Contents Preface vii 1 Maxwell’s Equations 1 1.1 Maxwell’sEquations, 1 1.2 LorentzForce, 2 1.3 ConstitutiveRelations, 3 1.4 BoundaryConditions, 6 1.5 Currents,Fluxes,andConservationLaws, 8 1.6 ChargeConservation, 9 1.7 EnergyFluxandEnergyConservation, 10 1.8 HarmonicTimeDependence, 12 1.9 SimpleModelsofDielectrics,Conductors,andPlasmas, 13 1.10 Problems, 21 2 Uniform Plane Waves 25 2.1 UniformPlaneWavesinLosslessMedia, 25 2.2 MonochromaticWaves, 31 2.3 EnergyDensityandFlux, 34 2.4 WaveImpedance, 35 2.5 Polarization, 35 2.6 UniformPlaneWavesinLossyMedia, 42 2.7 PropagationinWeaklyLossyDielectrics, 48 2.8 PropagationinGoodConductors, 49 2.9 PropagationinObliqueDirections, 50 2.10 ComplexWaves, 53 2.11 Problems, 55 3 Propagation in Birefringent Media 60 3.1 LinearandCircularBirefringence, 60 3.2 UniaxialandBiaxialMedia, 61 3.3 ChiralMedia, 63 3.4 GyrotropicMedia, 66 3.5 LinearandCircularDichroism, 67 3.6 ObliquePropagationinBirefringentMedia, 68 3.7 Problems, 75 ii CONTENTS iii 4 Reflection and Transmission 81 4.1 PropagationMatrices, 81 4.2 MatchingMatrices, 85 4.3 ReflectedandTransmittedPower, 88 4.4 SingleDielectricSlab, 91 4.5 ReflectionlessSlab, 94 4.6 Time-DomainReflectionResponse, 102 4.7 TwoDielectricSlabs, 104 4.8 Problems, 106 5 Multilayer Structures 109 5.1 MultipleDielectricSlabs, 109 5.2 AntireflectionCoatings, 111 5.3 DielectricMirrors, 116 5.4 PropagationBandgaps, 127 5.5 Narrow-BandTransmissionFilters, 127 5.6 EqualTravel-TimeMultilayerStructures, 132 5.7 ApplicationsofLayeredStructures, 146 5.8 ChebyshevDesignofReflectionlessMultilayers, 149 5.9 Problems, 156 6 Oblique Incidence 159 6.1 ObliqueIncidenceandSnell’sLaws, 159 6.2 TransverseImpedance, 161 6.3 PropagationandMatchingofTransverseFields, 164 6.4 FresnelReflectionCoefficients, 166 6.5 TotalInternalReflection, 168 6.6 BrewsterAngle, 174 6.7 ComplexWaves, 177 6.8 GeometricalOptics, 185 6.9 Fermat’sPrinciple, 187 6.10 RayTracing, 189 6.11 Problems, 200 7 Multilayer Film Applications 202 7.1 MultilayerDielectricStructuresatObliqueIncidence, 202 7.2 SingleDielectricSlab, 204 7.3 AntireflectionCoatingsatObliqueIncidence, 207 7.4 OmnidirectionalDielectricMirrors, 210 7.5 PolarizingBeamSplitters, 220 7.6 ReflectionandRefractioninBirefringentMedia, 223 7.7 BrewsterandCriticalAnglesinBirefringentMedia, 227 7.8 MultilayerBirefringentStructures, 230 7.9 GiantBirefringentOptics, 232 7.10 Problems, 237 iv ElectromagneticWaves&Antennas–S.J.Orfanidis 8 Waveguides 238 8.1 Longitudinal-TransverseDecompositions, 239 8.2 PowerTransferandAttenuation, 244 8.3 TEM,TE,andTMmodes, 246 8.4 RectangularWaveguides, 249 8.5 HigherTEandTMmodes, 251 8.6 OperatingBandwidth, 253 8.7 PowerTransfer,EnergyDensity,andGroupVelocity, 254 8.8 PowerAttenuation, 256 8.9 ReflectionModelofWaveguidePropagation, 259 8.10 ResonantCavities, 261 8.11 DielectricSlabWaveguides, 263 8.12 Problems, 271 9 Transmission Lines 273 9.1 GeneralPropertiesofTEMTransmissionLines, 273 9.2 ParallelPlateLines, 279 9.3 MicrostripLines, 280 9.4 CoaxialLines, 284 9.5 Two-WireLines, 289 9.6 DistributedCircuitModelofaTransmissionLine, 291 9.7 WaveImpedanceandReflectionResponse, 293 9.8 Two-PortEquivalentCircuit, 295 9.9 TerminatedTransmissionLines, 296 9.10 PowerTransferfromGeneratortoLoad, 299 9.11 Open-andShort-CircuitedTransmissionLines, 301 9.12 StandingWaveRatio, 