TMM Using Scattering Matrices.pdf

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Lecture5 Slide1

EE4395/5390 SpecialTopicsComputationalElectromagnetics

Lecture#5

TransferMatrixMethodUsingScatteringMatrices Thesenotesmaycontaincopyrightedmaterialobtainedunderfairuserules.Distributionofthesematerialsisstrictlyprohibited

InstructorDr.RaymondRumpf(915)[email protected]

Outline

Alternatematrixdifferentialequation(nomodesorting) Scatteringmatrixforasinglelayer Multilayerstructures Calculatingreflectedandtransmittedpower NotesImplementation

Advanced scattering matrices

Alternatives to scattering matrices

Lecture5 Slide2

Bonus

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Lecture5 Slide3

AlternateMatrixDifferentialEquation

Lecture5 Slide4

RecallDerivationfromLastLecture

0

0

0

0

0

0

yzr x

x zr y

y xr z

yzr x

x zr y

y xr z

EE k Hy zE E k Hz x

E E k Hx y

HH k Ey z

H H k Ez x

H H k Ex y

StartwithMaxwellsequationsfrom

Lecture2.

AssumeLHI.

0

0

0

0

0

0

yy z r x

xx z r y

x y y x r z

yy z r x

xx z r y

x y y x r z

dEjk E k H

dzdE jk E k Hdz

jk E jk E k H

dHjk H k E

dzdH jk H k Edz

jk H jk H k E

Assumedeviceisinfiniteanduniforminx andy directions.

x yjk jkx y

yy z r x

xx z r y

x y y x r z

yy z r x

xx z r y

x y y x r z

dEjk E H

dzdE jk E Hdz

jk E jk E H

dHjk H E

dzdH jk H Edz

jk H jk H E

Normalizez andwavevectorskx,ky,

andkz.0

0 0 0

yx zx y z

z k zkk kk k k

k k k

2

2

2

2

x yx xx r y

r r

y y x yr x y

r r

x yx xx r y

r r

y y x yr x y

r r

k kdE kH Hdz

dE k k kH H

dz

k kdH kE Edz

dH k k kE E

dz

Eliminatelongitudinal

componentsEz andHz bysubstitution.

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Lecture5 Slide5

DerivationofTwo22MatrixEquations

2

2

x yx xx r y

r r

y y x yr x y

r r

k kdE kH Hdz

dE k k kH H

dz

2

2

x yx xx r y

r r

y y x yr x y

r r

k kdH kE Edz

dH k k kE E

dz

Wecanwriteourtwosetsoftwoequationsinmatrixformas

2

2

1x x y r r x xy yr y r r x y

E k k k HdE Hdz k k k

2

2

1 x y r r x xxyy r y r r x y

k k k EHdEHdz k k k

Note:Theseequationsarevalidregardlessofthesignconvention.

Lecture5 Slide6

CompactPQFormWecanwriteourtwomatrixequationsmorecompactlyas

2

2

1x x y r r x xy yr y r r x y

E k k k HdE Hdz k k k

2

2

1 x y r r x xxyy r y r r x y

k k k EHdEHdz k k k

x x

y y

E HdE Hdz

P

xx

yy

EHdEHdz

Q

2

2

1 x y r r xr y r r x y

k k k

k k k

P

2

2

1 x y r r xr y r r x y

k k k

k k k

Q

Note:WewillseethissamePQformagainforothermethodslikeMoL,RCWA,andwaveguideanalysis.TMM,MoL,andRCWAareimplementedthesameafterPandQarecalculated.

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Lecture5 Slide7

MatrixWaveEquationOurtwogoverningequationsare

Wecannowderiveamatrixwaveequation.First,wedifferentiateEq.(1)withrespecttoz.

x x

y y

E HdE Hdz

P

xx

yy

EHdEHdz

Q

Eq.(1) Eq.(2)

22

2

2

00

x x

y y

E EdE Edz

PQ

Second,wesubstituteEq.(2)intothisresult.

2

2 x xx x

y yy y

E EH Hd d d d dE EH Hdz dz dz dz dz

P P

2

2x x

y y

E EdE Edz

P Q

Lecture5 Slide8

NumericalSolution(1of3)Thesystemofequationstobesolvedis

22 2

2

0

0x x

y y

E EdE Edz

PQ

Thishasthegeneralsolutionof x z zy

E ze e

E z

a a proportionality constant of forward waveproportionality constant of backward wave

aa

Note:Herewearefinallyassumingasignconventionofejz forforwardpropagationinthe+z direction.

Nomodesorting! Here,wesolvedasecondorderdifferentialequationwherethemodeswecalculateareoneway.Wesimplywritethemtwiceforforwardandbackwardwaves.Beforewesolvedafirstorderdifferentialequationthatlumpedforwardandbackwardmodestogether.

