Physics of Light and Optics JustinPeatross MichaelWare BrighamYoungUniversity 2013Edition May21,2014 Preface Thiscurriculumwasoriginallydevelopedforafourth-yearundergraduateoptics courseintheDepartmentofPhysicsandAstronomyatBrighamYoungUniversity. Topicsareaddressedfromaphysicsperspectiveandincludethepropagationof lightinmatter,reflectionandtransmissionatboundaries,polarizationeffects, dispersion, coherence, ray optics and imaging, diffraction, and the quantum natureoflight.Studentsusingthisbookshouldbefamiliarwithdifferentiation, integration, and standard trigonometric and algebraic manipulation. A brief reviewofcomplexnumbers,vectorcalculus,andFouriertransformsisprovided inChapter0,butitishelpfulifstudentsalreadyhavesomeexperiencewiththese concepts. Whiletheauthorsretainthecopyright,wehavemadethisbookavailablefree ofchargeatoptics.byu.edu.Thisisourcontributiontowardafutureworldwith freetextbooks!Thewebsitealsoprovidesalinktopurchaseboundcopiesofthe bookforthecostofprinting. Acollectionofelectronicmaterialrelatedtothe textisavailableatthesamesite,includingvideosofstudentsperformingthelab assignmentsfoundinthebook. Thedevelopmentofopticshasarichhistory. Wehaveincludedhistorical sketchesforaselectionofthepioneersinthefieldtohelpstudentsappreciate someofthishistoricalcontext. Thesesketchesarenotintendedtobeauthor- itative; theinformationformostindividualshasbeengleanedprimarilyfrom Wikipedia. [email protected]. Weenjoyhearing reportsfromthoseusingthebookandwelcomeconstructivefeedback.Weocca- sionallyrevisethetext.Thetitlepageindicatesthedateofthelastrevision. Wewouldliketothankallthosewhohavehelpedimprovethismaterial.We especiallythankJohnColton,BretHess,andHaroldStokesfortheircarefulreview andextensivesuggestions. ThiscurriculumbenefittedfromaCCLIgrantfrom theNationalScienceFoundationDivisionofUndergraduateEducation(DUE- 9952773). iii Contents Preface iii TableofContents v 0 MathematicalTools 1 0.1 VectorCalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 ComplexNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.3 LinearAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 0.4 FourierTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendix0.A TableofIntegralsandSums . . . . . . . . . . . . . . . . 19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 ElectromagneticPhenomena 25 1.1 Gauss’Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2 Gauss’LawforMagneticFields . . . . . . . . . . . . . . . . . . . . 27 1.3 Faraday’sLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 Ampere’sLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Maxwell’sAdjustmenttoAmpere’sLaw. . . . . . . . . . . . . . . . 31 1.6 PolarizationofMaterials . . . . . . . . . . . . . . . . . . . . . . . . 34 1.7 TheWaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 PlaneWavesandRefractiveIndex 43 2.1 PlaneWaveSolutionstotheWaveEquation . . . . . . . . . . . . . 43 2.2 ComplexPlaneWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 IndexofRefraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 TheLorentzModelofDielectrics . . . . . . . . . . . . . . . . . . . 49 2.5 IndexofRefractionofaConductor . . . . . . . . . . . . . . . . . . 52 2.6 Poynting’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.7 IrradianceofaPlaneWave . . . . . . . . . . . . . . . . . . . . . . . 56 Appendix2.A Radiometry,Photometry,andColor . . . . . . . . . . . 58 Appendix2.B Clausius-MossottiRelation . . . . . . . . . . . . . . . . 61 Appendix2.C EnergyDensityofElectricFields . . . . . . . . . . . . . 64 Appendix2.D EnergyDensityofMagneticFields . . . . . . . . . . . . 66 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 v vi CONTENTS 3 ReflectionandRefraction 71 3.1 RefractionatanInterface . