Prolate Plane

Electromagnetic Plane Wave Scattering by Uniformly Lossy

Dielectric Prolate Spheroids

By

© ^{Soumya Nag, B. Eng.}

A thesissubmitt ed to the Schoolof Graduate Studi es in partialfulfillment ofthe requirem ents forthedegr ee of

Masterof Engineerin g

Facul ty of Engin eer ing and AppliedScience Mem orial Unive rsity of Newfoundland

August1994

St.John's Newfoundl and Canada

.+.

N3lional library

ctceoaca ~~b1~el\aliOnale DifectiondP.sllCQUiSilionsel desserv'.cesl:Jib{lOQfaphiques

...

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~~I

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Abstract

Using modal ser ies ex p ansions ofelect roma gnet ic fieldsin termsof prolatespheroidalvectorwav efunctions ,anexactsolut ionj"obtained forthescatter ing by:(a) a singleuniformlylossy dielectricprolate spheroid,(b)a syst em of twouniform lylossydiele ctric!>rolatllsphe ro ids in arbitrary orient a t ion,and(e) asanim port ant special case asystem of two uniformly lossy dielectricprolate sphe roi ds in parallel config u ra- tion embe ddedin freespace.Inalltheabove cases,theexcita ti on being a mon ochromatic pla ne electrom agneticwaveofarbit ra rypola riza t ion andan gle of incid en ce.Sincethediel ectricmate rial s of the ecett e rers areofcom plex rela tiv epermittivitie s,compl ex eige nvalues areeva lu- atedCor thespheroidalscalar wave functi on s of transmitted co m po - nents of E-field and H-fieldex pansions.Rotational- Tr an slationalAd- ditionTheoremsand 1'l'anslationalAdditionTheo r e ms fornphe eoid nl vecto rwavefunctions areusedto studythe scatteringby a system of twospheroids inar bit rary configurationand asa:peciaJ caseby a syst emof two sphe r oids in parallelconfigu ra t io n re spectivel y.Thes e theorems are used totransform the out going wavefromonesphero id intothe incoming wave at theotherspheroid.The field solutio nde- terminesthecolumn vectoroftheunknown coefficients of theseries expansionsof thescattered and transmittedfields ex pre ssed.in terms ofthe columnvectoroCtheknown coefficients of~.heser ies expenetcne ofthe incident field and the system matrixwhichis indep endentofthe directionand polarizationof theincident wave. Numerical resultB in the form of plotsfornorm ali sed bistatic and monostaticradar cross

sectionsaregiven foravariet y of uniforml y lossy dielect r icpro late spheroidswit hresonan t or nea rreso nantleng ths. Als o for two- b ody ecat t e elng, differentar b itraryconfigurationsind ud ingparalle l config- ur a t ion s of the sphe ro ids at diffe rentdista ncesofseparation have been considered.

Acknowledgment

Iwouldlike tothan k&lidexp ressmygra titude tomylupe rvifOfDr. B.P.

Sinha for givingmeanexcellentopportu nitytoworkunderhi.lupervi .ion.Hi.

conatanttechnicalfeedback anduseful suggestion. helped me immen selyin my thesi. workandhavemad e my researehreallyenjoyable.

I wouldall Olike toexprcumythanksto SchoolofGraduateStudie.andDr.

J.J.Sharp,FacultyofEngineeringandAppliedScience,MemorialUniveraity ofNewfound landfor givingmefinancialsupportthrough out the periodof my gladuatestudi es.

Luthutnottheleast,Iamthankfulto myfriendsMr.Bibhu Bhattacharya and Mr.Tapu Banerjee{or giving mevaluableluggestion.in the development of,oftware pertainingtomy thesiswork.

iii

Contents

Abdract

Acknowledge me nt

1 Introduction 1.1 Literature Review.

1.2 Organization of theThesis.. . . .. . . . .. •

iii

2 ProlateSphe roidalCoordinate&and Prolat~SpheroidaJWa ve Fun ctions

2.1 Introd~ction..• ..• •. . . ...• • •.•.. • •..•. .•

2.2 ProlateSpheroidalCoordinates .

2.3 VectorHelmholtzEquations . ...•... . . .... 13 2.4 ProlateSpheroidalDifferentialEquations .. ... . .... .... 15 2.4.1 ProlateAngle Functions ••....•.. .•.• • •. .•. 17 2.4.2 Prolate RadiAlFunct ions... • .. ..•.. . .•..• 18 2.5 ProlateSpheroidal Vector Wave Funct ions . 20

3 ElectromagneticPlaneWaveScatteringbySingleUnirormlyLos sy

DielectricProlateSp he ro id 2S

3.1 Introduction.. ... ... . .... . . .. .

iv

25

3.2 Expansion of the Incident ElectricField in term. of Normalized ProlateSpheroidal Vector Wavc Functions

3.3 Expansionof the ScatteredElectric Field in termsof Normalized Prolate SpheroidalVectorWaveFunctions

3.4 Expansion of the 'IreuemittedElectricField in termsofNormal- izedProlateSpheroidal Vector WaveFunctions . 3.5 Expansion ofH-field interms of E-field .

3.6 Expansion oftheIncidentMagntticField in terms of Normalized ProlateSpheroidalVectorWave Functions

3.6.1 LimitingCase:"(p=0 and 8;_'/t/2..

3.7 Expansion ofthe ScatteredMagnetic Fieldin termsof Normalized Prolate SpheroidalVectorWave Functions

3.8 Expansion ofthe TransmittedMagnetic Fieldin termsofNormal- izedProlat eSpheroidalVectorWaveFunctions... 3.9 Applica.tionsofBoundary Conditions, _

3.9.1 ljI-matchingand 'I'1.matching 3.10 SystemMatrix[GI

3.11Far-Field Expansionsand Scatte ringCrosssections 3.12 Resultsof NumericalComputation

40 Electrom agn eticPlaneWaveScatteringbya Syst em of Two Uni- formlyLossyDielectric Pro late Spheroid sin ArbitraryConftgu- ra tion

4.1 Introduct ion .

4.2 Expansion oftheIncident ElectricFieldinterm' ofNormalized ProlateSpheroidal VectorWave Functions

27

29

30 32

33 34

34

35 35 36 37 38 40

4 .

49

50

4.3 Expansion ofthe ScatteredElectric Fieldin termsof Normalized

ProlateSpheroidalVector WaveFunctions 56

4.4 Expansion of theTransmitt edElect ricFieldinterms ofNormal- ized ProlateSpheroidal Vector WaveFunctions. 60 4.5 Expansionof Incident,Scatte redandTransmi tt edMagneticField,

in termsof NormalizedPfolat~SpheroidalVector WaveFunctions 62 4.6 Application ofBound ary Conditions.

4.7 SystemMatrix(GI ...,..

4.8 Far-FieldExpansionsand ScatteringCrosssections 4.9 Resultsof NumericalComputation ,

63 66 67 71

5 Electromag n et icPlan e Wave Scatte rin g by aSyst em of Two Par- allel Uni form lyLossyDielectricProlateSp he roids 82

5.1 Iatroducticn;. 82

5.2 Formulationofthe Problem .. 8'

5,3 Applicationof BoundaryCondition s.. 87

5.4 Far-Field Expansionsand Scattering Crosssections. 89

5.5 Result. of NumericalCompu t at ion 90

6 Conclusion 99

Bibli ogr aph y 101

A Co m p u t at ionofComp lex Eigen Values 107

B Defin iti onofElements of[GI 120

vi

D Defini t ion or Int egr als E Rotationa l-Tr an slati onal Ccefficients

130 13.

E.l TheEulerAngles... .. .•.• •. ... 138 B.2 Rotational-Tr&llslationalCoefficients •. .•.••..••. . .. .139 E.3 SpecialCase:Tr&llslationa1Coefficients... . . .. • . .•.143

vii

List of Figures

2.1 Prolate SpheroidalCoordinateSy.tem.. . .... .... ... 11 2.2 Prolate spheroidal geometryand cartcsiancoordinate... . 12

3.1 Scat~eringgeometry foralingleuniformlylouydielectr icprolate spheroidwit h arbit raryincidence andpolari zationofaplanedec-

tromagnetic wave .