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Lecture5 Slide9

NumericalSolution(2of3)RecallfromLecture4

1 1 z z z ze e e e W W W W

Sotheoverallsolutioncannowbewrittenas 1 1x z zy

E ze e

E z

W W a W W a

2

2 2

Eigen-vector matrix of Eigen-value matrix of

W

21

22

2N

z

zz

z

e

ee

e

1f f A W WWecanusethisrelationtocomputethematrixexponentials.

Lecture5 Slide10

NumericalSolution(3of3)Sotheoverallsolutioncannowbewrittenas

1 1x z zy

E ze e

E z

ccW W a W W a

Thecolumnvectorsa+ anda areproportionalityconstantsthathavenotyetbeendetermined.TheeigenvectormatrixWmultipliesa+ anda togiveanothercolumnvectorofundeterminedconstants.Tosimplifythemath,wecombinetheseproductsintonewcolumnvectorslabeledc+ andc .

x z zy

E ze e

E z

W c W c

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Lecture5 Slide11

AnalyticalExpressionsforW and

Usingthisrelation,wecansimplifythematrixequationfor2.2 2 2

2 22 2 2

010

x y r r x x y r r x zz

r r y r r x y y r r x y z

k k k k k k kk

k k k k k k k

PQ I

Sincethisisadiagonalmatrix,wecanconcludethat

2 2

1 00 1

W I

Thedispersionrelationwithnormalizedwavevectorsis2 2 2

r r x y zk k k

0

0z

zz

jkjk

jk

I

0

0

z

z

jk zz

jk z

ee

e

1 0 identity matrix

0 1 I

Wedontactuallyhavetosolvetheeigenvalueproblemtoobtaintheeigenmodes!

Lecture5 Slide12

SolutionfortheMagneticField(1of2)

Sincetheelectricandmagneticfieldsarecoupledandnotindependent,weshouldbeabletocomputeV fromW.First,wedifferentiatethesolutionwithrespecttoz.

Themagneticfieldwillhaveasimilarsolution,butwillhaveitsowneigenvectormatrixV todescribeitsmodes.

x z zy

H ze e

H z

V c V c

x z zy

H zd e eH zdz

V c V c

Weputtheminussigninthesolutionheresothatbothtermsinthedifferentiated equationwillbepositive.

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Lecture5 Slide13

SolutionfortheMagneticField(2of2) x z zy

H zd e eH zdz

V c V c

Wenowhave

xx yy

E zH zdE zH zdz

Q

Recallthat x z zy

E ze e

E z

W c W cand

Combiningtheseresultsleadsto z z z z

z z

e e e e

e e

V c V c Q W c W c

QW c QW c

Comparingthetermsshowsthat1 V QW V QW

Lecture5 Slide14

CombinedSolutionforEandHElectricFieldSolution

1 0

0 1x z z

y

E ze e

E z

W c W c Wamplitude coefficients of forward waveamplitude coefficients of backward waveeigen-vector matrix

ccW

CombinedSolution

1 x z zy

H ze e

H z

V c V c V QW

xz

yz

x

y

E zE z e

zH z eH z

W W 0 cV V 0 c

MagneticFieldSolution

Doesthisequationlookfamiliar?ThisisthesameequationwehadattheendofLecture4.

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Lecture5 Slide15

TwoPathstoCombinedSolution

0

0

r

r

E k H

H k E

MaxwellsEquations FieldSolution

2

2

yz yz zy x y yz zx yz zyzx x

y x x yx yyzz zz zz zz zz zz zz zz

zy yxz zx xzxy x y

zz zz zz zz zzy

x

y

k k kj k k jk k

kE jk j k kEHzH

2

2

x y xz zyxz zxxx xy

zz zz zz

x y yz zx yz zy yz yz zyx zxyx yy y x x

zz zz zz zz zz zz zz zz

y x yxz zxxx

zz zz z

k k

k k k j k k jk

k k k

x

y

x

y

xz zy zyxz zx xzxy y x y

z zz zz zz zz zz

EEHH

jk j k k

2

2

2

2

1

1

x y r r x

r y r r x y

x y r r x

r y r r x y

k k k

k k k

k k k

k k k

P

Q

44Matrix SortEigenModes

PQ Method

Nosorting!Isotropicordiagonallyanisotropic

Anisotropic

E E

H H

zz

z

ee

e

W WW

W W

0

0

x

y

x

y

EE

HH

zE E

zH H

z

z

ez

e

ez

e

0W W cV V c0

W W 0 cV V 0 c

Lecture5 Slide16

ScatteringMatrixforaSingleLayer

R.C.Rumpf,ImprovedFormulationofScatteringMatricesforSemiAnalyticalMethodsThatisConsistentwithConvention,PIERSB,Vol.35,pp.241261,2011.

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Lecture5 Slide17

MotivationforScatteringMatricesScatteringmatricesofferseveralimportantfeaturesandbenefits:

Unconditionallystablemethod. Parametershavephysicalmeaning. Parameterscorrespondtothosemeasuredinthelab. Canbeusedtoextractdispersion. Verymemoryefficient. Canbeusedtoexploitlongitudinalperiodicity.Matureandprovenapproach.Muchgreaterwealthofliteratureavailable.