304 9.13 DetermininganUnknownLoadImpedance, 306 9.14 SmithChart, 310 9.15 Time-DomainResponseofTransmissionLines, 314 9.16 Problems, 321 10 Coupled Lines 330 10.1 CoupledTransmissionLines, 330 10.2 CrosstalkBetweenLines, 336 10.3 WeaklyCoupledLineswithArbitraryTerminations, 339 10.4 Coupled-ModeTheory, 341 10.5 FiberBraggGratings, 343 10.6 Problems, 346 11 Impedance Matching 347 11.1 ConjugateandReflectionlessMatching, 347 11.2 MultisectionTransmissionLines, 349 11.3 Quarter-WavelengthImpedanceTransformers, 350 11.4 Quarter-WavelengthTransformerWithSeriesSection, 356 11.5 Quarter-WavelengthTransformerWithShuntStub, 359 11.6 Two-SectionSeriesImpedanceTransformer, 361 CONTENTS v 11.7 SingleStubMatching, 366 11.8 BalancedStubs, 370 11.9 DoubleandTripleStubMatching, 371 11.10L-SectionLumpedReactiveMatchingNetworks, 374 11.11Pi-SectionLumpedReactiveMatchingNetworks, 377 11.12Problems, 383 12 S-Parameters 386 12.1 ScatteringParameters, 386 12.2 PowerFlow, 390 12.3 ParameterConversions, 391 12.4 InputandOutputReflectionCoefficients, 392 12.5 StabilityCircles, 394 12.6 PowerGains, 400 12.7 GeneralizedS-ParametersandPowerWaves, 406 12.8 SimultaneousConjugateMatching, 410 12.9 PowerGainCircles, 414 12.10UnilateralGainCircles, 415 12.11OperatingandAvailablePowerGainCircles, 418 12.12NoiseFigureCircles, 424 12.13Problems, 428 13 Radiation Fields 430 13.1 CurrentsandChargesasSourcesofFields, 430 13.2 RetardedPotentials, 432 13.3 HarmonicTimeDependence, 435 13.4 FieldsofaLinearWireAntenna, 437 13.5 FieldsofElectricandMagneticDipoles, 439 13.6 Ewald-OseenExtinctionTheorem, 444 13.7 RadiationFields, 449 13.8 RadialCoordinates, 452 13.9 RadiationFieldApproximation, 454 13.10ComputingtheRadiationFields, 455 13.11Problems, 457 14 Transmitting and Receiving Antennas 460 14.1 EnergyFluxandRadiationIntensity, 460 14.2 Directivity,Gain,andBeamwidth, 461 14.3 EffectiveArea, 466 14.4 AntennaEquivalentCircuits, 470 14.5 EffectiveLength, 472 14.6 CommunicatingAntennas, 474 14.7 AntennaNoiseTemperature, 476 14.8 SystemNoiseTemperature, 480 14.9 DataRateLimits, 485 14.10SatelliteLinks, 487 14.11RadarEquation, 490 14.12Problems, 492 vi ElectromagneticWaves&Antennas–S.J.Orfanidis 15 Linear and Loop Antennas 493 15.1 LinearAntennas, 493 15.2 HertzianDipole, 495 15.3 Standing-WaveAntennas, 497 15.4 Half-WaveDipole, 499 15.5 MonopoleAntennas, 501 15.6 Traveling-WaveAntennas, 502 15.7 VeeandRhombicAntennas, 505 15.8 LoopAntennas, 508 15.9 CircularLoops, 510 15.10SquareLoops, 511 15.11DipoleandQuadrupoleRadiation, 512 15.12Problems, 514 16 Radiation from Apertures 515 16.1 FieldEquivalencePrinciple, 515 16.2 MagneticCurrentsandDuality, 517 16.3 RadiationFieldsfromMagneticCurrents, 519 16.4 RadiationFieldsfromApertures, 520 16.5 HuygensSource, 523 16.6 DirectivityandEffectiveAreaofApertures, 525 16.7 UniformApertures, 527 16.8 RectangularApertures, 527 16.9 CircularApertures, 529 16.10VectorDiffractionTheory, 532 16.11ExtinctionTheorem, 536 16.12VectorDiffractionforApertures, 538 16.13FresnelDiffraction, 539 16.14Knife-EdgeDiffraction, 543 16.15GeometricalTheoryofDiffraction, 549 16.16Problems, 555 17 Aperture Antennas 558 17.1 Open-EndedWaveguides, 558 17.2 HornAntennas, 562 17.3 HornRadiationFields, 564 17.4 HornDirectivity, 569 17.5 HornDesign, 572 17.6 MicrostripAntennas, 575 17.7 