Electromagnetic Fields: Restrictions and Approximation

229 Pages·2003·6.66 MB·English

by  Boris Z. Katsenelenbaum

Boris Z.Katsenelenbaum
Electromagnetic Fields –
Restrictions and Approximation
Electromagnetic Fields – Restrictions and Approximation.(cid:13) Boris Z. Katsenelenbaum
Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN:978-3-527-40388-2
Boris Z.Katsenelenbaum
Electromagnetic Fields –
Restrictions and Approximation
WILEY-VCH GmbH & Co.KGaA
Author:
Prof.Dr.Boris Z.Katsenelenbaum
Ha’alya str.20,Apr.8,22383 Nahariya,Israel
This book was carefully produced.Nevertheless,author and publisher do not warrant the information
contained therein to be free of errors.Readers are advised to keep in mind that statements,data,
illustrations,procedural details or other items may inadvertently be inaccurate.
Coverfigure:
Zero lines of the auxiliary field for rectangular object reconstruction
Library of Congress Card No.applied for
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Bibliographic information published by
Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic
data is available in the Internet at <http://dnb.ddb.de>.
© 2003 WILEY-VCH Verlag GmbH & Co.KGaA,Weinheim
All rights reserved (including those of translation into other languages).No part of this book may be
reproduced in any form – by photoprinting,microfilm,or any other means – nor transmitted or translated
into a machine language without written permission from the publishers.Registered names,trademarks,etc.
used in this book,even when not specifically marked as such are not to be considered unprotected by law.
Printed in the Federal Republic of Germany
Printed on acid-free paper.
Printing: betz-druck GmbH,Darmstadt
Bookbinding: Großbuchbinderei J.Schäffer GmbH & Co.KG,Grünstadt
ISBN3-527-40388-4
Preface
Thesubjectofthisbookfallswithinthescopeofthreedisciplines–theantennastheory,high-
frequency electrodynamics and mathematical physics. There are problems in which these
disciplinesarefundamentallyconnected.Theyaretheinverseproblemsofthehigh-frequency
field theory,moreprecisely–the connectionbetweentheshapeofthe domainin whichthe
monochromaticcurrentsarelocated,andthepossibilityofapproximatinganygivenfieldby
the fields of these currents. This connection gives rise to a number of profoundquestions.
Manyofthemareformulated,andsomeareevensolvedinthisbook.
Thefollowingproblemsareconsidered:
1. Antennasynthesis. Surfaces exist, which have the defect that the fields generatedby
the current located on these surfaces cannot even approximate many given fields (or their
patterns). Therearealotofthesesurfaces. Asarule,theantennasurfaceshouldnotbeclose
tothem.
2.Optimalcurrentsynthesis.Thecaseisconsideredwhenthefieldsgeneratedbycurrents,
located on the given surface, do not approximate a given field, or else this approximation
requiresaverylargecurrent. Oneshouldthenfindthefieldwhichis(ifpossible)closetoa
givenone,andsocanberealizedbythecurrentofthesmallestpossiblenorm.
3. Solvability of the first-kind integral equations. The existence of solutions to inverse
electrodynamicproblemsdependsontheintegrationdomain.Thisdependenceisinvestigated
withdifferentdefinitionsoftheterm“solution”.
4. Noncompleteness of the set of functions generated by the operator acting over the
completesetoffunctions. Iftheoperatorwhichtransformsthecurrentintothefield,actson
thecurrentlocatedontheabove-mentioneddefectsurfaces(throughoutthebookwewillrefer
to such surfaces or lines as specific ones), then the kernel of the conjugate operator is not
empty. This means that the generatedfield is orthogonal(in some metrics) to the so-called
orthogonalcomplementfunctions.Thesetofsuchfunctionsmaybehighlyrichandtheactual
degreeofthesurfacedefectivenessessentiallydependsonitsrichness.