ExcellentalternativestoSmatricesdoexist!

Lecture5 Slide18

GeometryofaSingleLayer

thfield within layeri z i th

th

mode coefficients inside layer

mode coefficients outside layeri

i

ii

cc

Indicatesapointthatliesonaninterface,butassociatedwithaparticularside.

0i

0i

iL

i

i

cc

1

1

0i ik L

0i ik L

2

2

cc

1

1

cc

2

2

i iz

i iz

Medium 1 Layer i Medium 2

z

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Lecture5 Slide19

FieldRelationsFieldinsidetheith layer:

,

,

,

,

i i

i i

x i iz

y i i i i ii i z

x i i i i i

y i i

E zE z ezH z eH z

W W c0V V c0

Boundaryconditionsatthefirstinterface: 1

1 1 1

1 1 1

0i

i i i

i i i

W WW W ccV VV V cc

Boundaryconditionsatthesecondinterface:

0

0

0 2

2 2 2

2 2 2

i i

i i

i i

k Li i i

k Li i i

k L

ee

W W W Wc c0V V V Vc c0

Note:k0 hasbeenincorporatedtonormalizeLi.

Lecture5 Slide20

DefinitionofAScatteringMatrix11 121 1

21 222 2

S Sc cS Sc c

1c

1c

11S

12S

22S

21S

11

21

reflectiontransmission

SS

Thisisconsistentwithexperimentalconvention.

reflection

transmission

2c

2c

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Lecture5 Slide21

DerivationoftheScatteringMatrix

11 1 1

1 1 1

i ii

i ii

W W W Wc cV V V Vc c

0

0

12 2 2

2 2 2

i i

i i

k Li ii

k Li ii

ee

W W W Wc c0V V V Vc c0

Solvebothboundaryconditionequationsfortheintermediatemodecoefficientsand.ic ic

Bothoftheseequationshave1 1 1

1 11 2

j j ij iji i ij i j i j

j j ij iji i ij i j i j

W W A BW W A W W V VV V B AV V B W W V V

Wesetsubstitutethisresultintothefirsttwoequationsandsetthemequal.

0

0

1 1 2 21 2

1 1 2 21 2

1 12 2

i i

i i

k Li i i i

k Li i i i

ee

A B A Bc c0B A B Ac c0

Wewritethisastwomatrixequationsandrearrangetermsuntiltheyhavetheformofascatteringmatrix.

1 1

2 2

? ?? ?

c cc c

Lecture5 Slide22

TheScatteringMatrixThescatteringmatrixSi oftheithlayerisdefinedas:

Aftersomealgebra,thecomponentsofthescatteringmatrixarecomputedas

1 12 2

i

c cS

c c

11 12

21 22

i ii

i i

S S

SS S

1 1

1 1

ij i j i j

ij i j i j

A W W V V

B W W V V

0i ik Li e

X

11 111 1 2 2 1 2 2 1 1

11 112 1 2 2 1 2 2 2 2

11 121 2 1 1 2 1 1 1 1

11 122 2 1 1 2 1 1 2 2

ii i i i i i i i i i i i

ii i i i i i i i i i i



S A X B A X B X B A X A B

S A X B A X B X A B A B



iS

i isthelayernumber.j iseither1or2dependingonwhichexternalmediumisbeingreferenced.

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Lecture5 Slide23

ScatteringMatricesintheLiteratureForsomereason,thecomputationalelectromagneticscommunityhas:(1)deviatedfromconvention,and(2)formulatedscatteringmatricesinefficiently.

1ic

1ic

ic

ic

11 121

21 22 1

i i

i i

S Sc cS Sc c

12S

11S

21S

22S

reflection

transmission

Heres11 isnotreflection.Instead.itisbackwardtransmission!

Heres21 isnottransmission.Instead,itisareflectionparameter!

Scatteringmatricescannotbeinterchanged. Scatteringmatricesarenotsymmetricsotheytaketwicethememorytostoreandaremoretimeconsumingtocalculate.

Lecture5 Slide24

LimitationofConventionalSMatrixFormulationNotethattheelementsofascatteringmatrixareafunctionofmaterialsoutsideofthelayer.

Thismakesitdifficulttointerchangescatteringmatricesarbitrarily.Forexample,thereareonlythreeuniquelayersinthemultilayerstructurebelow,yet20separatecomputationsofscatteringmatricesareneeded.

Threeuniquelayers

20layerstack

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Lecture5 Slide25

SolutionTogetaroundthis,wewillsurroundeachlayerwithanexternalregionsofzerothickness.Thisletsusconnectthescatteringmatricesinanyorderbecausetheyallcalculatefieldsthatexistoutsideofthelayersinthesamemedium.Thiswillhavenoeffectelectromagneticallyaslongaswemaketheexternalregionshavezerothickness.

iL

Layer iGap Medium Gap Medium

Lecture5 Slide26

VisualizationoftheTechnique

Threeuniquelayers

Wecalculatethescatteringmatricesforjusttheuniquelayers.