3.2 Normalized bistaticere ectionforTE polarization ofincident wave, as afunction ofIcatt eringaogle for .ingleuniformlylony dielectric prolate spheroid withaxialratio2 ADd10,andof relati ve sizeskill

=

1,2,3and4for complexrelative permitt ivityf,.=

26

2-jO.5 43

3.3 Normalizedbilt aticeros.aection forTEpolarisationof incident wave,as " (ullction ofscatteringanglefor.insleuniformlylos.y dielect ric prolat espheroidwith axial ratio 2 &lid 10,and ofrelative .ize. kto

=

^{1,2,3and4forcomplexrelative}permi tt ivity e,= 4 -jO.5..•.

v1ii

44

3,4 Normalize d bista tic cross sect ion forTEpolarizationofincident wave,as a functionof scatteri ogangle for single uniformlylossy dielectr ic prolatespheroid withaxialrati o10,relat ive sizekill=2 and withdifferentvaluesofcomplexrelative permittivity:e,= 4 -;0;f~=4 - jO.2;f.=4-jO,4;f.=4 -jO.6;e,

=

4 - jO.8;

f.

=

^{4- j1i}f~

=

4 -j1.2 .. .

3.5 Norm alized monosta ticcross sect ion,asa functionof aspect angle forsingleuniformlylossydielectr icprolate spheroidwith axialra- tio 2 and 10,and ofrelative sizeskill=1,2 , 3 and 4 forcomplex relative perm itt ivityf.=2 -jO.5..

3.6 Norm alized monostati c crosssectio n,as a functionof aspect angle forsingleuniformlylossy dielect ricprolatespheroid with axialra- tio2 and10, and ofrelat ivesizesk14=1,2,3and 4 forcomplex relat ivepermitt ivityf.

=

4 - jO.5..

3.7 Normali zedmonostaticcrosssection,asa functionofaspect an- gle forsingleuniformly louy dielectric prolat e sphe roid with axial rat io10,relati vesizeki ll

=

3,andfor differentvaluesof complex relati ve permittivitye,=4-jO;f.=4-jO.4;t.

=

4 -jO.8;

t,.=4-j 1.2.

4.1 Scattering geometr y for a systemoftwouniformlylossydielect ric prolatespheroid s in general configuration witharbitr ary incidence andpolari zation of a plane elect rom agneti c wave.

ix

45

46

47

48

51

4.2 Normalizedbilt&tie crollsectionlorTE polwu.tionolincident wave,ualunctionolsc&ttenntangle lortwoidentic&!lossydi- elect ricprol&tespheroidswithaxialr&tios2 and10, each with semi-major&Xii lec.gthaA

=

^liS

=

^{>'/4 ,}complex relativepermit- tivity4A=f~B

=

2- jO.5.Euler &lIgles0:

=

3rr ,fJ

=

^45·,,.

=

^W,

ane

^displacedalong z·axis(90

=

Oo, ~=0·)by:(0)d=>./2,( b) d=>.. . . .•. . . •.. .• • ••..•. . . . • . • .. 75 4.3 Nerm alieed bistat it cross sectionlor TE polarin.tionol incident

wave,ualunction of scatter inganglelortwoidenticallouydi- electricprolatespheroidswithaxialrat ios2and10,eachwith semi-major axil lengthGA

=

4B=>'/4,complex relativepermit- tivit yf~A=f.B

=

^4-jO.5,Euleranglesa

=

30",fJ=4S",,.=6{l0, and displacedalongz-axis (80

=

0·,~=0")by:(a)d.

=

>./2,(6) d=A•• ••••

4.4 Normalizedmonostaticerosl sectionasafunctionof upedangle (9;)fortwoident ical lossy dielectric:prolate IpheroidAwithaxial ratios2and 10,eAChwith semi-major axil lengthaA =as

=

>'/4,complex relat ivepermittivity4A

=

48

=

^{2 -};0.5,Euler anglesa=30' ,1J""45",,.

=

^{60",anddisplacedalongz-axis(9}0

=

76

0',,,

=

0' )by,(0)d

=

A/2,(&)d

=

A. . . ... • ... . . .. .. 77

4.5 Normalized monostaticcross sectionILSa function of a.spect&Ogle' (8,) fortwo identical lossy dielectric prolatespheroidswithaxial ra t ios2 andto,ea ch withsemi-majoraxis lengthtlA=4B

=

>'/4,complexrelativepermittivityf ..A=f..a

=

^{4- jO.5,Euler}

anglesQ==30°,,8

=

^45·,'7

=

60· ,end displacedalong e-sxie (80

=

4.6 Normalizedmonostaticcrosssectionasa function ofaspect angle (8,:)for twoident ical lossy dielectricprolatespheroidswithaxial ratios2and10,eechwithsemi-major axislengthaA

=

⁴⁸=>'/4, EuleranglesQ=30°,{J=45",1=60·, and displacedalong e-

78

axis(80=90· ,~o

=

0·)byd

=

^{>./2:(a)t:..A}

=

^t:,8

=

2 -jO.5,

(6)E,,,,=,;..8=4-jO.5. 79

4.1 Normalizedmonostaticcross sectionasafunctionofupectangle (8;:)fortwo lossydielectric prolatespheroids with axial ratios2and 10 respectively,eachwithsemi-major axislengthaA=a8=>'/4, Eulerangles a::::: 30",{J

=

45·11==60· ,anddisplacedalong the dir ectiond

=

),,/2,80

=

tiO°,~o

=

20°;(a) (..A

=

2 -jO.5, f..B

=

3 - jO.5,(6) t:,,,,==3-jO.5,£.-8"" 4-jO.5,(c)t:..A=2- j0.1, f' 8= 3-jO.1,(d)E.A

=

3- jO.l,e..8=4 - jO.1.

xi

80

4.8 Norm a.lizedmonostat iccross section as a functionof aspect ang le (6,)for twolOll ydielectricprolatespheroidswith axialra tios2 and 10respectiv ely,each withsemi-major&Xis lengthBA ::Cs::>'14 andcomp le x relative permit tivit yf, A ::3-;0.5,f..s==,4-;0.5, anddisplace d alongthedirectiond:: ),,/2,90

=

60", ~0

=

20";

Eule r anglesaregivenby:(11) 0::0",,8==45","'Y== 0";(&) 0 ""

5.1 Scattering geometry forasystemof two parallelun iformly lou y di- .electricprolate sphe roid.with arbit raryinciden ce and polarizat ion

ofaplaneelectromagneticwave .. . .

5.2 Normali zed bist aticcrosssect ion1ru(6)/),2forTEpola rization of incide ntwave, asa functionofscattering angle(6)fortwo identical 10l sydielect ri cprolatespheroidswith axialratios2and10,ea ch wit hsemi-major llXis lengthlJA ::BB::),/4, complex relative perm it tivit ye'A ::frS ::2-;0.5,and dir plac ed alongz·axis (80=0"'0=0·)by,(e)d=lf2,( b) d=l.

5.3 Normalizedbistaticcrosssectio ntru(6)/),,2forTEpola rizat ionof incident wave,a..>afun ctionof scatteringangle(8)for two identi cal lossy dielectricprola te spheroidswith axial ratios2and10,each wit h semi-major axislengt hCA=4S,=)'/4, complexrelati ve permit tivit yfrA

=

^f,B

=

^{4 - ;0.5,}and displacedalo ng z-axis (So=0·"0=O·)b y,(e)d=lf2, (b)d=l.

xii

81

83

93

94

5.4 Normalizedmonosta ticcrosssection1I'0'(8;,O)/ ..Vasa function of aspect angle(8; ) fortwo identi cal lossy dielectricproillte spheroids with axialratios 2 and 10, eachwithsemi-majoraxislengthl1A :::

as :::),/4 ,complexrelative permitti vityipA

=

ira""2 -;0.5, and displacedalong z-axis(80

=

O\fo;.:0⁰)by:(a) d=.\/2,(b )d

=

>.. 95 5.5 Normalizedmonosh.ti ccross section'11"0'(8;,O)/ )'~as a {unctionof

aspect angle (8;) {or two identicallnssydielectric prolatespheroids with axial ratio s2 and 10, each withsemi-major axis lengthIIA

=

as

=

>'/4, complex relativepermittivityirA "" irS

=

4-;0.5, and displacedalong z·ax b (80

=

00,¢o""00) by:(a)d=.\/2 ,(b) d"">..96 5.6 Normalizedmonostati ccrosssectionasa function of aapect angle

(8;) fortwoidentiealloaeydielectricprolatespheroid3withaxial rat ios 2 and10, each witheemi-majoraxialengthIIA ""as ""

>'/4, and displaced along ;r:·axis(80

=

90°,fU

=

0°) byd

=

^>./2:

(a) irA=irs=2 - ;0.5;(b)irA=t,.s

=

^{4 - ;0.5. •.. . .. .. . 97}

5.7 Normalizedmonosta ticcross section as afunctionof aspectangle (8;) fortwo lossy dielectricprolate spheroids with axialratios 2end 10 respecti vely,each withsemi-majoraxislengthaA==ee

=

>'/4 , and displacedalong the directiond

=

^),/2,80=60°,¢o=20°:

(a) i pA ""2-; 0.5, 4 B=3-; 0.5(b)irA

=

2 - jO.1,48

=

3-

jO.1(C)irA

=

3-jO.5,t,.8

=

4-jO.5(d) irA=3-;0.1 .48=4-;0.1.98

E.l The Euler Angles.•.

xiii

...139

Chapter 1 Introduction

1.1 Lit er a t ur e Revi ew

Thestudy orelectromagnet ic tCatt ering byit..y.t emoftwo (ormore)spheroidal objects hasgained considera ble inte rest in the lastfew decades. Thisis because the geomet ric. of manyobjec.hof practical interest can beapproximate dby .phcroid,.