5. ConstructionofarealsolutiontotheHelmholtzequation(ortheMaxwellequations)
byitszeroline(zerosurface).Thedefectlines(surfaces)arezerolines(surfaces)ofsomereal
solutionstotheseequations;theasymptoticsofthesolutionsatinfinitycontainsthefunction
oftheorthogonalcomplement. Theinvestigationofsuchsolutionsisanefficientmethodof
studyingadefectline.
6.Investigationofpseudo-solutionstothefirst-kindequationsoriginatingfromtheinverse
problemsfor the fields obtainedby the two-dimensionalFouriertransformationin arbitrary
domains.
Theemphasesinthebookareapproximatelyintheaboveorder. Themathematicaltech-
6 Preface
niqueisnotmoredifficultthanthatusedindiffractiontheoryproblems. Themajorityofthe
intermediatederivationsarereplacedbytheirdescription.Themethodsoffunctionalanalysis
andtheanalyticaltheoryofthesolutiontotheHelmholtzequationareusedverylittle.
The first edition of the book entitled “The approximability problem of electromagnetic
field”waspublishedinRussian(Nauka,Moscow1996)[1]. TheEnglisheditionisextended
withnewresults(partlyunpublished),inparticular,withthenewchapter“Longnarrowbeam
ofelectromagneticwaves”. Theappendix“Antennasynthesisbyamplituderadiationpattern
andmodifiedphaseproblem”,writtenbyN.N.Voitovich,isalsoincluded.
TheauthorisdeeplygratefultoN.N.Voitovichforhiscarefultranslationofthemanuscript
intoEnglishandforhiscontributioninthescientificediting.SpecialthanksareduetoUlrike
WernerandMelanieRohnforeditingthebookthorouglyandthepermanentattentiontothe
project.TheauthoralsothanksOlegKusyiforhistechnicalassistance.
B.Z.Katsenelenbaum
Israel,July5,2003
Contents
1 Introduction 9
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 SubjectandMethodofInvestigation . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Realizability,Approximability,AmplitudeApproximability . . . . . . . . . . 15
1.4 OutlineoftheBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 NonapproximabilityofPatterns 25
2.1 NonapproximabilityandZeroLinesoftheRealWaveField . . . . . . . . . . 25
2.2 ExamplesofSpecificLines.“Prohibited”AntennaShapes . . . . . . . . . . 33
2.3 AmplitudeNonapproximability. . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 ProbabilisticDescriptionofBodyShapebytheLikenessPrinciple . . . . . . 54
2.5 BodyShapeReconstructionbyitsScatteredPatterns . . . . . . . . . . . . . 63
2.6 PropertiesofSpecificLines . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 NonapproximabilityofNearFields 75
3.1 ApproximabilityConditionforNearFields . . . . . . . . . . . . . . . . . . 75
3.2 ConstructionofaFieldbyitsZeroLine.CaseoftheCircleArc. . . . . . . . 81
3.3 AnalyticalExtensionoftheEigenoscillationFieldoutsidetheBoundaryofthe
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 TheNormoftheCurrent 99
4.1 TheMinimalCurrentNormataGivenAccuracyofApproximation . . . . . 99
4.2 GeneralizedFunctionsofDoubleOrthogonality.NonapproximabilityandEx-
istenceofNonradiatingCurrents . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 OptimalCurrentSynthesis.TheGeneralCase . . . . . . . . . . . . . . . . . 119
4.4 DomainofSpecificLineInfluence . . . . . . . . . . . . . . . . . . . . . . . 131
5 ElectromagneticField.TheMaxwellEquations 141
5.1 TrivialGeneralizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2 PropertiesofSpecificSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 LongNarrowBeamofElectromagneticWaves 159
6.1 Two-dimensionalFourierTransformation . . . . . . . . . . . . . . . . . . . 159