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2 TheFresnelCoefficients . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 ReflectanceandTransmittance . . . . . . . . . . . . . . . . . . . . 76 3.4 Brewster’sAngle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5 TotalInternalReflection . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6 ReflectionsfromMetal . . . . . . . . . . . . . . . . . . . . . . . . . 81 Appendix3.A BoundaryConditionsForFieldsatanInterface . . . . 82 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 MultipleParallelInterfaces 87 4.1 Double-InterfaceProblemSolvedUsingFresnelCoefficients. . . 88 4.2 TransmittancethroughDouble-InterfaceatSubCriticalAngles . 92 4.3 BeyondCriticalAngle:TunnelingofEvanescentWaves . . . . . . 95 4.4 Fabry-PerotInstrument . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 SetupofaFabry-PerotInstrument . . . . . . . . . . . . . . . . . . 98 4.6 DistinguishingNearbyWavelengthsinaFabry-PerotInstrument 100 4.7 MultilayerCoatings . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.8 PeriodicMultilayerStacks . . . . . . . . . . . . . . . . . . . . . . . 107 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Review,Chapters1–4 115 5 PropagationinAnisotropicMedia 121 5.1 ConstitutiveRelationinCrystals . . . . . . . . . . . . . . . . . . . 121 5.2 PlaneWavePropagationinCrystals. . . . . . . . . . . . . . . . . . 123 5.3 BiaxialandUniaxialCrystals . . . . . . . . . . . . . . . . . . . . . . 127 5.4 RefractionataUniaxialCrystalSurface . . . . . . . . . . . . . . . 128 5.5 PoyntingVectorinaUniaxialCrystal . . . . . . . . . . . . . . . . . 129 Appendix5.A SymmetryofSusceptibilityTensor . . . . . . . . . . . . 131 Appendix5.B RotationofCoordinates . . . . . . . . . . . . . . . . . . 133 Appendix5.C ElectricFieldinaCrystal . . . . . . . . . . . . . . . . . . 135 Appendix5.D Huygens’EllipticalConstructforaUniaxialCrystal . . 138 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6 PolarizationofLight 143 6.1 Linear,Circular,andEllipticalPolarization . . . . . . . . . . . . . 144 6.2 JonesVectorsforRepresentingPolarization . . . . . . . . . . . . . 145 6.3 EllipticallyPolarizedLight . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 LinearPolarizersandJonesMatrices . . . . . . . . . . . . . . . . . 147 6.5 JonesMatrixforaPolarizer . . . . . . . . . . . . . . . . . . . . . . . 150 6.6 JonesMatrixforWavePlates . . . . . . . . . . . . . . . . . . . . . . 151 6.7 PolarizationEffectsofReflectionandTransmission . . . . . . . . 153 Appendix6.A Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . 155 Appendix6.B PartiallyPolarizedLight . . . . . . . . . . . . . . . . . . 156 CONTENTS vii Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7 SuperpositionofQuasi-ParallelPlaneWaves 169 7.1 IntensityofSuperimposedPlaneWaves . . . . . . . . . . . . . . . 170 7.2 Groupvs.PhaseVelocity:SumofTwoPlaneWaves . . . . . . . . 172 7.3 FrequencySpectrumofLight . . . . . . . . . . . . . . . . . . . . . 174 7.4 WavePacketPropagationandGroupDelay . . . . . . . . . . . . . 178 7.5 QuadraticDispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.6 GeneralizedContextforGroupDelay. . . . . . . . . . . . . . . . . 183 Appendix7.A PulseChirpinginaGratingPair . . . . . . . . . . . . . . 187 Appendix7.B CausalityandExchangeofEnergywiththeMedium . . 189 Appendix7.C Kramers-KronigRelations . . . . . . . . . . . . . . . . . 194 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8 CoherenceTheory 201 8.1 MichelsonInterferometer . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 CoherenceTimeandFringeVisibility. . . . . . . . . . . . . . . . . 205 8.3 TemporalCoherenceofContinuousSources . . . . . . . . . . . . 207 8.4 FourierSpectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.5 Young’sTwo-SlitSetupandSpatialCoherence . . . . . . . . . . . 