Physics of Light and Optics - Optics Education - Brigham Young

344 Pages·2014·17.72 MB·English

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.