Theoretical optics: an introduction

362 Pages·2005·2.14 MB·English

by  Hartmann Römer

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
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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