6.2 TransmissionofFieldbyWaveBeam:PossibilitiesandRestrictions . . . . . 165
6.3 ShapeofAntennaandRectenna . . . . . . . . . . . . . . . . . . . . . . . . 182
8 Contents
BibliographyforChapters1–6 191
Appendix:AntennaSynthesisbyAmplitudeRadiationPatternandModifiedPhase
Problem(byN.N.Voitovich) 195
A.1 SynthesisofAntennasbyAmplitudeRadiationPattern . . . . . . . . . . . . 195
A.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.1.2 Problemformulationforthecurvilinearantenna. . . . . . . . . . . . 196
A.1.3 ReducingtotheLagrange–Eulerequation . . . . . . . . . . . . . . . 197
A.1.4 Caseoflinearantenna.Mainpropertiesofsolutions . . . . . . . . . 199
A.2 ModifiedPhaseProblem.ContinuousCase . . . . . . . . . . . . . . . . . . 201
A.2.1 Modifiedphaseproblemandrelatedmathematicalandphysicalprob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2.2 Analytical solutions to the Lagrange–Euler equation for linear an-
tenna.Theoreticalresults . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2.3 Solutionbranching . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
A.2.4 Numericalexample.Problemwithsymmetricaldata . . . . . . . . . 214
A.2.5 Problemswithnonsymmetricaldata . . . . . . . . . . . . . . . . . . 221
A.3 ModifiedPhaseProblem.DiscreteCase . . . . . . . . . . . . . . . . . . . . 224
A.3.1 Problemformulationforlinearantennaarray. Lagrange–Eulerequa-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
A.3.2 Theoreticalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
A.3.3 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
BibliographyforAppendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Index 235
1 Introduction
1.1 Introduction
1.Inthefollowingexampleswebrieflyillustratetheproblemsconsideredinthebookandthe
resultsobtained.
A.Acylindricalmirrorisgiven;itsdirectorisanarcofacircle.Themirrorisilluminated
by a field with the polarization parallel to the cylinder’s axis. The induced current has the
same direction. Forsimplicity,weassume thatboththefieldandthecurrentdonotdepend
onthecoordinateinthis direction. Canthis currentgenerate(atasuitableillumination)the
pattern ,orapatternclosetoit?
The(cid:0)an(cid:1)sw(cid:2)(cid:3)e(cid:4)r(cid:0)tothisquestiondependsonthevalueofthecircleradius . Itisimpossibleto
generatethispatternoreventheoneclosetoitif ,where i(cid:1)stheBesselfunction
and isthewavenumber( isthefreque(cid:2)n(cid:0)c(cid:5)y(cid:3))(cid:1).(cid:6)Th(cid:7)e(cid:8)closestpa(cid:2)t(cid:0)terntothegivenoneis
(cid:3).T(cid:7)h(cid:4)is(cid:5)r(cid:6)esultdoesnotdependo(cid:4)nthemirrorwidth.
(cid:2)(cid:3)(cid:4)(cid:0)Thepatterncanbearthestampoftheregioninwhichthecurrentislocated. Thisstamp
cannotbeerasedbychangingthecurrentdistribution.Ifthecondition holds,then
itisimpossibleforthepatterngeneratedbythecurrent,locatedonsuc(cid:2)h(cid:0)a(cid:5)(cid:3)m(cid:1)i(cid:6)rr(cid:7)or(cid:8),tobeequal
(evenapproximately)toapattern,theFourierseriesofwhichcontainstheconstantterm. If
isnotequaltoazeroofthefunction ,thenitispossibletoapproximatesuchapatternor
(cid:3)ev(cid:1)enrealizeit,butif isclosetothis(cid:2)z(cid:0)ero,thenthecurrentgeneratingthepatternmustbe
verylarge. (cid:3)(cid:1)
B.Ametalscreenisapartofasphere.Itisilluminatedbyanelectromagneticimpulse.Is
itpossibletoverifywhetherthescreenisreallyapartofasphere,andtofinditsradiusbythe
measuredpatternofthescatteredfield? Boththespatialandtimestructuresoftheimpulseas
wellastheshapeofthemirrorcontourareunknown.