SolvingIcalarHelmholt z equationin spheroidalcoordin atesi.the Iluting pointinsolvingproblem.of electromagnetic Icd teri ngby.pheroids.Flam m er 11] andStrattonet&I..121havestudiedextensivelythe spheroidalwave function•.

Solution of.calar Helmholtzequation in termsoC sph eroidalcoordinate.ystem and expressions of .phcroidalvector wave function.arc availahle in detail. inII) ead (2J.

The exact analytical solution totheelectromagnetic scat te ringbyconducting prolate spheroid.(oraxialincidencehubeengivenbySchult z {3J.LaterSiegelet el.14]obtained lome nume rical relultlforIcatteringby athin prolatespheroid (axial ratio10:1) by varyingthe relat ive lizeof theIpheroid from 0.1toti.

The exactlolutiontothe problemotelect roml gnet icIcatt ering by a eca duct- ingprolateIpheroid torarb itraryangleof iaeideceeand polarisation wu given

byReitlinger[5\.But hisworkinvolved complexityin numericalcomp utat ion.

Nevertheless, hiswork providedafoundation in findinganalyticallolu t ions to more complexproblems involvingscatterin gby spheroidalobjects.

Becauseofthe complexityof thespheroidal geometr y,non-orthogon alityof spheroidalwave functionsand increasingdifficultiesinnumericalcomp utations beyond the Rayleigh region[61. no progress was made intheresonance region afterthework ofSiegeletal. [41,although extensive works have been carrie dout at both low frequencies and highfrequencies.It wasnot until1974tba tSinha obtainedfor thefirst timean exactanalyticalsolutionto the electromagne tic statteringby a conductin g prolate spheroidin resonance region [6\. Healso developed fast, accurate and simplified algorithmsforcomp ut ing spheroidal eigen values and spheroidalwave{unctionsthereby overcoming the shortcomi ngs in Reitlinger'swork.Furt hermore,he provedthat resultsobtai ned by Siegelwere accuratefor valuesof axial rat io upto2.5.LaterSinha andMacPhiepresented numericalresults ofbackscatte ringradarcrosssection atdifferentvaluesof axial ratio for a singleconductingspheroid[7\.

DalmasandDeleuilalsost udied theelectromagneticscatteringbyaperfectly conduct ing spheroidby using

Mr

and

iJr

spheroidal vect or wavefunctions[aj-[IO].

Usingthese vector wavefunction s,Asano and Yamamoto present edtheanalytical solutionoflightscatteringbyaspheroidal particleat arbitr ar y incidence and polarization of incidentwave(11).

Scatt ering of e.!ect romagneticwavesby r.rbit rarilyshap ed dielectri c bodies lying in theresonanceregion hasbeen studied byBerber and Vel:[12J.In[12}

plots of differentialscatteringcross sectionVI .scatt eri ng angle arepresentedin azimuthalplane andequato rial plane forbc diee like~pheres.prolateIpheroids,

oblate spheroidsand cylinders having differentgeometriesand dielectric con- stants.Asano andSato/I31analyzed the problem oflight scatteringby randomly oriented identicaldielectricspheroidal particles and computedextinction, scat- teringand absorptioncronsections andasymmetryfact or of prolate and oblate spheroids.Asanoalso investigated the light lCatteringpropertiesof spheroidal particlesorientedrandomly with theirlong axes horizontal 114J.Latereoora.y /15J worked on the"atteringproblemby&perfectdielectric prolatespheroid by employingmcdeleeeieeexpans ion of electromagnetic waves.

Extensiveresearch isavailableinliteratureregarding scattering of electr o- magneticwaveby a system of two (or more) spherical andspheroidal objects.

BruningandLa[16J studied extensively thescattering of electromagneticwave by a system of twospheres by applying thetransla.tionllladdition theorem for spherical vectorwave functions [17). whichis an extens ionof the theoremfor scalar wr.vefunctionsdeveloped by Stein [181.

Itwas not until the mid 19805 that Sinhaand MacPhie[191 developedthe translational additiontheorems for scalar andvectorspheroidal wave functions in whicbthe outgoingwavefrom one spheroidi'transformedinto the incoming wave to theother spheroid assuming thesimplest form forthe vector wave function., as they translatelike a scalerwave function.Utilizing these theorems, Sinha and MacPhieobtainedthe exact solution for the scattering behaviorof plane electro- magnetic wave hy two parallel conducting prolate ephercide [20J.InAppendix II of 120) the computationwas simplifiedby obtaining the spheroidal translational additioncoefficients interms of the spherical translationaladditioncoefficients.

DalmAlJ ead Deleuil also studied the multiplescattering of electromagnetic waves by two infinitely conducting prolatespheroids in [211usingTranslationalAddition

Theorems for prolatespheroidal vectorwave functions

AIr

and

iJr

[22], Lately, Coorayetel.123]worked on the scatteringproblem by twoperfectdielectr ic prolat e spheroidsinparallel orientation.

Furthergeneralia atlcnof thetwo-spheroid scatteringproblem bubeen possi- ble dueto formul atio nof Rot at ioOilI.Trand ation,,1Addition Theorem s for vect or spheroidalwavefunctionsin[24J-{25],Utilizingthesetheorems , electroma gnetic scatt er ing by a system oftwo perfectlycondudingandtwo lossless dielectric spheroids in arbit raryorientationwere analytically studiedby CoorayandCiric in[27J, [281end 129).

By means of modalseries expansionsof electromagnetic fieldsin term s ofpro- lat e spheroidalveetor weve funeticne,anexact enelytical eolution for~

~or electroma gneticplane wave Icattering by uniformlylouydielectric prolatespheroids has beengiven in the present thesil, Scatt ering ofplaneelec- tromagnetic wave by (i)asingleuniformlylossy dielectricprolate spheroid ,(ii) by a syste mof two uniformlylossy dielectr ic prolat eepberoids inarbit rary con- figuration,and (iii) at a special case scatteringby asystem of two unifor mlylossy dielectri c prolatespheroidsin parallel orientation ,have beenseparat elystudied inresonance region.The parallel configurationof two ormore prolateIpheroids hat practic alimpor tance, Hence it isfelt thatanalyt ica.lstudyofthisspecialcaee ofthe moregeneralcr-eof arbit rAryorient ationofthesphe roidsbe developed sepa ra tely by using translat ional additiontheorems ,10that thoseinterestedin thisepecielcase maynotnecessarilybeinvolved inthe comp lexit yof the general Icrmula tion.

Inthis thesis, the analysisfor singlebody iscarried outbyusinglti-·v.- andfJ-,V,1vector wavefunctionsfollowing theprocedure employedbySinhaand

MacPhie[71 for conductingprolate spheroid. The scattering byasingle lossy dielectric prolate spheroidbu also beenanalyzedby Zimmer [30]who employed Airandfirvector wave functions.

In the presentwork the dielectricmediaofthe scetterersareassumed\.:l1be uniformly(orhomogeneous)lossy andnon-ferromagnetic.Since theprcpaga- tion constantof the mediainside the ICa.Uerenis complex innat ure , complex eigenvaluesareevaluatedfor the spheroidal scalar wavefunctions of transmitted component! ofE·field andH·field expansionl.Oguchihucalculat ed the eigen value.of spheroidalwavefunction.forcomplexvalues of propagationconltanh in[31}. Zimmeralso has computedcomplexeigenvalues forsphercidelwave functionsusuminge-j ""timevariationof tlme-bermcnic electromagneticfield 130J. Subsequently Sebakand Sinha.have calculatedcomplexeigecvalues cor- respondingto prolatespheroidal functionl in orderto studythe scattering by a conductingspheroidal objectwithImJ~ydielectriccoatingat axialincidence [32J.In thepresent work, the algorithmdeveloped in[33] forrealeigenvalue computationhas been usedto computecomplex eigen values.