ParabolicReflectorAntennas, 581 17.8 GainandBeamwidthofReflectorAntennas, 583 17.9 Aperture-FieldandCurrent-DistributionMethods, 586 17.10RadiationPatternsofReflectorAntennas, 589 17.11Dual-ReflectorAntennas, 598 17.12LensAntennas, 601 17.13Problems, 602 CONTENTS vii 18 Antenna Arrays 603 18.1 AntennaArrays, 603 18.2 TranslationalPhaseShift, 603 18.3 ArrayPatternMultiplication, 605 18.4 One-DimensionalArrays, 615 18.5 VisibleRegion, 617 18.6 GratingLobes, 618 18.7 UniformArrays, 621 18.8 ArrayDirectivity, 625 18.9 ArraySteering, 626 18.10ArrayBeamwidth, 628 18.11Problems, 630 19 Array Design Methods 632 19.1 ArrayDesignMethods, 632 19.2 Schelkunoff’sZeroPlacementMethod, 635 19.3 FourierSeriesMethodwithWindowing, 637 19.4 SectorBeamArrayDesign, 638 19.5 Woodward-LawsonFrequency-SamplingDesign, 643 19.6 Narrow-BeamLow-SidelobeDesigns, 647 19.7 BinomialArrays, 651 19.8 Dolph-ChebyshevArrays, 653 19.9 Taylor-KaiserArrays, 665 19.10MultibeamArrays, 668 19.11Problems, 671 20 Currents on Linear Antennas 672 20.1 Hall´enandPocklingtonIntegralEquations, 672 20.2 Delta-GapandPlane-WaveSources, 675 20.3 SolvingHall´en’sEquation, 676 20.4 SinusoidalCurrentApproximation, 678 20.5 ReflectingandCenter-LoadedReceivingAntennas, 679 20.6 King’sThree-TermApproximation, 682 20.7 NumericalSolutionofHall´en’sEquation, 686 20.8 NumericalSolutionUsingPulseFunctions, 689 20.9 NumericalSolutionforArbitraryIncidentField, 693 20.10NumericalSolutionofPocklington’sEquation, 695 20.11Problems, 701 21 Coupled Antennas 702 21.1 NearFieldsofLinearAntennas, 702 21.2 SelfandMutualImpedance, 705 21.3 CoupledTwo-ElementArrays, 709 21.4 ArraysofParallelDipoles, 712 21.5 Yagi-UdaAntennas, 721 21.6 Hall´enEquationsforCoupledAntennas, 726 21.7 Problems, 733 viii ElectromagneticWaves&Antennas–S.J.Orfanidis 22 Appendices 735 A PhysicalConstants, 735 B ElectromagneticFrequencyBands, 736 C VectorIdentitiesandIntegralTheorems, 738 D Green’sFunctions, 740 E CoordinateSystems, 743 F FresnelIntegrals, 745 G MATLABFunctions, 748 References 753 Index 779 1 Maxwell’s Equations 1.1 Maxwell’s Equations Maxwell’sequationsdescribeall(classical)electromagneticphenomena: ∂B ∇∇∇×E=− ∂t ∂D ∇∇∇×H=J+ ∂t (Maxwell’sequations) (1.1.1) ∇∇∇·D=ρ ∇∇∇·B=0 The first is Faraday’s law of induction, the second is Amp`ere’s law as amended by Maxwelltoincludethedisplacementcurrent∂D/∂t,thethirdandfourthareGauss’laws fortheelectricandmagneticfields. Thedisplacementcurrentterm∂D/∂tinAmp`ere’slawisessentialinpredictingthe existenceofpropagatingelectromagneticwaves. Itsroleinestablishingchargeconser- vationisdiscussedinSec.1.6. Eqs. (1.1.1) are in SI units. The quantities E and H are the electric and magnetic field intensities and are measured in units of [volt/m] and [ampere/m], respectively. ThequantitiesDandB aretheelectricandmagneticfluxdensities andareinunitsof [coulomb/m2]and[weber/m2],or[tesla]. B isalsocalledthemagneticinduction. The quantities ρ and J are the volume charge density and electric current density (charge flux) of any external charges (that is, not including any induced polarization chargesandcurrents.) Theyaremeasuredinunitsof[coulomb/m3]and[ampere/m2]. Theright-handsideofthefourthequationiszerobecausetherearenomagneticmono- polecharges. Thechargeandcurrentdensitiesρ,J