Theanswerispositive. Todoit,thevectorpatterngenerated(atanyillumination)bythe
currentonthescreenshouldbemultipliedbyacertainweightvectorfunction,andtheproduct
shouldbeintegratedoverthesolidangle andFourier-transformedovertime. Ifthescreen
isapartofasphere,thenatsomefrequen(cid:9)ci(cid:7)estheFouriertransformwillbeequaltozero.The
sphereradiuscanbecalculatedbyvaluesofthesefrequencies.
Theaboveexamplesconcerningthepartsofcylindricalorsphericalsurfacesarenotexotic.
There are many such surfaces, moreover, in their neighborhood infinite numbers of other
surfacesofthiskindexist.Thisfactmakesthestudyoftheproblemreasonable.
Theshapeofthesurface,wherethecurrentsarelocated,canbedecisivefortheapprox-
imationnotonlyofanypatternbythepatternsofthesecurrents,butitisalsorelevanttothe
fieldsinthenearzone.
Electromagnetic Fields – Restrictions and Approximation.(cid:13) Boris Z. Katsenelenbaum
Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN:978-3-527-40388-2
10 1 Introduction
C.Aplaneisilluminatedbyabeamincomingfromtheantennalocatedinanotherplane
paralleltothegivenone. Cansomegivenfieldbecreatedonit? Theanswerispositiveonly
ifthefieldfulfilstheconditionthatsomeexpression,containingthisfield, equalszero. The
problem is a two-dimensional generalization of the problem on the existence condition for
the field, located on the finite straight-line segment, which creates a given pattern. In both
casestheproblemisreducedtoafirst-kindintegralequation.Theaboveconditionmeansthat
the pseudo-solutionto this equationsolves it, that is, the solution exists. In the case of the
one-dimensionalproblemon the patternof a linear current, the aboveconditionmeans that
thegivenpatternbelongstotheclassoffunctionsdefinedbythePaley–Wienertheorem.
2.Usingsimplereasoning,onecaneasilyexplaintheimpossibilityofapproximatingsome
classofpatternsfromthefirstexample. Letusgivesuchanexplanation,emphasizingthatit
isnotuniversal–ingeneral,thephysicalexplanationismorecomplicated.
As in most of the book, we will consider here the two-dimensional scalar formulation,
becauseitisshorterandmoredemonstrative. Inessence,thethree-dimensionalvectorprob-
lemsarenotmorecomplicated,buttheyaremuchmorecumbersome.Theseproblemswillbe
consideredinChapter5.
First,weprovethatif ,thennocurrentonthewholecircle generatesa
patternhavingaconstantte(cid:0)r(cid:0)m(cid:0)(cid:1)i(cid:2)n(cid:1)th(cid:2)eF(cid:3)ourierseries.Atthisfrequencytheree(cid:3)x(cid:2)ists(cid:2)thesolution
tothehomogeneousHelmholtzequationfortheelectricfield,whichequalszeroonthecircle
andhasnosingularityinsideit. Thissolutionis . Inotherwords,thecircle
isaresonantone. (cid:4)(cid:0)(cid:3)(cid:5)(cid:6)(cid:1) (cid:2) (cid:0)(cid:0)(cid:0)(cid:1)(cid:3)(cid:1)
Consideranauxiliaryinteriorproblemonthefield inahollowmetalcylinderof
(cid:0)
givenradius .Attheresonantfrequencytheeigeno(cid:4)s(cid:0)c(cid:3)i(cid:5)ll(cid:6)at(cid:1)ion canexist
insuchavol(cid:3)um(cid:2)e.(cid:2)Thecurrentonthewallsisproportionalto (cid:4)(cid:0)(cid:3)a(cid:5)n(cid:6)d(cid:1)d(cid:2)oe(cid:0)s(cid:0)n(cid:0)o(cid:1)t(cid:3)d(cid:1)ependon
(cid:0)
theangle . Thiscurrentgeneratesafieldequaltozerooutsi(cid:7)de(cid:4)(cid:8)th(cid:7)e(cid:3)cylinder. Belowwewill
(cid:0)
useonlyth(cid:6)efactthatthecurrent,independentoftheangle ,isnotradiatingatthisfrequency.