By applyingappropriateboundary conditions,the fieldsolution is obtained in the formS=[GIl, whereSandIarerespectivelythe colum nvector of the unknown Coefficientsof the seriesexpansions ofthe scatteredandtransmitted fielda takentogetherand thecolumn vectorof the known coefficientsoftheseries expansionsoftheincidentfield.{OJ isthe system matrixthat dependsonly on thescattering system and the frequencyof the incident radiat ion, and is inde- pendent ofthe direction andpolarizationofthe incidentwave.Thesolution in theabove formeliminatesthe need forrepeatedly solving a new sctof simul.

teneoueequations in orderto obtainthe expansionco.:f6cienhofecett ered and

transmitted fields(or anew a.o.gleof incidence.

It.is wort hwhiletonote that the oblatespheroidal vectorwavefunct ionscan be obt ained £romtheprolate onesbythetransformations{_j{and h _ -j h , where { isthespheroidalradialcoordinate;h=kF ,Fbeing eemi-ia terfccal dista nceof thespheroidandkbeingthepropagationconstant of the medium

ill.

1.2 Organization of the Thesis

Thisthesis deals with the exactanalyticalsolutionofscattering by uniformly lossy dielectricprolate spheroids- the incident excitationbeing unit amplitu de monochromatic plane electromagnetic wave ofarbitraryincidenceand polariaa- tionandhaving wavelength.t Electromagnetic .catteringbyasingleuniformly lossydielect ricprolate spheroid, byasystem of two uniformlylonydielectric prolatespheroids inarbitrary configuration, andby asystemoftwouniformly lossydielectr ic prolatespheroidsinparallel orientationhave beenconsidered sep- arat ely. InSection1.1wehave discussedtheavailable research inlitera ture pert ainingtoelectromagnetic scatte ringby spheroidal objectsand have given the general outlineo"the problempresented inthisthesis.The organizationof ot her chapte nisatfollows:

• InChapter 2, a briefidea of spheroidal coordinate system ispresented first.Thensolutionofscalar waveequation inspheroidal coord inate s and expressions ofvariousspheroidalwavefunctions areconsidered. Finally, prolat espheroidalvectorwavefunctionsaredhcueeed.

•Chapter3 deal.with the analyticalsolut ion of.cattering of planeelectr o- magnet icwaveby.ingle uniformlylou ydielectricprolatespheroid. Inci-

dent,scatteredand transm itt edcomponen t!of electricand magneticfields areexpressedintermsof normaliz edprolatespheroidal vectorwavefunc- tions.Appropr iateboundar ycond itioDs arethen applied on thesu rface of thesp heroid tosolvea tet of simultan eous linear algebraicequation s reo lat ing the unknownexpansioncoefficieut ecorresponding toscatt ered and tran smitt ed fieldsexpressed in termsof known expansion coefficientsof the incidentfield. Finally the scattere dfieldisealeulatediIi.the far zoneand numericalresult sare presentedinthe form of curves of normalizedbistatic and monostati cradarcroRisectionsfora varietyof uniformlylossydielectric prolate spheroids.

•InChapter4,the exact solutionforthe problem of electro magn et icscatter- ingbytwo uniformlylossydielect ricprolate spheroids in arbitrar yorienta- tionh8JIbeendiecueeed.Rot ational·Tranda.tionalAddit ionTheoremsfor spheroidal vect or wavefunctionshavebeenused here- a vectorspheroidal wavefunctiondefinedinonespheroidalcoordinate system(t ,'1,ifJ)hasbeen expressedin termsofa seriesexpansionofvectorspheroidalwavefunctions definedinanother spheroidalcoordinate system ({', .,,',41'),which isrotat ed and tramJatedwithrespect tothe firstone. Applicationsof appropri- ate bound aryconditionsand derivation of the system equation have been discussedthen. Finallynumerical resultsintheform ofcurvesfor normal- ized biatetieandmonostatic radar erosesectionsare given for a variety of two-body.ylt em ofuniformlylc say dielectricprolate.pheroid. in a.rbitrary orient at ionhaving resonantor near resonant length. anddifferent distances ofeep aretlcn.

• In Chapter5 wepresentan exact solution for electromagneticscatt eringby two uniformlylossy dielectricprolat e spheroids in parallelorient at ion.The 'I'ranllat ional Addition Theorems,which transformtheoutgoing wavefrom onespheroidinto the incomingwave atthe otherspbercld, have beenueed here.In cidentallytranslat ionaladditiontheorems [19Jcanbe consider ed as a specialcase ofrotationa l-trans lat ional additiontheorems whenEuler anglesQ_ 0°,/3_0°and"(_ 0°.Appliclt ionlof appropriat eboun d- ary conditions and derivat ionofthesystem equationhavebeendiecuseed.

Numerical resultsin the formofcurves fornormalizedbidat ic and monos- taticradar crosssections have beenobtainedintheresonanceregionfor a varietyof two-bodysystem of uniformly lossydielectric prolat espheroids in parallelorientation havingdifferentdiatancesofseparation.

• Finally in Chapter6concludingrem ark! are presented.

Chapter 2

Prolate Spheroidal Coordinates and Prolate Spheroidal Wave Functions

2. 1 I n troduc t ion

Inthia chapterwegivea briefoverviewof prolatespheroida lcoordinate system followedbyderivationofvector Helmholtzequations fori-fieldILnd Ii-field of an electromagneticwave .Wethen diecuae the solutionofsealu wave equations in spheroidal coordinate •.Finallywe obtainaolution. ofvector wave equation by applyingcertainvectordifferent ialoperators to thescalar wave {unctions.The resultingsolution.are calledvect or wavefunction s.

2.2 Prolate Spheroid al Coordinates

There are two typeaof spheroida lcoordinatesystem:prolateand oblate. The prolate and oblate spheroida.lcoordinate system arefann e dby rotating thetwo- dimensional elliptic coordinatesystem, consisting of confocal ellipses and hyper- bolu,aboutthe majorand minoraxel of theellipses,reapee'avely,Itis cUltomary to makl:thez·ax iltheaxisof revolutionin eachcu e (II.

Sinceinthilthesisweare consideringthescatt ering problem fromprolate .pheroid.,we concentrate onthedescription ofprolate .pheroidal coordinate system; however description ofoblatespheroidal coordinatesystemcan be found in pp. 6- 7,[I].Fig.2.1shows theprolatespheroidalcoordinatesystem.

Let thesemi-interfoceldistanc~of theccafocel ellipsesbedenoted by F, as showninFig.2.2. Then lor asingle ellipsethe spheroidal coordinates, denoted by({,"I,f)of a point P in space distantIIandr3respectively from the fociFI endF3, are givenby:

(2.1) where {isradialcoordinat e,"Iisangularcoordinate and(Jis azimutbalcoordi- nat e. Itcanbe shown thatprolatespheroidal coordinates«,'1,¢) are given in terms ofre ct angular coordinates bythe following rela tionspp.17-18, [6J:

-iF

^[(:.:2+y3

+

(%

+

F)3)1/3+(:1:²+y2

+

(e_F)2)1/2) (2.2)

J] =

-iF

[(:1:²+y2+(%

+

F)2)1/2 _ (:1:2+y3

+

(%-F)2)1/2J (2.3)

¢ "' -'(Ylx) (2.4)

By inverse transformationwecanalso obtain:

F(I_'12)1/2W_ l)1/2COIq, Y= F(l_1'/2)1/2

(e -

1)1/²lin1,6

(2.5) (2.6) (2.7)

with-1

s

1'/

s

I,1

s es

00,0

s

¢

s

2-w.

In prola tespheroidal system thesurface

t

=conlt ant>1i.anelcnget- d ellipsoidof revolutionwith majoraxis oflength2Ftand minor axil of length

10

r "

L I

Figure2.1:ProlAteSpheroidal CoordinateSYI~

11

Figure2.2:Prolate spheroidalgeometry and C&11eIian coordinates.

12

2F({ - 1)1/2•The degenerateeurface

e =

1i.a .traightline alongthe.z-axi.

fromz

=

-Ftoz

=

+F.The.urface1'1/1=cou. tant<1i.&hyperboloid of revolution of two .heett withan uymp toti c conewhose generr.t ingline paste.

through the originand isinclinedatan angle 8

=

cos-¹'1/ tothe .z·ax is.The degeneratesurfaceI'll=1is thatpart of thez-axisfor whichIzi>F.The surface1$=const ant is a plane throughthe z·uis formingthe anglefwiththe :r-zplane[1/.

ltis to benotedthat the confocalquadricsurfacesin space intersecteach other atrightangle., i.e. the tangent planesofthethree surfacespaningthrough anygivenpointinap aceare mutuallyperpendicular .Thueprolate spheroidal coordinat e syst emisa systemoforthogonal curvilinearcoordinates.Ineach case thecoordinat es(e,'I,1$)form a left. handed system , since

i

^xIi=_-~;Iix

¢ = -I;

~x

1

= - •.