Thenwejustmanipulatethesesamethreescatteringmatricestobuildtheglobalscatteringmatrix.

Gapsbetweenthelayersaremadetohavezerothicknesssotheyhavenoeffectelectromagnetically.

Faster! Simpler! Less memory needed!

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Lecture5 Slide27

RevisedGeometryofaSingleLayer

0i

0i

iL

i

i

cc

1

1

0i ik L

0i ik L

2

2

cc

1

1

cc

2

2

i iz

i iz

Layer iGap Medium Gap Medium

,

,

r h

r h

,

,

r h

r h

Lecture5 Slide28

CalculatingRevisedScatteringMatricesThescatteringmatrixSi oftheithlayerisstilldefinedas:

Buttheelementsarecalculatedas

1 12 2

i

c cS

c c

11 12

21 22

i ii

i i

S S

SS S

1 1

1 1i i h i h

i i h i h

A W W V VB W W V V

0i ik Li e

X

11 111

11 112

21 12

22 11



i i

i i



S S

S S

Layersaresymmetricsothescatteringmatrixelementshaveredundancy. Scatteringmatrixequationsaresimplified. Fewercalculations. Lessmemorystorage.

iS

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Lecture5 Slide29

ScatteringMatricesofLosslessMediaIfascatteringmatrixiscomposedofmaterialsthathavenolossandnogain,thescatteringmatrixmustconserveenergy.Thatis,allincidentenergymusteitherreflectortransmit.Thisimpliesthatthescatteringmatrixisunitary.Ifthescatteringmatrixisunitary,itmustobeythefollowingrules:

1

1 1

H

H H

S SS S SS S S SS I

HintsAboutStabilityinTheseFormulations DiagonalelementsS11 andS22 tendtobethelargestnumbers.Dividebytheseinsteadofanyoffdiagonalelementsforbestnumericalstability.

X describespropagationthroughanentirelayer.DontdividebyX oryourcodecanbecomeunstable.

Lecture5 Slide30

11 12

21 22

S S

SS S

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Lecture5 Slide31

MultilayerStructures

Lecture5 Slide32

SolutionUsingScatteringMatricesThescatteringmatrixmethodconsistsofworkingthroughthedeviceonelayeratatimeandcalculatinganoverallscatteringmatrix.

1S 2S 3S 4S 5S

device 1 2 3 4 5 S S S S S SRedheffer starproduct.NOTmatrixmultiplication!

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Lecture5 Slide33

Redheffer StarProductTwoscatteringmatricesmaybecombinedintoasinglescatteringmatrixusingRedheffers starproduct.

A A

11 12

A A21 22

A S S

SS S

B B

11 12

B B21 22

B S S

SS S

AB A B S S S

Thecombinedscatteringmatrixisthen

AB AB

11 12

AB AB21 22

AB S S

SS S

1AB A A B A B A11 11 12 11 22 11 21

1AB A B A B12 12 11 22 12

1AB B A B A21 21 22 11 21

1AB B B A B A B22 22 21 22 11 22 12

S S S I S S S S

S S I S S S

S S I S S S

S S S I S S S S

R. Redheffer, Difference equations and functional equations in transmission-line theory, Modern Mathematics for the Engineer, Vol. 12, pp. 282-337, McGraw-Hill, New York, 1961.

Lecture5 Slide34

DerivationoftheRedheffer StarProductWestartwiththeequationsforthetwoadjacentscatteringmatrices.

A A B B11 12 11 122 21 1A A B B

3 32 221 22 21 22

S S S Sc cc c

c cc cS S S S

Weexpandtheseintofourmatrixequations.

A A B B1 11 1 12 2 2 11 2 12 3

A A B B2 21 1 22 2 3 21 2 22 3

Eq. 1 Eq. 3

Eq. 2 Eq. 4

c S c S c c S c S c

c S c S c c S c S c

WesubstituteEq.(2)intoEq.(3)togetanequationwithonly.2cWesubstituteEq.(3)intoEq.(2)togetanequationwithonly.2c

B A B A B

11 22 2 11 21 1 12 3

A B A A B22 11 2 21 1 22 12 3

Eq. 5

Eq. 6

I S S c S S c S c

I S S c S c S S c

WeeliminateandbysubstitutingtheseequationsintoEq.(1)and(4).Wethenrearrangetermsintotheformofascatteringmatrix.

2c 2c

1 1

3 3

? ?? ?

c cc c

Overall,thisisjustalgebra.Westartwith4equationsand6unknownsandreduceitto2equationswith4unknowns.