209 Appendix8.A SpatialCoherenceforaContinuousSpatialDistribution 213 Appendix8.B VanCittert-ZernikeTheorem . . . . . . . . . . . . . . . 214 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Review,Chapters5–8 221 9 LightasRays 227 9.1 TheEikonalEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9.2 Fermat’sPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.3 ParaxialRaysandABCDMatrices . . . . . . . . . . . . . . . . . . . 234 9.4 ReflectionandRefractionatCurvedSurfaces . . . . . . . . . . . . 236 9.5 ABCDMatricesforCombinedOpticalElements . . . . . . . . . . 238 9.6 ImageFormation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.7 PrincipalPlanesforComplexOpticalSystems . . . . . . . . . . . 244 9.8 StabilityofLaserCavities . . . . . . . . . . . . . . . . . . . . . . . . 246 Appendix9.A AberrationsandRayTracing . . . . . . . . . . . . . . . . 248 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10 Diffraction 257 10.1 Huygens’PrincipleasFormulatedbyFresnel . . . . . . . . . . . . 258 10.2 ScalarDiffractionTheory . . . . . . . . . . . . . . . . . . . . . . . . 260 10.3 FresnelApproximation . . . . . . . . . . . . . . . . . . . . . . . . . 262 10.4 FraunhoferApproximation . . . . . . . . . . . . . . . . . . . . . . . 264 10.5 DiffractionwithCylindricalSymmetry . . . . . . . . . . . . . . . . 265 Appendix10.A Fresnel-KirchhoffDiffractionFormula . . . . . . . . . . 267 viii CONTENTS Appendix10.B Green’sTheorem . . . . . . . . . . . . . . . . . . . . . . . 270 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 11 DiffractionApplications 275 11.1 FraunhoferDiffractionwithaLens . . . . . . . . . . . . . . . . . . 275 11.2 ResolutionofaTelescope . . . . . . . . . . . . . . . . . . . . . . . . 280 11.3 TheArrayTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 11.4 DiffractionGrating . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11.5 Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.6 DiffractionofaGaussianFieldProfile . . . . . . . . . . . . . . . . 288 11.7 GaussianLaserBeams . . . . . . . . . . . . . . . . . . . . . . . . . 290 Appendix11.A ABCDLawforGaussianBeams . . . . . . . . . . . . . . 292 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 12 InterferogramsandHolography 301 12.1 Interferograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 12.2 TestingOpticalSurfaces . . . . . . . . . . . . . . . . . . . . . . . . 302 12.3 GeneratingHolograms . . . . . . . . . . . . . . . . . . . . . . . . . 303 12.4 HolographicWavefrontReconstruction . . . . . . . . . . . . . . . 304 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Review,Chapters9–12 309 13 BlackbodyRadiation 315 13.1 Stefan-BoltzmannLaw . . . . . . . . . . . . . . . . . . . . . . . . . 316 13.2 FailureoftheEquipartitionPrinciple . . . . . . . . . . . . . . . . . 317 13.3 Planck’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 13.4 Einstein’sAandBCoefficients . . . . . . . . . . . . . . . . . . . . . 322 Appendix13.A ThermodynamicDerivationoftheStefan-Boltzmann Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Appendix13.B BoltzmannFactor . . . . . . . . . . . . . . . . . . . . . . 326 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Index 331 PhysicalConstants 336 Chapter 0 Mathematical Tools Beforemovingontochapter1whereourstudyofopticsbegins,itwouldbegood tolookoverthischaptertomakesureyouarecomfortablewiththemathematical toolswe’llbeusing.Thevectorcalculusinformationinsection0.1isusedstraight awayinChapter1,soyoushouldreviewitnow.InSection0.2wereviewcomplex numbers.Youhaveprobablyhadsomeexposuretocomplexnumbers,butifyou arelikemanystudents,youhaven’tyetfullyappreciatedtheirusefulness. Your lifewillbemucheasier