In the limitwhen theinterfocal distance2F becomeszero,boththe prola.te andoblate spheroidalsystemsreduce to theapherieal coordinat e syl te m. For finite2F,the surface

e

⁼constant ineach case becomessphericalas { approaches to infinity; thul

F(_r,'I ...co.8.as { ...cc

wherer and 8 are spherieel radiala.nd angularcoordinate.respectively,

2.3 Vector H elmholtz Equations

(2.8)

We knowthat atime-harmonicelectromagnet ic field(ellAtime ,-ariation ,w being angular frequency )sati sfiesMaxwell'sequations:

VxE

=

-jwjJH

13

(2.9)

where

f' = f- jU /W

(2.10)

(2.11) iscalledcomplcz pennitti vityof the medium;e,jJandCTarethepermitt ivity,per- meabilityand condudivity ofthemedium relpectively.jisvolumedistri bution ofelectric currentperunit area.

Let~obewavelengthoftheelectromagnetic waveinfreespacejfOand pobe the permitti vit yand permeability offree space respectively, andc be velocity of propagationof wave infreespace.Thenpropagationcoostant offree Spaceil givenby

Suhsti tutingwfrom(2.12) in(2.9) and(2.10) we get

v

xE

-j(JlO~)1/2~J1

VXH

j(JlO~)'J] koE

Taking curl00hoth sides of(2.13) weget

VV.E_ V

^2E

= (J't...)kJ

_/lato

^E

Forcharge free space

'l.B

= 0

Prom (2.15)and(2.16) we have

'l'B+ (J't...) IlE =

⁰

"'"

14

(2.12)

(2.13) (2.14)

(2.15)

(2.16)

(2.17)

Tuingcurlon both.ide.of(2.14) weget

(2.18) FromMuweU'.equAtion

V.B= 0

From (2.18)&nd(2.19) we have

V2I1 + (~~) ~H =

0 Setting

(2.19)

(2.20)

(2.21)

( ")'"

k= L ko=n.ko

PoCo

where n,.=VlJoc'/lJOcoilthe complexrefractiveinJezofthe medium, vector Helmholtz equation-for

E

and

ii

fieldsof electromagneticwave an he givenu

v

^JE+1c^J

e

^{= 0}

v

^Jii+J:²ji = /)

(2.22) (2.23)

2.4 P ro la t e Sp heroid al Differenti al Equat io n s

ProlateIpheroid&1coordinateIyatemil one

or

^theelevencoordinate Iyit eml1in whichthe Icalarwaveequation

(2.24)

i••eparable,whereki,the mediumwave number.Thi.equation, in prolate .pberoidalcoordinate.y.t em,canbewritten&I

[

a;; -. a;;

8 (1 ,)8

+ Be

8(I -(,)8

Be + ((' - 1'-.'

1)(1_.')

8.'

8'

+ h '«(' - . ,)].,. . =

0 (2.25) Icarteaian, clulli arcylinder, ellipticcylinder,parabolicc}'lInder,Ipherical,collical, patabolic,pr olat.Iphcroidal, oblatelpheroidal,e llipeoidaIudparaboloidal.

15

where weseth

=

kF.

Usingthe separa tion of variablesprinciple,the solution of (2.25)can be ob- tained in theform11]

where~~(h,()andS",n(h,,,)are called prolderadialfunctionand prolate angle function respectively.R!:.~(h,OandS",..(h,!'1)satisfy theordinarydifferential equations

~ [(1-'l2)~S",

^..(h,'7)]

⁺ [~", ^..

^_h2'72-1

::2]

S",,,{h,'7) 0(2.27) fe

[w -l)fe~~ (h,()]

^-

[~m"-

^h'('

^+(,":' 1 ] ~~(h, ()

^0(2.28)

where~",,,(h)andmare separationconl tanh.Those valuel of~.",n(h)forwhich (2.27)admitssolut ions that are finite atlj= ±Iaretheeigenvaluesof the differentialequation(2,27). Heren=lml, Iml

+

1,Iml

+

2"",andmil any integer includingzero.

his realor compl exaccordingly the propagationconl tant ofthe medium under considera tionis real or complex .Computationsof eigenvaluesofspheroidal WAvefun d ions for Tealvaluesof propagation consb,ntsand for comple:z:valuesof propagationconstantsare respectively shown in[33]andApp endixA01tbe present thel is.

Itis worthwhileto notethattheoblat espheroidal scalarwave fun ctioncan be obtained fromtheprolate one by utilizingthe tran sforma ti ons

e

^--+

je

^and

h--+-jhin(2.26).

16

2.4.1 Prolat eAngleFunctions

The uJOciated eigenfunction,5...(11.,11 )are theprolate,pheroidalAngle functioD' oftin tkind, oforderm anddegree'1,torI"elpondingtocigenvaluell...(h)in (2.27).Thereare t",otype.ofangle (unetious:S<~(h,,,)- &llgIefunction o( tint kind,andst.!.!,(h,,,)- &IIglefunction of .econdkind.In moatbounda.ry·va1ue problelnt,thepbysicalquantities are definedover the entiudom ain -1$"'1 $1,

o

$",:52•.The usualrequirement thatthe"avefunctionbefiniteat'I

=

^±l

con finesthen-dependeneeof thewavefunct iontothatoftheanglefunctionof firttkind,becausetheangle function ofsecondkind,SJ!l(h,'1) aresingularat these point..Hencefort hwewillusethenotation5...(11.,'1)torefer to angle funct ion offintkind.

WhenII.goestozero, angle functionof fint kindreduces tousoclat edLeg- endre functions of the fint kindof integralorderand degree.So we have

l...(O) :::n.(n

+

I},n~m WhenII.i.not equalto zero,anglefunctionof tintkindi. ofthefonn

S~+(h, .)

⁼

. f:. .. ^{' r,+(h lP.::•• (. l}

(2.29)

(2.30)

whered;.""(h)areprolatespheroidalexpansion coefficienta.The prime onthe aum matioD.ymbolindie-ate.thil.t the.ummatioDi.overonlyevenvaluea ofr when(n - m}ilevenandoveronlyodd Va1uCl ofr wben(n - m) iaodd.P:.~(1J) iaulociated Legendre functionof firstkind.Theexpansion coefliclentare-"(h) are given bythe recurre nce fonnulain equa tion(3.1.4)of [1].Examinationof equation(3.1.4) in [1) reveals that asr _ 00,either~"(h.)/cr."_~(h)increase, as_4r'Jlh²,orgoestozeroU -h^2f(4r').WechcceethelatterUitleada tc a ecnvergeetaerin.Numericalcomputat ion of anglefunctioni.given in[61.

17

From generaltheory of Sturm-Liouvilledifferentialequations, it followsS...(h,11) form an orthogonal eet in theinte rval(-1,1)[11:

wher e

c.. .. '

isKronecker deltafunctionand Nm.(h) is tbenorma lizationfactor,

2

f; ,(, +

2m)!(d;"(h )):

~,,0,1 (2r + 2m + 1)rl ^(2.32)

(2.33) (2.34) 2.4.2 Prolat eRadi alFunct ions

Prolatespheroidalradialfunctionsm:.~(h,{)(i

=

1,2,3,4)satisfythe differen tial equation (2.28), where1:5(:5lXI,The eigenvalues .\",..(h)wbich'c ccurin (2.28) are those to whichangle functionsS.....(h,l1)belong.

In physical problems one usuallyrequiresboththe radialfunct ionsof the tint kind,R.UHh,(),andthoseof the second kind,~l(h,e).Two useful combina- tions of these func tionsare knownasradialfunctions ofthethird kind and radial functions fourthkinds,given by~Hh,e)andJ4:l(h,e)respectively.

Theradialfunctions ofthe first kindR~l(h,()and radial functionsof second kind~l(h,e)arerespectivelygiven by the seriesl2J ;

( e_

1,...

'2

^coo

R!.:!(U)

T)

.~.' .~·(h)jm,.(h()

R!:!(h,() ({'

e ~ ,)m

^l'.,,0,1

f; _' .~·(h)n".,.(h()

wherei...+~(h{)andn".+~(h()are sphericalBessel functionandNeumannfunc- tionrespectively,and4:;'''(h)are convergentexpan sion coefficientssuchthat as r- 0lXI4~"(h)/4:;'_i(h)-.h²/ (4r²)__O.The expansion coefficients4:;""(h )are given bytherecurr encerelat ion inequation (3,24) of[6J.