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Lecture5 Slide35

PuttingitAllTogether

global 1 2

Device in gap me

r

dium

f trne N SS SS S S

Wehaveoneremainingproblem.Intherevisedframework,theglobalscatteringmatrixplacesthedeviceinfreespace.Inmanyapplications,wemaywantsomethingotherthanfreespaceoutsidethedevice.Weconnecttheglobalscatteringmatrixtotheexternalmaterialsbysurroundingitbyconnectionscatteringmatricesthathavezerothickness

deviceS

Deviceexistswithingapmedium

Lecture5 Slide36

Reflection/TransmissionSideScatteringMatricesThereflectionsidescatteringmatrixis

ref 111 ref ref

ref 112 ref

ref 121 ref ref ref ref

ref 122 ref ref

2

0.5

S A B

S A

S A B A B

S B A

1 1ref ref ref

1 1ref ref ref

h h

h h

A W W V VB W W V V

trn 111 trn trn

trn 112 trn trn trn trn

trn 121 trn

trn 122 trn trn

0.5

2

S B A

S A B A B

S A

S A B

1 1trn trn trn

1 1trn trn trn

h h

h h

A W W V VB W W V V

Thetransmissionsidescatteringmatrixis

,I

,I

r

r

,

,

r h

r h

0limL

,II

,II

r

r

,

,

r h

r h

refs

trns

0limL

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Lecture5 Slide37

CalculatingTransmittedandReflectedPower

Lecture5 Slide38

RecallHowtoCalculateSourceParameters

inck

TEaTMa

inc 0 inc

sin cossin sin

cosk k n

0 0

1n

incTE

inc

0 0

ya

k nak n

TE incTM

TE inc

a kaa k

IncidentWaveVector SurfaceNormal UnitVectorsAlongPolarizations

CompositePolarizationVectorTE TMTE TM P p p aa

z

x

y

Righthandedcoordinatesystem

1P InCEM,wemake

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Lecture5 Slide39

SolutionUsingScatteringMatricesTheexternalfields(i.e.incidentwave,reflectedwave,transmittedwave)arerelatedthroughtheglobaltransfermatrix.

globalref inctrn

c c

Sc 0

Thismatrixequationcanbesolvedtocalculatethemodecoefficientsofthereflectedandtransmittedfields.

global global

ref 11 12 incglobal global

trn 21 22

c S S cc 0S S

global

ref 11 inc

globaltrn 21 inc

c S c

c S c

,inc1inc ref

,inc

x

y

EE

c W

inc c

rightinc not typically usedc

,inc

,inc

x x

y y

E PE P

WegetEx,inc andEy,inc fromthepolarizationvectorP.

Lecture5 Slide40

CalculationofTransmittedandReflectedFieldsTheproceduredescribedthusfarcalculatedcref andctrn.Thetransmittedandreflectedfieldsarethen

ref incglobal global 1

ref ref ref 11 inc ref 11 refref inc

trn incglobal global 1

trn trn trn 21 inc trn 21 reftrn inc

x x

y y

x x

y y

E EE E

E EE E

W c W S c W S W

W c W S c W S W

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Lecture5 Slide41

CalculationoftheLongitudinalComponentsWearestillmissingthelongitudinalfieldcomponentEz oneithersizeofthelayerstack.ThesearecalculatedusingMaxwellsdivergenceequation.

0, 0, 0,0, 0, 0,

0, 0, 0,

0, 0, 0,

0, 0,0,

0

0

0

0

jk r jk r jk rx y z

jk r jk r jk rx x y y z z

x x y y z z

z z x x y y

x x y yz

z

E

E e E e E ex y z

jk E e jk E e jk E ek E k E k Ek E k E k E

k E k EE

k

ref refref

ref

trn trntrn

trn

x x y yz

z

x x y yz

z

k E k EE

k

k E k EE

k

Note:

0 reduces to

0 when is homogeneous.

E

E

Lecture5 Slide42

CalculationofPowerFlow

2

trn ,ref ,trn2

,trn ,incinc

Re r zr z

E kT

kE

1 materials have loss 1 materials have no loss and no gain

1 materials have gain R T

Reflectanceisdefinedasthefractionofpowerreflectedfromadevice.2

ref2

inc

ER

E 2 22 2x y zE E E E

Transmittanceisdefinedasthefractionofpowertransmittedthroughadevice.

Itisalwaysgoodpracticetocheckforconservationofenergy.

Note:WewillderivetheseformulasinLecture7.

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Lecture5 Slide43

ReflectanceandTransmittanceonaDecibelScaleDecibelScale

dB 1020 logP A dB 1010 logP P

Howtocalculatedecibelsfromanamplitudequantity.

Howtocalculatedecibelsfromapowerquantity. 2 2dB 10 10 10 log 20logP A P A A

ReflectanceandTransmittanceReflectanceandtransmittancearepowerquantities,so

dB 10

dB 10

10 log

10log

R R

T T

Lecture5 Slide44

NotesonImplementation

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Lecture5 Slide45

StoringtheProblemHowisathedevicedescribedandstoredforTMM?Wedontuseagridforthismethod!Storethepermittivityforeachlayerina1Darray.Storethepermeabilityforeachlayerina1Darray.Storethethicknessofeachlayerina1Darray.