ifyouunderstandthematerialinsection0.2byheart. Complexnotationispervasivethroughoutthebook,beginninginchapter2. YoumaysafelyprocrastinatereviewingSections0.3and0.4untiltheycome upinthebook.ThelinearalgebrarefresherinSection0.3isusefulforChapter4, whereweanalyzemultilayercoatings,andagaininChapter6,wherewediscuss polarization.Section0.4providesanintroductiontoFouriertheory.Fouriertrans- formsareusedextensivelyinoptics,andyoushouldstudySection0.4carefully RenØ Descartes(1596-1650,French) wasbornininLaHayeenTouraine beforetacklingChapter7. (nowDescartes),France. Hismother diedwhenhewasaninfant. Hisfather wasamemberofparliamentwhoen- 0.1 VectorCalculus couragedDescartestobecomealawyer. Descartesgraduatedwithadegreein lawfromtheUniversityofPoitiers Eachpositioninspacecorrespondstoauniquevector r≡xxˆ+yyˆ+zzˆ,where in1616. In1619,hehadaseriesof dreamsthatledhimtobelievethathe xˆ,yˆ,andzˆareunitvectorswithlengthone,pointingalongtheirrespectiveaxes. shouldinsteadpursuescience. Descartes Boldfacetypedistinguishesavariableasavectorquantity,andtheuseofxˆ,yˆ, becameoneofthegreatestmathemati- andzˆ denotesaCartesiancoordinatesystem. Electricandmagneticfieldsare cians,physicists,andphilosophersof alltime. Heiscreditedwithinventing vectorswhosemagnitudeanddirectioncandependonposition,asdenotedby thecartesiancoordinatesystem,which E(r)orB(r). AnexampleofsuchafieldisE(r)=q(r−r )(cid:177)4π(cid:178) |r−r |3,which isnamedafterhim. Forthe(cid:28)rsttime, 0 0 0 geometricshapescouldbeexpressedas isthestaticelectricfieldsurroundingapointchargelocatedatpositionr .The 0 algebraicequations. (Wikipedia) absolute-valuebracketsindicatethemagnitude(orlength)ofthevectorgivenby |r−r |=(cid:175)(cid:175)(x−x )xˆ+(cid:161)y−y (cid:162)yˆ+(z−z )zˆ(cid:175)(cid:175) 0 0 0 0 (cid:113) (0.1) = (x−x )2+(cid:161)y−y (cid:162)2+(z−z )2 0 0 0 1 2 Chapter0 MathematicalTools Example0.1 Computetheelectricfieldatr=(cid:161)2xˆ+2yˆ+2zˆ(cid:162)Åduetoapositivepointchargeq positionedatr =(cid:161)1xˆ+1yˆ+2zˆ(cid:162)Å. 0 Solution:Asmentionedabove,thefieldisgivenbyE(r)=q(r−r )(cid:177)4π(cid:178) |r−r |3. 0 0 0 Wehave r−r =(cid:161)(2−1)xˆ+(2−1)yˆ+(2−2)zˆ(cid:162) Å =(cid:161)1xˆ+1yˆ(cid:162) Å 0 and (cid:112) (cid:112) |r−r |= (1)2+(1)2Å = 2Å 0 Theelectricfieldisthen q(cid:161)1xˆ+1yˆ(cid:162) Å E= (cid:112) 4π(cid:178) (cid:161) 2Å(cid:162)3 0 Figure0.1Theelectricfieldvec- Inadditiontoposition,theelectricandmagneticfieldsalmostalwaysdepend torsaroundapointcharge. ontimeinopticsproblems. Forexample, acommontime-dependentfieldis E(r,t)=E cos(k·r−ωt).Thedotproductk·risanexampleofvectormultiplication, 0 andsignifiesthefollowingoperation: k·r=(cid:161)k xˆ+k yˆ+k zˆ(cid:162)·(cid:161)xxˆ+yyˆ+zzˆ(cid:162) x y z =k x+k y+k z (0.2) x y z =|k||r|cosφ whereφistheanglebetweenthevectorskandr. Proofofthefinallineof(0.2) (cid:48) (cid:48) Considertheplanethatcontainsthetwovectorskandr.Callitthex y -plane.In thiscoordinatesystem,thetwovectorscanbewrittenask=kcosθxˆ(cid:48)+ksinθyˆ(cid:48)and r=rcosαxˆ(cid:48)+rsinαyˆ(cid:48),whereθandαaretherespectiveanglesthatthetwovectors makewiththe x(cid:48)-axis. Thedotproductgivesk·r=kr(cosθcosα+sinθsinα). Thissimplifiestok·r=krcosφ(see(0.13)),whereφ≡θ−αistheanglebetween the vectors. Thus, the dot product between two vectors is the product of the magnitudesofeachvectortimesthecosineoftheanglebetweenthem. Anothertypeofvectormultiplicationisthecrossproduct,whichisaccom- plishedinthefollowingmanner:1 (cid:175) (cid:175) (cid:175) xˆ yˆ zˆ (cid:175) E×B=(cid:175)(cid:175) Ex Ey Ez (cid:175)(cid:175) (cid:175) (cid:175) (0.3) (cid:175) B B B (cid:175) x y z =(cid:161)E B −E B (cid:162)xˆ−(E B −E B )yˆ+(cid:161)E B −E B (cid:162)zˆ y z z y x z z x x y y x 1Theuseofthedeterminanttogeneratethecrossproductismerelyaconvenientdevicefor rememberingitsform.