18

The radialfunction of third kindandradialfunctionof fourthkinderetepee- tlvely givenas:

~! (h,() R!.:!(h,()

.'l!:!( h,( )

+

iR!;!(h,( )

~!(h,()-iR!;!(h,()

(2.35) (2.36) Thefollowing asymptoticproperties ofthesphericalBeeeel,Neumann,and Han- kel Iunctionsareworth noting,p.31, [1/:

i_(hO n_(hO

hi"(h( ) hi" (h( )

(2.37)

Thu,theASymp totic behaviorof~l(h,O,R~l(h,o.~l(h,Oand

R$:l(h,e)

is readilyobtainedlU

~!(h,() ~

~ cos (he - _~(n+

1)1r) ^(2.38) R!;!(h,(j

It:> ~ sin (he- ~(n +

^{1)1r) (2 19)}

R!;!(h,() "!=: ~,["'- ll-H"J ^(2.40) R!.:!(h,() It::' ..!..e-h( [II(- ~(n+ll·1 (2.41) At very largedislance from the spheroid~l(h,Oand R:!!l(h.e) 'havetheprop- erties of divergingspherical waves,As~l(h,e)hu the propertyof outgoing sphericalwave for

ile _

^00,itwill be used to describe the scatte redfields of electromagnet ic wave,Seriesrepre, ent ation of

.RUl(h,

e)show. fut convergence, whereu,eriesfor~l(h,

0

doesnot converge rapidlyfor ,mallvalueof

h e

or

19

he

se1.Anintegral method introducedby Sinha and MacPhie in[34Jimproves the convergenceof~l(h,

0.

2.5 Prolate Spheroidal Vector Wave Functions

Beforedefiningprolatespheroidalvectorwave funct ions letus discuu in brief the fund ament al setof solutions of anyvectorwave equation, pp. 392 - 393, [37J.

Within any closeddomain 01a homogeneous,isotropicmedium from which sourceshavebeenexcluded,all vectors characterizingtheelectr omagnetic field - thefieldvect orsE,ii,jjandH,the vectorpotentialand the Hertzian vectors - satisfy oneand the same differentialequation.If

6

denotesan)·such vector, then

(2.42) wherek2=flJl.;Ja-japw (e^j...¹timevariationof

6

isassumed).

By the operatorVa actingon a vector we mean"1³="1'\7.- VxV x; thereforeinplace of(2.42) wecan write

(2.43) Nowthe vector equationin(2.43)can always bereplacedby a simultaneous sys·

tern ofthre escalar equations,but the solut ion of this systemforanycomponent of

6

is impractical inmeetcases.Itisonlywhen

iJ

isresolvedinto ihrect angular compoDeDts that threeindependentequll.tionsareobtained.Thul, in thiscue

(2.44) Let thescalarfunctiont/Jbe1\solutionofthe equat ion

(2.45) 20

(2.46) andletQ.be any COD.t antvectorofunit length.We nowcan con.tructthree independentvect or solution.of (2.43)AIfollow.:

~ ^: ~~a~ I

fI = (l/kjV xM

If

C

i.placed equalto

i,MorN

^weun verifythat(2.43)i••atiJficdident ically by (2.46) subjectto (2.4S).SinceIii.a conltan tvectorMcan ....0be wriUe.au

M

=

L

xa =

(l/k)VxfI (2.47)

Foroneandthelame generatingfunction1/Jthe vector

Ai

i. perpendicular to the vector

i;

or

i.M= 0

The vector funct ion.L,MandNhavethefollowing propert ies:

AltoAiandNaresolenoidal ,i.e.

V.M=O, V.N= O

(2.48)

(2.49)

(2.50)

LetUJDOWdefinethe .pheroidalvector wavefunction•.The .cala.rfunction tati.lyingthe.calar HelmholtzequationexpreJledinterm. of .pheroid alccordi- nate. b.. theform siven in (2.26):

where the .upencripheaDd0refer toeven~dependencyandodd;-depen dency re.p edively andi=1,2,3, 4.

21

The spheroidalvector wave functionscanbe defined as[1]:

Ml~"(h;(,•••) Nl~"(h;(,••"

V~~2(h; (••• •JxG

.-'17

^xMl~"(h;(•••')

(2.52) (2.53)

(2.54) whereIiis arbitraryunitconstantvectororradius(position) vector.In spheroidal coordinate syst em, noneofthe unitvectors

e,

^~,~huthe propertiesrequiredfor Ii,Hencewechoosecartesiancoordinat e system in which each of the unit vectors oi,y,

z

^{is a}constant unitvector,Theunit vectors in cartesiancoordinates ere relat ed tothe spheroidalcoordinate unit vectorsbythe relations[II:

((' -I) '" (1-.')'" - _

:i:=-'1 (3_1]3 cos~~+ ( l3~ cos ~(- sin~~

((' -I)'" (1 -.,) " , - -

Y=-1/

(3_1/3 sin ~ ~+ ( ^{e -1]3} sio,(+cos~tP

_ (1-")''' _ ((' -1)"' -

~=( (3_1/3 '1+1/ (3_'13 (

Thusthe cartesian unit vectorsin(2,54)generate threedistinct classesof prolate spheroidal vectorwave functionsMandiJ,givenby:

M,;:!"(h;(••••) N';:!"(h;(,•••)

V~~2(h;(•• ,.)xG••

= '.Y.'

(2.55) k-'V xMl~i)(hi(,J'lltP),^a=:?:, Y. ~ (2.56) Also withregard to the positionvector

r,

theMandiJvect ors aJ'e expressedas

M;:!"(h;(•••• ) V~~2(h;(••• •) xi' (2.57)

N;:!" (h;(,•••)

.-'17

xM;:!"(h;(,••') (2.58) Inthis thesis theanalysisfor singlebody is carriedoutbyusingMe.II,'andiJcol/"

vectorsfollowing the procedure employed by Sinha and M&CPhieforconducting prolate spheroid[1J.3

3Zlmmnemployedii-andii-vecton toaoalYlethe ICatteriplproblem by Ilinlie IO*IY dieledricproll r.eIpberold130).

22

various componentsisequal tothe product of eithercos~orsin~withcosm~

orsin_m~.Thereforeit is convenienttodefinethe followingadditionalspheroidal vector wavefunctions, [wherethe components of those labeled withthe index m+ 1 have either aco.(m +l )~or.in(m+1)</1 ';'dependence,whilethe component.

of thoselabeledwithm -1 bave eitheracos(m-1)~orsin(m - l)~aile)p. 70

M;.+,g~

..

(hj{,7J,,p) M~-_(;!"( h:I , ',') Nl~(:~

..

(h;{,7J,tP) J'I~:(;!"(h:I,',')

~ [M~~'l(h: I,',o)

^T

~Ml'J')(h:I" " )l

^(2.59)

HM~~')(h:I,',')

^±

~M~~')(h:I" ")l

^(2.60)

k-¹V'xMl~(;~

..

(hjt,7J,tP) (2.61) k- '<JxM~:(;!"(h:I, ',') (2.62) Explicit expressions ofthe vector wave functionsdefined above are given in Table v,pp.74-76, [1[.

According toSin~aand MacPhie [20). it is possibleto expren the sinusoidal variation of ¢ in equation(2.51)as complexexponential variation.Also for any integer n>0since -n:5m:5n,it is possibletonormalize spheroidal vector wave functionsMandNinterm.ofIml.Normalization ofspheroid al vector wave functionsand their representationsin exponential form.are explicitly given in [20J and AppendixA of

IUil.

Ol·thogonalitypropertyof complex exponential, given below,will be used later:

(2.63) where 5...,is Kroneckerdelta function.

Itisworthwhile to note here thatnot ations for

it

andiJfollowed through thisthesis are the onesusedby Sinha and MacPhie [20).Plemmee'eM.:~t

..

^and

23

N.::~t

..

^{become /;1.:$:1and}

N :.L'l

ⁱⁿ[20) respectively,and Flammer'.M;~L

..

and N.:~L

..

become

M.:1'l

^andN.:~lin{2O} respectively10that/;1.;1'1andN~)have (m± 1); ;'dependence.HoweverFlammer'.Jol:~1andif:;}remain...me&Itb ...t in120)havingm~f.dependence.

2.