ER = [ 2.50 , 3.50 , 2.00 ];UR = [ 1.00 , 1.00 , 1.00 ];L = [ 0.25 , 0.75 , 0.89 ];

Wewillalsoneedtheexternalmaterials,andsourceparameters.er1,er2,ur1,ur2,theta,phi,pte,ptm,andlam0

Inputarraysforthreelayers

Lecture5 Slide46

StoringScatteringMatricesWeoftentalkaboutthescatteringmatrixS asasinglematrix.

11 12

21 22

S S

SS S

However,weneveractuallyusethescatteringmatrixS thisway.WeonlyeverusetheindividualtermsS11,S12,S21,andS22.So,scatteringmatricesareactuallystoredasthefourseparatecomponentsofthescatteringmatrix.

11 12

21 22

S S

SS S 11 12 21 22

, , , and S S S S

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24

Lecture5 Slide47

CalculatingXi = exp(-ik0Li)

0

0

0

0

0

z i

i i

z i

jk k Lk L

i jk k L

ee

e

X

Recallthecorrectanswer:

Itisincorrecttousethefunctionexp() becausethiscalculatesapointbypointexponential,notamatrixexponential.

X = exp(-OMEGA*k0*L);X =

0.0135 + 0.9999i 1.00001.0000 0.0135 + 0.9999i

Approach#1:expm() Approach#2:diag()X = expm(-OMEGA*k0*L);X =

0.0135 + 0.9999i 00 0.0135 + 0.9999i

X = diag(exp(-diag(OMEGA)*k0*L));X =

0.0135 + 0.9999i 00 0.0135 + 0.9999i

Lecture5 Slide48

EfficientStarProductAfterobservingtheequationstoimplementtheRedheffer starproduct,weseetherearesomecommonterms.Calculatingthesemultipletimesisinefficientsowecalculatethemonlyonceusingintermediateparameters.

1AB A A B A B A11 11 12 11 22 11 21

1AB A B A B12 12 11 22 12

1AB B A B A21 21 22 11 21

1AB B B A B A B22 22 21 22 11 22 12

S S S I S S S S

S S I S S S

S S I S S S

S S S I S S S S

1A B A12 11 22

1B A B21 22 11

D S I S S

F S I S S

AB A B S S S

A B A11 11 11 21

B12 12

A21 21

B A B22 22 22 12

AB

AB

AB

AB

S S DS S

S DS

S FS

S S FS S

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Lecture5 Slide49

UsingtheStarProductasanUpdateVeryoftenweupdateourglobalscatteringmatrixusingastarproduct.

Whenweusethisequationasanupdate,weMUSTpaycloseattentiontotheorderthatweimplementtheequationssothatwedontaccidentallyoverwriteavaluethatweneed.

1global12 11 22

1global global21 22 11

global global22 22 22

global21 21

global global12 12

global global11 11

global1

1

2

1 21

i i

i

i

i

i i

D S I S S

F S I S S

S S FS

S FS

S DS

S S DS S

S

1global global12 11 22

1global21 22 11

global global global11 11 11 21

global12 12

global global21 21

global global22 22 22 12

i

i i

i

i

i i

D S I S S

F S I S S

S S DS S

S DS

S FS

S S FS S

global globali S S S global global i S S S

reverseo

rder

standard

orde

r

Lecture5 Slide50

SimplificationsforTMMinLHIMediaInLHImedia,

1 00 1i W I ,i z ijk I

1 0 identity matrix0 1 Iand

Nowwedonotactuallyhavetocalculate becausei i

Givenallofthis,theeigenvectorsforthemagneticfieldscanbecalculatedas1 1

i i i i i i V Q W Q

Whencalculatingscatteringmatrices,theintermediatematricesAi andBi are1 1 1

1 1 1

i i h i h i h

i i h i h i h

A W W V V I V V

B W W V V I V V

Thefieldsandmodecoefficientsarenowrelatedthroughref trn

1inc ref ref 11 inc 11 inc trn 21 inc 21 incref trn

x x x x

y y y y

P P E EP P E E

c W W S c S c W S c S c

3/28/2014

26

Lecture5 Slide51

InitializingtheGlobalScatteringMatrixBeforeweiteratethroughallthelayers,wemustinitializetheglobalscatteringmatrixasthescatteringmatrixofnothing.Whataretheidealpropertiesofnothing?

1. Transmits100%ofpowerwithnophasechange.

2. Doesnotreflect.

global global12 21 S S I

global global11 22 S S 0

Wethereforeinitializeourglobalscatteringmatrixas global

0 IS

I 0ThisisNOTanidentitymatrix!Lookatthepositionofthe0sandIs.

