Titelei_Römer 18.10.2004 17:08 Uhr Seite 3 Hartmann Römer Theoretical Optics An Introduction WILEY-VCH Verlag GmbH & Co. KGaA Titelei_Römer 18.10.2004 17:08 Uhr Seite 4 Author All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher Prof. Dr. Hartmann Römer do not warrant the information contained in these University of Freiburg books, including this book, to be free of errors. Institute of Physics Readers are advised to keep in mind that statements, [email protected] data, illustrations, procedural details or other items may inadvertently be inaccurate. 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KG., Grünstadt ISBN 3-527-40429-5 Contents PrefacetotheGermanedition IX PrefacetotheEnglishedition XIII 1 Ashortsurveyofthehistoryofoptics 1 2 Theelectrodynamicsofcontinuousmedia 15 2.1 Maxwell’sequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Molecularvs.macroscopicfields . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Asimplemodelfortheelectriccurrent . . . . . . . . . . . . . . . . . . . . 20 2.4 Dispersionrelationsandthepassivitycondition. . . . . . . . . . . . . . . . 23 2.5 Electricdisplacementdensityandmagneticfieldstrength . . . . . . . . . . 27 2.6 Indexofrefractionandcoefficientofabsorption . . . . . . . . . . . . . . . 33 2.7 Theelectromagneticmaterialquantities . . . . . . . . . . . . . . . . . . . . 35 2.8 Theoscillatormodelfortheelectricsusceptibility . . . . . . . . . . . . . . 39 2.9 Materialequationsinmovingmedia . . . . . . . . . . . . . . . . . . . . . . 40 3 Linearwavesinhomogeneousmedia 45 3.1 Elasticwavesinsolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Isotropicelasticmedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Wavesurfacesandraysurfaces . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Crystaloptics 55 4.1 Thenormalellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Planewavesincrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Opticallyuniaxialcrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Opticallybiaxialcrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Reflectionandrefractionatinterfaces . . . . . . . . . . . . . . . . . . . . . 66 4.6 Fresnel’sequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.7 TheFabry–Perotinterferometer . . . . . . . . . . . . . . . . . . . . . . . . 72 5 Electro-,magneto-andelastoopticalphenomena 75 5.1 Polarizationeffectsuptofirstorder–opticalactivity . . . . . . . . . . . . . 75 5.2 Polarizationeffectsofhigherorder . . . . . . . . . . . . . . . . . . . . . . 79 5.2.1 Dependenceondistortions . . . . . . . . . . . . . . . . . . . . . . 80 VI Contents 5.2.2 Dependenceonshearflows . . . . . . . . . . . . . . . . . . . . . . 80 5.2.3 Influenceofelectricfields . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.4 Dependenceonmagneticfields . . . . . . . . . . . . . . . . . . . . 81 6 Foundationsofnonlinearoptics 83 6.1 Nonlinearpolarization–combinationfrequencies . . . . . . . . . . . . . . 83 6.2 Nonlinearwavesinamedium . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3 Surveyofphenomenainnonlinearoptics . . . . . . . . . . . . . . . . . . . 89 6.4 Parametricamplificationandfrequencydoubling . . . . . . . . . . . . . . . 91 6.5 Phasematching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.6 Self-focussing,opticalbistability,phaseself-modulation . . . . . . . . . . . 95 6.7 Phaseconjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.8 Fiberopticsandopticalsolitons . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Short-waveasymptotics 107 7.1 Introductoryremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Short-waveexpansionofMaxwell’sequations . . . . . . . . . . . . . . . . 109 7.3 Thescalarwaveequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 Phasesurfacesandrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.5 Fermat’sprinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.6 Analogybetweenmechanicsandgeometricaloptics . . . . . . . . . . . . . 116 8 Geometricaloptics 121 8.1 Fermat’sprincipleandfocalpoints . . . . . . . . . . . . . . . . . . . . . . 