Chapter 3

Electromagnetic Plane Wave Scattering by Single Uniformly Lossy Dielectric Prolate

Spheroid

3.1 Introduction

In thischa pter we studyscatte ring of planeelectromagnetic wavebysingleani- Cormlylossydielect ricprolate spheroid.Incident,scatteredandtransmi~tedcom- ponenteof electric andmagneticfield. are expreeeedin lenniof normalizedpro- latespheroida lvectorwavefunct ions (definedin Append ix AoC(15]).Sincethe dielect ric mediumof the scattererisof complex relativepermittivity,complex eigenvaluesareevaluat ed for the 5pheroidalscalarwavefunction softransmit ted components ofE·fieldand H·field expansions.Approp ria te bounda ryconditio ns arethen appliedon thelun aceofthe spheroid tosolve asetofsimultaneous linear algebraicequatio nsrelating theunknownexpansioncoefficients com espon din g to sca Uere d and transmitte d field.expre ssed interm.of known e:t'pan. ionccef- ficientsoftheincidentfield.Finallyweelaborateonthe comput atio nofrs der croll sectionsin farfield for lingleuniform lylOlly dielect ricprolate spheroid.

25

Figure3.1:Scatteringgeometry lor alingleunilonnlyIouydielectric prolate .pheroidwith arbitraryincidence andpolarization ofa planeelectromagnetic wave.

26

3.2 Expansion of the Incident Electric Field in terms of Normalized Prolate Spheroidal Vector Wave Functions

LetUIconsider amonochromaticplaneelectroma gnetic wave ofwavelengt h>.

and of unit amplitude propagating in free space.Thiswavei.propagat ing inthe

.:I: -z plane(¢.=0)atan angle 6i(S 1(/2) madewith theZ-axi' land is incident

on singleuniform lylouy dielectricprolatespheroidallebcwn inFig. 3.1.Let ( =

e o

bethe value of

e

on the eurfsce of the spheroid,andabethe length of semi major axis of the spheroid. The mediacuteideand insidetheseetterer arcassumedto beDon-ferro magnetic[i.e.their permeabilitiesare equaltotbe permeability of free space/lo),

Let the electric fieldEiof the inddent plane wave be linearly polarized inan arbitrarydirection.Thiscan be decomposed into two ortho gonallypolarl eedE vectorsEiTEandE'TM.

ii;

vector is decomposedinto orthogonaily polarizedjJ vectorsH;TEandH;TU. E'TEandHiTUlieperpendicula r1.0the plane of inci- dence whereeaEiTUand

ii"l

&lie intheplane of inciden ce.Thus the polarization angle1,(the angle which theincident electric field makeswith thenormalto the plane of incidence(:I: -z plane»)issuch that (orTEpolarization 1,=0and (or TMpolarization"Y,

=

7r/2.

Accordin g toFlamme.: an incidentplanewavecanbe exp andedinterm s of M':~l)andM':;;~I I.Theelect ric field of unit amplitudeincidenton..spheroidis givenu[20Jand(151

E,

=

_...=-..,

f f

_....,"'1

[p~"M~~')+p;"M;~")

^(3.1)

27

where

:i 2j"-1 (COS7, " )

p...== -N (h)51...1..(hl , cos 8;) - ,."=FJSID1'.

Iml" 1 COl . (3.2)

whereht

=

(2'Wj)' )F.Nlml,,(hl)is theDormalization fa.ctorgiven hy equation (2.32);Slmln(hi, cos8;)isthe prolateepheroldalangle fundion (definedin Chl.pter 2).

lfthe expansionofE.givenin(3.1)i. arrangedinthe t/rlequence(O)4J. (±I)4J, (±2)4J,''',thenthe seriesexpansion in(3.1)can bewritten in the form of associatedmatrixfield expa.nsions givenin[20Jand[15J:

whereTindicate tran sposeof a matrix,and

[

lii;,

1 [P' ]

M

^~. -^{.- ~'I}Mi3 ^,' ^1-- ^PIP3

:

;

with

[M~P lTMj(l)Tj

[M:~;)T M;~;lTiM~l;~lJM:!;~J]'^f12:

1

where

(3.3)

(3.4)

(3.5) (3.6)

Also

(p!fp iT]

Ip;':l

P;JI;

P!fr+l)P :f.-ll]'

^(f~

1

28

(3.8) (3.9)

with

(3.10) wherep;'..(n :,:Iml,lml+1,lml+2,· ··) isgiven by equation(3.2).

3.3 Expansion of the Scattered Electric Field in terms of Normaliz ed Prolate Spheroidal Vector Wave Functions

In responseto the electromagneticfieldincident on spheroid,there will be scat·

tered component of electric fieldoutside the spheroid

<e

>Co) whiChmustsatisfy the radiationcondition.Knowing

(3.11) we can expand thescatteredE·fieldin terms of normalized prolatevector wave functions.Alsothecomponents of thescatt ered fieldmust havethe same,p..

dependenceasthe correspondingelements of the incidentfield. Thulaccording to[20Jwe can write

whereall M.vectotl in the aboveequation are evaluated wit h respectto hi

t-

2~FI),,)which is real.a+ .,, a-·s and a'••arethe unknownexpansioncoeffi- cientscorrelponding to theseriesexpanelonof scatteredE·field that have tobe evaluated.

Ifsame ;'sequenceof azimuthalharmonicsused forthe incidentfieldilused in this cue, then thescat teredfield fromthe~pheroidcan be written in the

29

generalized column vectorproductsimilarto that forii.[20]

(3.13) where

with

[M~~4)TM~4}Tj

(M:i~)TM;(4 }TM:!:~I)

M: I:ITj ,

a~

1

with

M;(4 }

definedin (3.7)fori""4,and

Also

[ a!i afl

[a;~~

a ;T 0:[. _1)

a~~] ,

a;::

¹

with

a; T""

[a ;'I"10;'1" 1+1o;'I"IH •••

j

Q~T^""[OI~,I'" Q~,r"l+l o~,I"IH^•••,

(3.14)

(3.15) (3. 16)

(3.17)

(3.18) (3.19 )

(3.20) (3.21)

3.4 Expansion of the Transmitted Electric Field in terms of Normalized Prolate Spheroidal Vector Wave Function s

Since the mediuminsidetheecatterer(Ipheroid)i, composed of dielectricrna- terial,therewillallObea tr&.llllmittcd component of E·fieldlaside the .pheroid

30

whichi.non-planewavetype.

Now imidea lossydielectric medium, propagation constant or wavenumber Ieis given by

• =

(2'IA).,pJ;. (3.22)

wheref.'

=

t-jrr/w,t.ispermittivity of freespace,and conductivity(0)ofthe mediuminsidethespheroidis notequal to zero.f,.

=

~/fDi.calledcomplex relative permittivityof the medium.Thus inside theloslly dielectric spheroidIe is complex,and is ofthe formIt ::::le' -jk",wherele'and k" arerealquantities.

For the expansion ofthe tra.nsmittedE·field1the vectorwavefunctionsof firstkindareto be considered and also theyhave tobe evaluated with respect toh2(=(21rF/),,).;4 ),whichis complex.Thulthetransmit ted E-fieldimide thespheroid expressedintermsofnormalized prolate spheroidal vectorwave functionsisgiven as (231

P+·s,p-.sandfJ~-.arethe unknown expansioncoefficientscorresponding to theseries expansionof transmittedE-fieldthat have to beevalua~ed.All the Ai-vectors in theequation(3.23)are evaluated withrespect toh2⁰

IfthetermJin theexpansion ofElarearrangedin the 4>-sequenceof(0)';1 (±1);,(±2);,...then we can have

(3.24)

31

where

with

iM~11)7'M~(1)7'1

[M:~lr^M;(ljTM=!~~lM~(~)TI, a?:1 with M;(I)defined in(3.7)Cori

=

1, and

Also

[J3!;J3;T )

[13;':113;7'

13:(..-lll3~~l, u?:1 with

13;7'

=[.8;'r"II3;'I,.I+1'o;'I,.I+1·

/3~T=[.8:.I,.,I3:,I,.I+l'o:....,H.·

(3.25)

(3.26) (3.21)

(3.28)

(3.29) (3.30)

(3.31) (3.32)

3.5 Expansion of H-field in terms of E-field

WeknowfromMaxwell', equations fora time harmnnicelectromagnetic field

" (i) "

H

= z:;;

'VxE (3.33)

Againweknowthatw

=

(27f/),)/(IloEo)1/3IJIdIe

=

(E/EO)1/1(2~/)'),An uming thatthemedia outsideandinsidethespheroidare non.ferromagnetic, we can

32

write

(3.31) whereI:isthewavenumber(or propagation constant) andtis thepermitt ivit y of the medium (realorcomplex).Wecan obtainexpansionsof thedifferentH-fields insideand outsidethe ecatterer in termsof appropriatenormalizedspheroidal vector wavefunctions Cromthoseof thecorrespondingE-fields. We do tbishy replacingAibyNand multiplyingeachexpansion by the appropriatevalue of j(<<IJJo)1/2.