Lecture5 Slide52

CalculatingtheParametersoftheHomogeneousGaps

2 2 2, , ,

,

1

z h r h r h x y

h

h z h

h h h h

k k k

jk

W I

IV Q W

Ouranalyticalsolutionforahomogeneouslayeris2

, ,

2, , ,

1 x y r h r h xh

r h y r h r h x y

k k k

k k k

Q

Wearefreetochooseanyr andr thatwewish.Wealsowishtoavoidthecaseofkz,h = 0.Forconvenience,wechoose

2 2, ,1.0 and 1r h r h x yk k

Wethenhave

2

2

1

1x y y

hx x y

k k k

k k k

Q

h

h hj

W IV Q

3/28/2014

27

Lecture5 Slide53

BlockDiagramofTMMUsingSMatrices

CalculateParametersforLayeri

2 2,

2

2

1,

1

z i i i x y

x y i i xi

i y i i x y

i z i i i i

k k k

k k k

k k k

jk

Q

I V Q

CalculateScatteringMatrixforLayeri

0

1 1

11 111 22

11 112 21

i i

i i h i i hk L

i

i ii i i i i i i i i i i i

i ii i i i i i i i i i i

e

A I V V B I V V

X

S S A X B A X B X B A X A B

S S A X B A X B X A B A B

UpdateGlobalScatteringMatrix

global global global global11 12 11 12 11 12

global global global global21 22 21 22 21 22

1global global global global global11 11 12 11 22 11 21

global global12 12 1

i i

i i

i i

S S S S S S

S S S S S S

S S S I S S S S

S S I S

1global1 22 12

1global global global21 21 22 11 21

1global global global22 22 21 22 11 22 12

i i

i i

i i i i

S S

S S I S S S

S S S I S S S S

Done?

no

yes

CalculateTransmittedandReflectedFields

ref

11 incref

trn

21 inctrn

x

y

x

y

EE

EE

S c

S c

CalculateLongitudinalFieldComponents

ref refref

ref

trn trntrn

trn

x x y y

zz

x x y yz

z

k E k EE

k

k E k EE

k

CalculateTransmittanceandReflectance

2

ref

trn2 reftrn ref

trn

Re zz

R E

kT Ek

CalculateTransverseWaveVectors

inc

inc

sin cos

sin sinx

y

k n

k n

InitializeGlobalScatteringMatrix

global 0 I

SI 0

Start

Finish

CalculateGapMediumParameters h h hj W I V Q

CalculateSource

TE TE TM TM

inc

1

x

y

P p a p a

P

PP

c

ConnecttoExternalRegions

global ref global

global global trn

S S S

S S S

Loopthroughalllayers

Lecture5 Slide54

HowtoHandleZeroLayersFollowtheblockdiagram!!Setupyourloopthisway

NLAY = length(L);for nlay = 1 : NLAY

...end

IfNLAY = 0,thentheloopwillnotexecutetheglobalscatteringmatrixwillremainasitwasinitialized.

global 0 I

SI 0

Forzerolayers:ER = [];UR = [];L = [];

3/28/2014

28

Lecture5 Slide55

CanTMMFail?Yes!

TheTMMcanfailtogiveananswerandbehavenumericallystrangeanytimekz = 0.Thishappensatacriticalanglewhenthetransmittedwaveisverynearitscutoff.Wefixedthisprobleminthegapmedium,butthiscanalsohappeninanyofthelayersorinthetransmissionregion.

2 2r r x yk k

Thishappensinanymediumwhere

Lecture5 Slide56

AdvancedScatteringMatrices

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29

Lecture5 Slide57

LongitudinallyPeriodicDevicesSupposewejustcalculatedthescatteringmatrixfortheunitcellofalongitudinallyperiodicdevice.

UnitCell1 UnitCell2 UnitCell3 UnitCell4 UnitCell4 UnitCell6 UnitCell7 UnitCell8

UnitCell

1 A B C S S S S

Thereexistsaveryefficientwayofcalculatingtheglobalscatteringmatrixofalongitudinallyperiodicdevicewithoutcalculatingandcombiningalltheindividualscatteringmatrices.

A B C

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 S S S S S S S S S S S S S S S S S S S S S S S S S

Bothareinefficient!!! 8 1 1 1 1 1 1 1 1 S S S S S S S S S

Lecture5 Slide58

CascadingandDoublingWecanquicklybuildanoverallscatteringmatrixthatdescribeshundredsandthousandsofunitcells.Westartbycalculatingthescatteringmatrixforasingleunitcell.

1 A B C S S S S A B C

Next,wekeepconnectingthescatteringmatrixtoitselftokeepdoublingthenumberofunitcellsitdescribes.

2 1 1 S S S 4 2 2 S S S 8 4 4 S S S

3/28/2014

30

Lecture5 Slide59

AlgorithmforArbitraryNumberofUnitCellsStep1 Calculatethescatteringmatrixfortheunitcell.

