond the scope of ths course.. Construct band dagrams. Lecture 5a lde 36 8

19 Alternatves to catterng Matrces Lecture 5a lde 37 Transmttance Matrces (T Matrces) The T matrx method s the transfer matrx method where forward and backward waves are dstngushed. left ctrn T T c nc rght cnc T T cref Benefts Much faster (5 to tmes) Uncondtonally stable Drawbacks Less memory effcent Cannot explot longtudnal perodcty Less popular n the lterature M. G. Moharam, Drew A. Pommet, Erc B. Grann, table mplementaton of the rgorous coupled-wave analyss for surfacerelef gratngs: enhanced transmttance matrx approach, J. Opt. oc. Am. A, ol., No. 5, pp , 995. Lecture 5a lde 38 9

20 Hybrd Matrces (H Matrces) The h matrx method s borrowed from electrcal two port networks. h hi I h h h I I h I I I h h I In the framework of felds, the h matrx s defned as E H x, x, E y, H H H y, H x, E x, H H H y, Ey, Clamed Benefts Improved numercal stablty More concse formulaton mpler to mplement Improved numercal effcency (3% better than ETM) Uncondtonally stable Eng L. Tan, Hybrd-matrx algorthm for rgorous coupled-wave analyss of multlayered dffracton gratngs, J. Mod. Opt., ol. 53, No. 4, pp , 6. Lecture 5a lde 39 R Matrces The R matrx method s essentally the mpedance matrx framework borrowed from electrcal two port networks. z z I z z I z I I I I z I I I I z z In the framework of felds, the h matrx s defned as E H x, x, E y, R R H y, Ex, H x, R R Ey, H y, Clamed Benefts Uncondtonally stable Improved numercal effcency Lfeng L, Bremmer seres, R-matrx propagaton algorthm, and numercal modelng of dffracton gratngs, J. Opt. oc. Am. A, ol., No., pp , 994. Lecture 5a lde 4