3.6 Expansion of the Incident Magnetic Field in terms of Normalized Prolate Spheroidal Vect or Wave FUnctions

Utilizingequation(3.34),theincidentH·field can beexpressedas (3.35) where1:1andEIare the propagationconstaat end permittivityofthe medium outsidethespheroidrepectively. Using (3.1) and the relation

(3.36) we get

ii, = ...=-"" ..

f: f:

",,",I

[p:'"N':~1 + p;"N;~11

^(3.37)

whereP;'..aredefined in (3.2).Equation (3.37)<:&0be writtenin the matrix form.imilar to tha torE•...

(3.38) 33

N!l jT has the same formas thatof

M P)T

definedin(3.4)-(3.7) bu t withAi replacedby

ii.

Ihasalso beendefinedin (3.4)and (3.9)- (3.10).

3.6 .1 Limitin g Case:"(p=0 an d0,_ 1r/ 2

The series expansionof incidentE·fieldand H·fieldbecomesindet ermi nat ewhen angle of incidence 8;... 90°forTE polarization[i.e.'l,

=

0) ofincidentexci- tat ion. The limitingexpressions forp;' .. for8; ...90·correspondin gtoT E polari zationofincident fieldsaregiven by Sinhaand MacPhie in

1 7J:

{

0, (n-Iml)even

t = 2j"-1(_ I)("-I"'HI/ 2(n+!ml+l )!

Pm. N,.,.·

2.(~)t(~)! '

(n-Iml)odd

N1"'lnis thenormalizat ionfactor givenbyequation (2.32).

(3.39)

3.7 Expansion of the Scattered Magnetic Fi eld in terms of Normalized Prolate Spheroidal Vector Wave Functions

U.ing(3.34)we canwriteforfi.

(3.40)

U. ing (3.l2) and therelationgivenin (3.36)

Equat ion (3.41) canhewritten in thematri xformlimil&:to tha.tofE.all

(3.42) 34

N~4lThuthe lameformuthat ofM~4lTdefinedin(3.14) - (3.17) butwit hM rep laced byN.a haaalso beendefinedin(3.14) aad (3.18)-(3.21).

3 .8 Expansion of the Transmitted Magnetic Field in terms of Normalized Prolate Spheroidal Vector Wave Functions

Using(3.34) we can write foriii

(3';3) wherek2andt2are the propagation constant andcomplex permittivity of the medium inside thespheroidrepective ly.

Using (3.23)and therelation given in (3.36)

Equation(3.44)can bewritten in thematrix formlimillLl'to that of

E ,

&I

(3.45)

N! I)T

h&llthe lame form&8thatofMP lT definedin (3.25) - (3.28)but with

Ai

replaced by

N. 13

has also beendefinedin (3.25)and (3.30)-(3.32).

3 .9 Applications of Boundary Conditions

Bou ndary conditions requirethat &CrollSthesurface of spheroid

«(

^=<0)the

tangential components ("and.,) oftheE·fieldand as well&8those ofH-field

35

(assumingnosurface current)must be continuous.Theseconditionscan be expressed equivalently :

~,,+~~ ~~" }

Ei4-

+

E..

=

EI~ (3.46)

iiiTf+ H", =HlTf Hi.

+

H•• =H,.

at

e=ec

^andforallvalues off} and¢ inthe ranges-1_~11:<:;1 and 0:5¢:<:;2'11' respectively.

Byexpandingeach E·fie\dand H·fieldin term. ofnormalizedprolate sp heroidal vect or wavefunctions, wecan rewrite (3.46)as

(M!,ITI+M~4)T0)xilf:(o (:Njt1TI

+ N!-)T

0)xilf:(o

Mp lT{jxilf=fo (3.47)

(e2/(,) '/ 2N plT,8x el fcto (3.48)

3.9 .1 ~matchingand'1-matching

In(3.47) and(3.48) thecoefficientsofthesame,.depend entexponentialterms on both sidesofeachequationshould beequaland the equality sbouldbold good for eachcorrespondingterm under the summation ofm.Aho thosetermson both sidesof(3.47) and (3.4g) that are under Bummatioo ofncannotbe matchedterm by term.

Thus according to[15]and(231for¢i-matchi ngand'1-matching bothsides of (3.47) and (3.48) arescalar multipliedby the vectorfunctioos

{::~}

^Siml.I"'HNe±jC"':tt)4>, N=O,I,2,,,.

andthe productsareintegrat edover thesurface of the spheroid withrespect to both,,(-1~1/S1) and; (nS;S211'),wherel~andl~/Ilegivenby

I" =j2F(e~-,,2)ln ) (3.49)

I. ==2F({g_1j2) 36

forequa tion(3A7)end hy

(3.50) forequ at ion(3.4B).Usi ngtheorthogonalitypropertiesofcomplex exponen ti als, givenin(2.63),8D.dangle functi on.,given in(2.31).we obtainaset of coupled algebrai cequa.tionsof theform:

(PM}13

+

[QMI '"

[PNII3+IQNI '"

(3.51) (3.52) where theelementsof{Pu]. [QuJ,

IRMJ.

^{{PH!,[QH]and[RHlaredefined in}

Append ixB.

3.10 System Matrix [G]

Combining (3.51)and(3.52) we can writ e

Equa.tion(3.53)canbewritteninthe for m

5 =

I GJI

where

(3.53)

(3.54)

5

=

[a]

(3.55)

(3.56)

[0 ]is the generalized Iy.tem matrixwhich is independentof the direct ionand polarizationoftheincident wave. Thusthe solutioDin the above form eliminates

37

the need forrepeatedly eolvieg a Dew,el ofsimu ltaneousequilotion.in orderto obtain the expansioncoefficientsof scatteredand transmittedfields fora new angle of incidence.

3.11 Far-Field Expansions and Scattering Cross sections

Oncethe unknownexpansioncoefficientscorrespondingto series expansionof scattered and transmittedfields are determinedbysolvin g thesystem equ at ion (3.M), wecan find themagnitudeof the scatteredfield at a particula rdistance fromthe spheroidby substitut ingthe valuesofcoefficienh correspond ing to .e ries expansionof eeetteredfieldin equation(3.12).However,of practical iatere etis the scatteredfield in thefar zoneforlrl...00,r being thedistance fromtLe spheroidtothe point of observation.

Atverylarge dist ance from the spheroidhie...00.So ashie_00,

h,1 - kr

1

11 -+ cos6

~

... - 8

AlsoiLlIhd _00,it can be shownthat[201and

1 1SI:

RJ:~(h"l)

~RJ:~ (h"O

(3.57)

(3.58) (3.59) The asymptotic forms of spheroidalvectorfunct ionsare obtained by Mglecting

e -

²and its higherinverse powerterms.Thusinthefar zone the scattered E·field with respect totho:origin0 of spheroidA,isgiven by(20]

(3.60) 38

where

- f: f: jft~l [~((a:'ft

^{-O;;'ft)'eos(m t}

1)~

",=(lft='" 2

t

j(o:'ftta;;.,,).siD(m + l)~}t*a~lftl fo,,~

^j"

[~'1S", ..

{(a:'"ta;;.,,)cos(m t

l)~

t j(a:.,, - a;;.,,) sin(mt l)~}- ~S"'~l,"

{(O:"~I,"ta~(m~I),,,)cos(mtl)~

+

j(o:" +l,..-Q~("'+l)^...)lin(m

+

l)iP}

t

~'1SI

^..

a~I"

^-

~SOna~]

(3.61)

(3.62)

8

and~are the unitvectors inthe directionof increasing(Jand ¢respectively.

The scatt eringcresssection isdefined as 4'11" timestheratioof the ecatt ered power delivered per unitsolid angleinthe directionof the receiver to the power per unitarea incidentatthe ecetterer. This can be Iho'RDto be independentof

The biltatic radar cron lect ion0'(8,16)is defined11.I

(3.63)

wherei representsthe polarizationof the receiveratthe observationpoint (r,6ItP ),With;' in thesamedirection asE,the normalized bistaticradlU'croll lectionilgivenby

(3.64) Thenormalizedbistaticradar crou sectioninE·Plane and H·Planeare obt ained by lubstituting16==."j2and16 :::0 in (3.64). For normalized backacatte ring

39