121 8.2 Perfectopticalinstruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3 Maxwell’sfish-eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4 Canonicaltransformationsandeikonalfunctions . . . . . . . . . . . . . . . 125 8.5 Imagingpointsclosetotheopticaxisbywidespreadraybundles . . . . . . 128 8.6 Lineargeometricalopticsandsymplectictransformations . . . . . . . . . . 131 8.7 Gaussianopticsandimagematrices . . . . . . . . . . . . . . . . . . . . . . 134 8.8 LensdefectsandSeidel’stheoryofaberrations . . . . . . . . . . . . . . . . 139 9 Geometrictheoryofcaustics 143 9.1 Short-waveasymptoticsforlinearpartialdifferentialequations . . . . . . . 143 9.2 Solutionofthecharacteristicequation . . . . . . . . . . . . . . . . . . . . . 146 9.3 Solutionofthetransportequation . . . . . . . . . . . . . . . . . . . . . . . 151 9.4 Focalpointsandcaustics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.5 Behaviorofphasesinthevicinityofcaustics . . . . . . . . . . . . . . . . . 156 9.6 Caustics,LagrangiansubmanifoldsandMaslovindex . . . . . . . . . . . . 158 9.7 Supplementaryremarksongeometricalshort-waveasymptotics . . . . . . . 161 10 Diffractiontheory 167 10.1 Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.2 TheprinciplesofHuygensandFresnel . . . . . . . . . . . . . . . . . . . . 167 10.3 Themethodofstationaryphases. . . . . . . . . . . . . . . . . . . . . . . . 171 Contents VII 10.4 Kirchhoff’srepresentationofthewaveamplitude . . . . . . . . . . . . . . . 175 10.5 Kirchhoff’stheoryofdiffraction . . . . . . . . . . . . . . . . . . . . . . . . 179 10.6 Diffractionatanedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.7 ExamplesofFraunhoferdiffraction . . . . . . . . . . . . . . . . . . . . . . 186 10.7.1 Diffractionbyarectangle . . . . . . . . . . . . . . . . . . . . . . . 187 10.7.2 Diffractionbyacircularaperture . . . . . . . . . . . . . . . . . . . 188 10.7.3 Arrangementsofseveralidenticalstructures . . . . . . . . . . . . . 189 10.8 OpticalimageprocessinginFourierspace . . . . . . . . . . . . . . . . . . 191 10.9 Morsefamilies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.10 OscillatoryfunctionsandFourierintegraloperators . . . . . . . . . . . . . 198 11 Holography 203 11.1 Theprincipleofholography . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11.2 Modificationsandapplications . . . . . . . . . . . . . . . . . . . . . . . . 205 11.2.1 Observingsmallobjectdeformations . . . . . . . . . . . . . . . . . 206 11.2.2 Holographicopticalinstruments . . . . . . . . . . . . . . . . . . . 206 11.2.3 Patternrecognition . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.3 Volumeholograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 12 Coherencetheory 211 12.1 Coherentandincoherentlight . . . . . . . . . . . . . . . . . . . . . . . . . 211 12.2 Realandanalyticalsignals . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.3 Thelightwavefieldasastochasticprocess . . . . . . . . . . . . . . . . . . 217 12.4 Gaussianstochasticprocesses . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.5 Thequasi-monochromaticapproximation . . . . . . . . . . . . . . . . . . . 222 12.6 Coherenceandcorrelationfunctions. . . . . . . . . . . . . . . . . . . . . . 224 12.7 Thepropagationofthecorrelationfunction . . . . . . . . . . . . . . . . . . 227 12.8 Amplitudeandintensityinterferometry . . . . . . . . . . . . . . . . . . . . 230 12.8.1 Amplitudeinterferometry:Michelsoninterferometer . . . . . . . . 230 12.8.2 Photoncorrelationspectroscopy . . . . . . . . . . . . . . . . . . . 231 12.9 Dynamicallightscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 12.10 Granulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 12.11 Imageprocessingbyfiltering . . . . . . . . . . . . . . . . . . . . . . . . . 237 12.12 Polarizationofpartiallycoherentlight. . . . . . . . . . . . . . . . . . . . . 239 13 Quantumstatesoftheelectromagneticfield 245 13.1 Quantizationoftheelectromagneticfieldandharmonicoscillators . . . . . . 