We returntotheproblemofthefieldgeneratedbyan(cid:6)arbitrarycurrentonthecirclearc.
Startingwiththecaseofthewholecircle,expandthecurrentintheFourierseries.Everyterm
oftheseriesisproportionalto or andgeneratesthefieldwiththe
sameangulardependence.But(cid:4)t(cid:5)h(cid:6)e(cid:9)z(cid:6)ero-o(cid:6)r(cid:7)d(cid:8)e(cid:9)rt(cid:6)er(cid:0)m(cid:9)d(cid:2)oe(cid:3)s(cid:5)n(cid:9)o(cid:5)t(cid:10)(cid:10)g(cid:10)(cid:1)enerateafieldoutsidethecircle,
thereforefor suchatermisabsentinthefieldexpansionofanycurrentaswellasinthe
pattern. (cid:3) (cid:11)(cid:2)
This result is also validforanyarcof the circle, in spite of the factthat the arc is not a
closedline and, therefore,nonradiatingcurrentcannotbe inducedonit. Assume that some
currentonthearcgeneratesapatternwithzero-ordertermintheFourierseries. Thenwecan
supply the arc to the whole circle and set the current to be zero on the supplementaryarc.
Inthiswaywehaveconstructedthecurrentonthewholecircle,generatingthepatternwith
nonzeroconstantterm.But,asitfollowsfromtheabove,itisimpossible.
The direct proofof the abovestatement is elementaryfor this example. If is a circle
arcofradius and isacurrent,locatedon ,thenthepatterngeneratedby(cid:12) is(with
accuracytoa(cid:2)nones(cid:13)s(cid:0)e(cid:14)n(cid:1)tialfactor) (cid:12) (cid:13)(cid:0)(cid:15)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:0)(cid:5)(cid:4) (1.1)
(cid:16)(cid:0)(cid:6)(cid:1)(cid:2) (cid:17) (cid:13)(cid:0)(cid:14)(cid:1)(cid:18)(cid:14)(cid:10)
(cid:1)
(cid:0)
1.1 Introduction 11
TheconstanttermintheFourierseriesforthepatternis(withthesameaccuracy)
(cid:0)(cid:0)
(1.2)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:2)(cid:1)(cid:2)(cid:3)(cid:3)(cid:4)(cid:1)(cid:0)(cid:5)(cid:6)(cid:1) (cid:7)(cid:0)(cid:8)(cid:1)(cid:2)(cid:8)(cid:9)
(cid:0) (cid:0)
(cid:1) (cid:1)
If ,thenthezerothtermintheseriesfor isabsent. Thispropertydoesnot
dep(cid:4)e(cid:1)n(cid:0)(cid:5)d(cid:6)o(cid:1)n(cid:2)bo(cid:4)ththecurrentandthelengthofarc . If(cid:0)(cid:0)(cid:1)i(cid:1)sthewholecircle,thenthisproves
oncemorethat,attheresonance,anycurrentont(cid:10)hewa(cid:10)llscreatesafieldwhichdoesnothave
thezerothFourierterm. However,themultiplier is factoredoutforthecircle arcas
well. (cid:4)(cid:1)(cid:0)(cid:5)(cid:6)(cid:1)
3. Letus state theprobleminmorespecific terms, but still withoutaspiringto anexact
formulation.Considerthecurrent locatedonthegivenline ( isthecoordinateon ).