maybethoughtofasthesourcesoftheelectro- magneticfields. Forwavepropagationproblems,thesedensitiesarelocalizedinspace; forexample,theyarerestrictedtoflowonanantenna. Thegeneratedelectricandmag- neticfieldsareradiatedawayfromthesesourcesandcanpropagatetolargedistancesto 2 ElectromagneticWaves&Antennas–S.J.Orfanidis thereceivingantennas. Awayfromthesources,thatis,insource-freeregionsofspace, Maxwell’sequationstakethesimplerform: ∂B ∇∇∇×E=− ∂t ∂D ∇∇∇×H= ∂t (source-freeMaxwell’sequations) (1.1.2) ∇∇∇·D=0 ∇∇∇·B=0 1.2 Lorentz Force Theforceonachargeqmovingwithvelocityv inthepresenceofanelectricandmag- neticfieldE,B iscalledtheLorentzforceandisgivenby: F=q(E+v×B) (Lorentzforce) (1.2.1) Newton’sequationofmotionis(fornon-relativisticspeeds): dv m =F=q(E+v×B) (1.2.2) dt wheremisthemassofthecharge. TheforceF willincreasethekineticenergyofthe chargeataratethatisequaltotherateofworkdonebytheLorentzforceonthecharge, thatis,v·F. Indeed,thetime-derivativeofthekineticenergyis: 1 dW dv W = mv·v ⇒ kin =mv· =v·F=qv·E (1.2.3) kin 2 dt dt Wenotethatonlytheelectricforcecontributestotheincreaseofthekineticenergy— themagneticforceremainsperpendiculartov,thatis,v·(v×B)=0. Volume charge and current distributions ρ,J are also subjected to forces in the presenceoffields. TheLorentzforceperunitvolumeactingonρ,J isgivenby: f=ρE+J×B (Lorentzforceperunitvolume) (1.2.4) where f is measured in units of [newton/m3]. If J arises from the motion of charges withinthedistributionρ,thenJ=ρv(asexplainedinSec.1.5.) Inthiscase, f=ρ(E+v×B) (1.2.5) ByanalogywithEq.(1.2.3),thequantityv·f=ρv·E=J·E representsthepower perunitvolumeoftheforcesactingonthemovingcharges,thatis,thepowerexpended by(orlostfrom)thefieldsandconvertedintokineticenergyofthecharges,orheat. It hasunitsof[watts/m3]. Wewilldenoteitby: dP loss =J·E (ohmicpowerlossesperunitvolume) (1.2.6) dV 1.3. ConstitutiveRelations 3 InSec.1.7,wediscussitsroleintheconservationofenergy. Wewillfindthatelec- tromagneticenergyflowingintoaregionwillpartiallyincreasethestoredenergyinthat regionandpartiallydissipateintoheataccordingtoEq.(1.2.6). 1.3 Constitutive Relations TheelectricandmagneticfluxdensitiesD,BarerelatedtothefieldintensitiesE,Hvia theso-calledconstitutiverelations,whosepreciseformdependsonthematerialinwhich thefieldsexist. Invacuum,theytaketheirsimplestform: D=(cid:3) E 0 (1.3.1) B=µ H 0 where(cid:3) ,µ arethepermittivity andpermeability ofvacuum,withnumericalvalues: 0 0 (cid:3) =8.854×10−12farad/m 0 (1.3.2) µ =4π×10−7henry/m 0 Theunitsfor(cid:3) andµ aretheunitsoftheratiosD/EandB/H,thatis, 0 0 coulomb/m2 coulomb farad weber/m2 weber henry = = , = = volt/m volt·m m ampere/m ampere·m m Fromthetwoquantities(cid:3) ,µ ,wecandefinetwootherphysicalconstants,namely, 0 0 thespeedoflight andcharacteristicimpedanceofvacuum: (cid:1) 1 µ c = √ =3×108m/sec, η = 0 =377ohm (1.3.3) 0 µ (cid:3) 0 (cid:3) 0 0 0 Thenextsimplestformoftheconstitutiverelationsisforsimpledielectricsandfor magneticmaterials: D=(cid:3)E (1.3.4) B=µH Thesearetypicallyvalidatlowfrequencies. Thepermittivity(cid:3)andpermeabilityµ arerelatedtotheelectricandmagneticsusceptibilities ofthematerialasfollows: (cid:3)=(cid:3) (1+χ) 0 (1.3.5) µ=µ0(1+χm) The susceptibilities χ,χm are measures of the electric and magnetic polarization propertiesofthematerial. Forexample,wehavefortheelectricfluxdensity: D=(cid:3)E=(cid:3) (1+χ)E=(cid:3) E+(cid:3) χE=(cid:3) E+P 0 0 0 0