245 13.2 Coherentandsqueezedstates . . . . . . . . . . . . . . . . . . . . . . . . . 251 13.3 Operators,orderingproceduresandstarproducts . . . . . . . . . . . . . . . 259 13.4 TheQ,P,andWignerfunctionsofadensityoperator . . . . . . . . . . . . 266 14 Detectionofradiationfields 273 14.1 Beamsplittersandhomodynedetection . . . . . . . . . . . . . . . . . . . . 273 14.2 Correlationfunctionsandquantumcoherence. . . . . . . . . . . . . . . . . 279 14.3 Measurementofcorrelationfunctions . . . . . . . . . . . . . . . . . . . . . 281 14.4 Anti-bunchingandsub-Poissonianlight . . . . . . . . . . . . . . . . . . . . 285 VIII Contents 15 Interactionofradiationandmatter 289 15.1 Theelectricdipoleinteraction . . . . . . . . . . . . . . . . . . . . . . . . . 289 15.2 Simplelasertheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 15.3 Three-levelsystemsandatomicinterference . . . . . . . . . . . . . . . . . 296 15.3.1 Electromagneticallyinducedtransparency . . . . . . . . . . . . . . 299 15.3.2 Refractiveindexenhancement . . . . . . . . . . . . . . . . . . . . 301 15.3.3 Lasingwithoutinversion . . . . . . . . . . . . . . . . . . . . . . . 301 15.3.4 Correlatedemissionlaser . . . . . . . . . . . . . . . . . . . . . . . 301 15.4 TheJaynes–Cummingsmodel . . . . . . . . . . . . . . . . . . . . . . . . . 302 15.5 Themicromaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 15.6 Quantumstateengineering. . . . . . . . . . . . . . . . . . . . . . . . . . . 310 15.7 ThePaultrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.8 Motionofatwo-levelatominaquantizedlightfield . . . . . . . . . . . . . 320 16 Quantumopticsandfundamentalquantumtheory 323 16.1 Quantumentanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 16.2 Bell’sinequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 16.3 Quantumerasersandmeasurementwithoutinteraction . . . . . . . . . . . . 332 16.4 Nocloningandquantumteleportation . . . . . . . . . . . . . . . . . . . . . 337 16.5 Quantumcryptography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 16.6 Quantumcomputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Selectedreferences 351 Index 355 Preface to the German edition Theideaforwritingthisbookcrystallizedfromaninterestinmethodsofshort-waveasymp- toticsandsymplecticgeometry.Laterfollowedacourseabouttheoreticaloptics,whichgave megreatpleasure.Thispleasureturnedouttobelasting,andsothepresentbookarosefrom severalrevisionsandextensionsoftheoriginalmanuscript. Indeed,therearemanyreasonstopresentthevenerableandtraditionalfieldofopticsina newform.Thediscoveryoflasers,thesteadyprogressofpowerfuldata-processingsystems, andthedevelopmentofnewmaterialswithunusualopticalpropertieshaveallrevolutionized thefieldofoptics. Lasers are light sources of unprecedented intensity and coherence, without which such new branches as nonlinear optics and holography would have been impossible. Computer technologyallowstheprocessingofopticalsignalsandtheconstructionofdiffractiveoptical devices,whichhavealreadytakentheirplacenexttocommonlensesandmirrors.Glass-fiber cables, compact disks, and simple holograms are proof that the products of the new kind of opticshavealreadyenteredourdailylife. However,notonlyhaveexperimentalphysicsandtechnologyexperiencedrapidprogress, butalsowehaveseenthedevelopmentandapplicationofnewtheoreticalmethodsinoptics: Themoderntheoryofnonlineardynamicalsystemshasfoundmanyapplicationsinnonlin- earoptics.Herewemeetbifurcationandchaoticbehavioraswellasdispersion-freesolutions ofnonlinearintegrablesystems. Theadoptionoftheconceptsandmethodsfromthetheoryofstochasticprocessesforthe description of fluctuating light wave fields turned out to be extremely fruitful. In this way, stochastic optics became a new branch of theoretical optics; the notion of coherence found a more profound formulation, and today applications for image processing and correlation spectroscopyhavebecomestandardroutines. Progressinthetheoryofshort-waveasymptoticswithintheframeworkofsymplecticge- ometrynotonlyledtoanimprovementoftheWKBmethodandtoabetterunderstandingof