Itgeneratesthepattern (cid:7)(cid:0)(cid:11)(cid:1) (cid:10) (cid:11) (cid:10)
(1.3)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:2) (cid:0)(cid:0)(cid:11)(cid:12)(cid:1)(cid:1)(cid:7)(cid:0)(cid:11)(cid:1)(cid:2)(cid:11)(cid:9)
(cid:0)
(cid:1)
Theformofasmoothkernel in(1.3)isnotimportanthere.(Inthethree-dimensionalvector
case,theline shouldberep(cid:0)lacedbyasurface, –byafunctionalmatrix,andsoon.)
Intheante(cid:10)nnasynthesistheory,equation(1.3)(cid:0)isconsideredastheintegralequationonthe
current . Weareinterestedintheproblemontheexistenceofasolutiontothisequation
ortoan(cid:7)e(cid:0)q(cid:11)u(cid:1)ationinwhich isreplacedbyanotherfunctioncloseto inthequadratic
metric. Thecurrent s(cid:0)ho(cid:0)(cid:1)ul(cid:1)dhaveafinitenorm. Firstofall,wewil(cid:0)l(cid:0)i(cid:1)nv(cid:1)estigatehowthe
existenceofasolutio(cid:7)n(cid:0)(cid:11)to(cid:1)thisequationdependsontheline .
For the validity of most of the results obtained below(cid:10), it is not necessary for the norm
of to be finite. Moreover, it is acceptable that the current may have singularities and
(cid:7)(cid:0)(cid:11)(cid:0)(cid:1)maynotbeintegrable.Itisonlysignificantthatthecurrentshouldbeintegrableitself,
(cid:1)so(cid:7)(cid:0)t(cid:11)h(cid:1)a(cid:1)t the integral on the right side of expression (2.6) exists (see below). This condition
isfulfilledalsoforthecurrentattheborderofasemi-plane(forbothpolarizations),andfor
, that is, for the approximation, usually used in the antenna array theory.
(cid:7)H(cid:0)o(cid:11)w(cid:1)e(cid:2)verÆ,(cid:0)f(cid:11)or(cid:3)si(cid:11)m(cid:1)p(cid:1)licity(particularlyinChapter4),wewillrequire (cid:0) tobeintegrable.
Thereexistsuchlines,forwhichequation(1.3)hasnosolution,a(cid:1)n(cid:7)d(cid:0)(cid:11)((cid:1)w(cid:1)hatisimportant)the
equation,inwhich isreplacedbyafunctionclosetoit,hasnosolution,either.Toobtain
thesolvableequatio(cid:0)n(cid:0),(cid:1)w(cid:1)eshouldchange byafinitevalue.Intermsoffunctionalanalysis
thismeansthatthecompletesetofcurren(cid:0)ts(cid:0)(cid:1)(cid:1) generatesanoncompletesetofpatterns .
Itturnsout,thatline possesses thisprop(cid:7)e(cid:0)r(cid:11)ty(cid:1),ifthereexistsasolutiontothehomoge(cid:0)n(cid:0)o(cid:1)u(cid:1)s
Helmholtz equation, e(cid:10)qualto zero on . In the aboveexample“A” this solution is .
The investigation of such a solution tu(cid:10)rns out to be a very efficient method for solv(cid:4)in(cid:1)g(cid:0)(cid:5)th(cid:13)(cid:1)e
problemofapproximabilityandtherelatedones.
Themost importantresult is that thereare “many”such lines and surfaces, and“many”
patterns ,forwhichequation(1.3)hasnosolutionevenif isreplacedbyanyfunc-
tion“clo(cid:0)se(cid:0)”(cid:1)t(cid:1)oit. Therefore,theeffectofnonapproximabilitya(cid:0)n(cid:0)d(cid:1)i(cid:1)tsconsequencesdeserves
adetailedanalysis.Wewillexplainbelow,what“many”and“close”mean.