Step2 Determinethebinarydigitsforthetotalnumberofrepetitions.

Chainof10unitcells 1010Step3 Performcascadingadoublingwhileupdatingtheglobalscatteringmatrixonlywiththescatteringmatricescorrespondingtobinarydigitsof1.

1 A B C S S S S

bin bin binb. c. repeat through all binary digits

S S SLoop#binarydigits

bin 1

N

0 IS

I 0

S S

a.Ifbinarydigitis1 binN N S S S

A B C

1.InitializeAlgorithm 2.Performmodifiedcascadinganddoubling

Lecture5 Slide60

BlockDiagramforModifiedCascadingandDoublingAlgorithmInputs

1 S-matrix of one unit cellNumber of times to repeat unit cellN

S

ConvertN tobinary10110

InitializeAlgorithm bin 1 N

0 IS S S

I 0

Done?

UpdateDoubling bin bin bin S S S

digit=1? UpdateS(N)

binN N S S Syes

no

no

Output NSLoopthroughallbinarydigits

startingwiththeleastsignificantdigit.

Example

N = 22 101100: No update to S(N)

S(bin) now 2 unit cells

1: S(N) now 2 unit cellsS(bin) now 4 unit cells


0: No update to S(N)S(bin) now 16 unit cells


3/28/2014

31

Lecture5 Slide61

DispersionAnalysis(1of2)Anoverallscatteringmatrixiscalculatedthatdescribestheunitcell.

uc uc

11 110 0uc uc

1 111 11N N

S Sc c

c cS S

Thetermsarerearrangedinalmosttheformofatransfermatrix.

uc uc12 111 0uc uc

1 022 21

N

N

0 S S Ic c

c cI S S 0

Ifthedeviceisinfinitelyperiodicinthez direction,thenthefollowingperiodicboundaryconditionmusthold.

1 0

1 0

zzjN

N

ke

c cc c

Herekz istheeffectivepropagationconstantofthemode.

uc 1same as on previous slideS S

Lecture5 Slide62

DispersionAnalysis(1of2)Wesubstitutetheperiodicboundaryconditionintoourrearrangedequationtoget

uc uc11 120 0uc uc

0 021 22

z zjke

S I 0 Sc c

c cS 0 I S

Thisisageneralizedeigenvalueproblem.

uc11 0uc

021

uc12

uc22

zzkje

S I cA x

cS 0Ax Bx

0 SB

I S

[V,D] = eig(A,B);Eigen vectors Eigen values

3/28/2014

32

Lecture5 Slide63

AlternativestoScatteringMatrices

Lecture5 Slide64

TransmittanceMatrices(TMatrices)TheTmatrixmethodisthetransfermatrixmethodwhereforwardandbackwardwavesaredistinguished.

lefttrn 11 12 inc

rightinc 21 22 ref

c T T cc T T c

BenefitsMuchfaster(5to10times)Unconditionallystable

Drawbacks LessmemoryefficientCannotexploitlongitudinalperiodicity Lesspopularintheliterature

M. G. Moharam, Drew A. Pommet, Eric B. Grann, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach, J. Opt. Soc. Am. A, Vol. 12, No. 5, pp. 1077-1086, 1995.

3/28/2014

33

Lecture5 Slide65

HybridMatrices(HMatrices)Thehmatrixmethodisborrowedfromelectricaltwoportnetworks.

1 11 12 1

2 21 22 2

V h h II h h V

2 1

2 1

1 111 12

2 20 0

2 221 22

1 20 0

V I

V I

V Vh hI V

I Ih hI V

Intheframeworkoffields,thehmatrixisdefinedas

, 1 , 1

, 1 11 12 , 1

, ,21 22

, ,

x i x ii i

y i y ii i

x i x i

y i y i

E HE HH EH E

H H

H H

ClaimedBenefits ImprovednumericalstabilityMoreconciseformulation Simplertoimplement Improvednumericalefficiency(30%betterthanETM)UnconditionallystableEng L. Tan, Hybrid-matrix algorithm for rigorous coupled-wave analysis of

multilayered diffraction gratings, J. Mod. Opt., Vol. 53, No. 4, pp. 417-428, 2006.

Lecture5 Slide66

RMatricesTheRmatrixmethodisessentiallytheimpedancematrixframeworkborrowedfromelectricaltwoportnetworks.

1 11 12 1

2 21 22 2

V z z IV z z I

2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

I I

I I

V Vz zI I

V Vz zI I

Intheframeworkoffields,thehmatrixisdefinedas

, 1 , 1

, 1 11 12 , 1

, ,21 22

, ,

x i x ii i

y i y ii i

x i x i

y i y i

E HE HE HE H

R R

R R

ClaimedBenefitsUnconditionallystable Improvednumericalefficiency

Lifeng Li, Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings, J. Opt. Soc. Am. A, Vol. 11, No. 11, pp. 2829-2836, 1994.

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