thequasi-classicallimitofquantummechanics,but“symplecticoptics”alsoallowsadeeper insightintothegeometricalstructuresofthe realmbetweenwaveopticsandrayoptics,into thenatureofcaustics,andintothetheoryofdiffraction. These overwhelming developments in the applications and the theory of optics have led to a considerable number of publications in recent years. However, these presentations, of- ten written in the form of an experimental textbook, either provide a summary of methods, phenomena,andapplicationsofmodernoptics,ortheydescribe,intheformofamonograph, specialpartsofexperimental,applied,ortheoreticaloptics. TheoreticalOptics.HartmannRo¨mer Copyright(cid:1)c 2005Wiley-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40429-5 X PrefacetotheGermanedition The present book attempts a coherent and concise presentation of optics, emphasizing theperspectiveofa theoretician. However,it is notmeanttobe a textbookonmathematical physics,buttriesinsteadtomediateamongdifferentpositions: Fortheexperimentalist,theappliedphysicist,andthetheoretician,thisbookaimstopro- vide a unification and a deeper understanding of the theoretical background, and, for those who are interested, a first access to the corresponding mathematical literature. The theoreti- callyor mathematically inclined reader is introduced to the “applications” and the manifold phenomena relatedto the theory of light. What, in my opinion, makes optics so particularly attractive is that in this field the path between theory and phenomena is shorter and more straightforwardthaninotherphysicalareas. In order to remain comprehensible for the above-mentionedgroup of readers, I have as- sumedaslittleaspossiblepreviousknowledge.Thebookrequiresonlybasicknowledgeabout Maxwell’sequationsandtheunderlyingelementaryvectoranalysis,acertainfamiliaritywith theessentialpropertiesofFouriertransformations,andthesimplestphenomenaofthepropa- gationofwavesandwavepackets,inparticularthenotionofgroupvelocity. Inshort,thisbookpursuesthefollowingthreeaims: • provisionofatheoreticaloverview; • theoreticalextensionandintroductiontothefundamentalmathematics; • presentationandinterpretationofmanyimportantopticalphenomena. Alookatthetableofcontentsrevealsthat,inparticular,thisthirdaimhasnotbeenneglected: thedescriptionofmanifoldphenomenaincrystaloptics,nonlinearoptics,geometricaloptics, diffraction theory, diffraction optics, as well as statistical optics and coherence optics will providethisbookwithasolidframe. Unfortunately,certainchoicesandlimitationofthematerialturnedouttobeunavoidable.I decidedtoincludeonlyashortpresentationofthoseopticalphenomenawhichdependmainly onquantumtheoryortheparticlepictureoflight.Inparticular,thisreferstoquantumoptics andthetheoryoflasers,aswellastheinteractionoflightwithmatter,likethephotoelectricef- fect,theComptoneffect,paircreationandbremsstrahlung,and,finally,applicationsofoptics inatomicandmolecularspectroscopy. Inthisway,thepresentationofthematerialinthisbookcouldbestructuredasastraight- forwarddevelopmentofthecontentofMaxwell’sequations: Chapter1containssomehistoricalremarks,withanemphasisonthedevelopmentofthe wavetheoryoflightandthedescriptionofNewtonianoptics. Chapter 2 develops in a brief and concise form the electrodynamics of media and it is shownhowtheinfluenceofpolarizablemediacanbetakenintoaccountbytheintroduction oftheadditionalfieldsD andH.Wewilldescribethecausalityandpassivityconditionsof mediaandtheirinfluenceontheconductivityandsusceptibilities. Thefollowingfourchaptersaredevotedtothepropagationofwavesinhomogeneousbut notnecessarilyisotropicmedia.Chapter3servesasanillustrativeintroductiontothegeneral theory of wave propagation in elastic media; in particular, we will explain the notions of a wavesurfaceandaraysurface,whichwillturnouttobeessentialforthecomingchapters. PrefacetotheGermanedition XI Chapter 4 summarizes the basic concepts of the theory of crystal optics. Among other things, we will discuss double refraction, conical refraction, and reflection and refraction at interfacesbetweenhomogeneousmedia. Chapter5dealswiththeinterestingphenomenarelatedtoelectro-,magneto-,andelasto- opticalproperties,whileChapter6isdevotedtononlinearoptics.Itcontainsasummaryofthe mostimportantnonlinearopticalphenomena,thetheoryofnonlinearwaves,andthecoupling of three waves.We will describe the phenomena of frequency doubling, parametric amplifi- cation, self-focussing, momentum contraction, phase conjugation, wave conduction in glass fibers,andopticalsolitons. The following five chapters deal with the propagation of wavesin isotropic but not nec- essarily homogeneous media. In general, this is a very difficult problem, which includes, amongstotherthings, the theory of mostopticalelements.The necessarytools will turn out tobetheshort-waveasymptoticsandthetheoryofdiffraction.Althoughthepresentationwill remain on an elementary level, we also want to provide the necessary foundations for the advancedmathematicaltheories. InChapter7,wedescribethetransitionfromwaveopticstogeometricalopticsanddevelop aformalanalogybetweenclassicalmechanicsandgeometricaloptics. Chapter8isdevotedtogeometricaloptics.Specialemphasisisputonthepresentationof matrixmethodsandtherelationtolinearsymplectictransformations.Additionalsubjectsare theimpossibilityofperfectopticalinstruments,andSeidel’stheoryofaberrations. Chapter9containsageneralgeometricaltreatmentofshort-waveasymptotics,inparticu- larageometricaltheoryofcaustics.Inthiscontextwewillexplainthenotionsofcharacteris- ticequations,transportequations,focalpoints,Lagrangiansubmanifolds,andMaslowindex. Theretheinterestedreaderwillfindashortintroductiontosymplecticgeometry. In Chapter 10, we will discuss the theory of diffraction, in particular the principles of Huygens and Fresnel, Fraunhofer diffraction, and image processing in Fourier space. Two sectionsattheendofthischaptercontainanintroductiontothetheoryofMorsefamiliesand Fourierintegraloperators. Chapter11describesbrieflythefoundationsandapplicationsofholography,aparticularly attractivebranchofdiffractionoptics. Chapter12concludesthisbookwithasummarizingpresentationofstatisticaloptics.The centralsubjectis a descriptionof the wavefield by stochasticprocesses,which, ina natural way,leadstothenotionsofthevariouscorrelationfunctions.Asapplications,wewillexplain correlationspectroscopy,dynamicallightscattering,speckleeffect,andimageprocessingby filtering. Itismysincerehopethatthisbookwillgivethebeginneracomprehensibleintroduction tothesubjectoftheoreticalobjectsand,atthesametime,willconveytotheadvancedreader manyinterestinginsights. Finally, it is my pleasant duty to thank all those who have helped me, often essentially, during work on this book: First of all I should like to thank the students who attended my lectures. Their attention and interest has always been encouraging for me, and I owe them thanksformanyhelpfulsuggestions.Furthermore,Iamverygratefultoformermembersofthe Institute for Theoretical Physics for their constructive criticism and comments, in particular J. Barth, W. Bischoff, C. Emmrich, M. Forger, T. Filk, D. Giulini, P. Glo¨ßner, C. Kiefer, M. Koch, A. Mu¨nnich, K. Nowak, H. Steger, and A. Winterhalder, as well as E. Binz from XII PrefacetotheGermanedition theUniversityofMannheim.C.HeinischfromKaiserslauternhas,withgreatpatience,setthe manuscriptinLATEX. TwopersonsIshouldliketomentionwithspecialthanks:E.Meinrenkenwasoneofthe attendeesatmylectures.Later,hedealtintensivelywiththetheoryofshort-waveasymptotics. The final mathematical sections in chapters 9 and 10 owe much to the presentations in his diplomathesis.Furthermore,hecarefullylookedthroughthepreliminaryversionsofchapters 7to10.IamalsogratefultoG.JerkefromtheVCHpublishingcompany.Hisconstantinterest inthedevelopmentofthisbook,hispersonalefforts,andcountlessvaluablesuggestionswent far beyond what one can usually expect during such a book project. Finally, I should like to thank the VCH publishing company, in particular R. Wengenmayr, for their pleasant and confidentialcooperation. Freiburg,April1994 HartmannRo¨mer