Orestes N. Stavroudis The Mathematics of Geometrical and Physical Optics k The -function and its Ramifications TheAuthor AllbookspublishedbyWiley-VCHarecarefully Prof.Dr.OrestesN.Stavroudis produced.Nevertheless,authors,editors,andpublisher CentrodeInvestigacionesenOptica, donotwarranttheinformationcontainedinthese Leo´n,Guanajuato,Mexico books,includingthisbook,tobefreeoferrors.Readers [email protected] areadvisedtokeepinmindthatstatements,data, illustrations,proceduraldetailsorotheritemsmay inadvertentlybeinaccurate. Coverpicture PersistenceofVisionRaytracerVersion3.5 LibraryofCongressCardNo.: SampleFile appliedfor Author:ChristopherJ.Huff BritishLibraryCataloguing-in-PublicationData Acataloguerecordforthisbookisavailablefromthe BritishLibrary. Bibliographicinformationpublishedby DieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationinthe DeutscheNationalbibliografie;detailedbibliographic dataisavailableintheInternetat<http://dnb.ddb.de>. (cid:1)c 2006WILEY-VCHVerlagGmbH&Co.KGaA, Weinheim Allrightsreserved(includingthoseoftranslationinto otherlanguages).Nopartofthisbookmaybe reproducedinanyform–byphotoprinting,microfilm, oranyothermeans–nortransmittedortranslatedintoa machinelanguagewithoutwrittenpermissionfromthe publishers.Registerednames,trademarks,etc.usedin thisbook,evenwhennotspecificallymarkedassuch, arenottobeconsideredunprotectedbylaw. PrintedintheFederalRepublicofGermany TypesettingSteingraeberSatztechnikGmbH, Ladenburg PrintingStraussGmbH,Mo¨rlenbach BindingScha¨fferGmbH,Gru¨nstadt ISBN-13:978-3-527-40448-3 ISBN-10:3-527-40448-1 Acknowledgements Many hands make the work light. It is my pleasure to acknowledge and thank those whose efforts made this work possible. These are (in alphabetic order): Maximiliano Avendan˜o- Alejo, Isidro Cornejo, Lark London, Christopher Stavroudis, Dorle Stavroudis, and many former students whose helpful comments and snide remarks were of immensurable value. Thanksarealsoduetomyprojecteditor,UlrikeWerner,whoskillfullyandtactfully,ledme intheproperdirection. Contents I Preliminaries 1 1 Fermat’sPrincipleandtheVariationalCalculus 3 1.1 RaysinInhomogeneousMedia . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 TheCalculusofVariations . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 TheParametricRepresentation . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 TheVectorNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 TheInhomogeneousOpticalMedium . . . . . . . . . . . . . . . . . . . . 10 1.6 TheMaxwellFishEye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 TheHomogeneousMedium . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 AnisotropicMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 SpaceCurvesandRayPaths 15 2.1 SpaceCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 TheVectorTrihedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 TheFrenet-SerretEquations . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 WhentheParameterisArbitrary . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 TheDirectionalDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 TheCylindricalHelix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 TheConicSection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.8 TheRayEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 MoreontheFishEye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 TheHilbertIntegralandtheHamilton-JacobiTheory 29 3.1 ADigressionontheGradient . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 TheHilbertIntegral. ParametricCase . . . . . . . . . . . . . . . . . . . . 33 3.3 ApplicationtoGeometricalOptics . . . . . . . . . . . . . . . . . . . . . . 34 3.4 TheConditionforTransversality . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 TheTotalDifferentialEquation . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6 MoreontheHelix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.7 Snell’sLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.8 TheHamilton-JacobiPartialDifferentialEquations . . . . . . . . . . . . . 41 3.9 TheEikonalEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 TheDifferentialGeometryofSurfaces 45 4.1 ParametricCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 SurfaceNormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 TheTheoremofMeusnier . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 TheTheoremofGauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 GeodesicsonaSurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6 TheWeingartenEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7 TransformationofParameters. . . . . . . . . . . . . . . . . . . . . . . . . 55 TheMathematicsofGeometricalandPhysicalOptics:Thek-functionanditsRamifications.O.N.Stavroudis Copyright(cid:1)c 2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40448-1 XII Contents 4.8 WhentheParametricCurvesareConjugates . . . . . . . . . . . . . . . . . 57 4.9 WhenF(cid:1)=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.10 TheStructureoftheProlateSpheroid . . . . . . . . . . . . . . . . . . . . 61 4.11 OtherWaysofRepresentingSurfaces . . . . . . . . . . . . . . . . . . . . 64 5 PartialDifferentialEquationsoftheFirstOrder 67 5.1 TheLinearEquation. TheMethodofCharacteristics . . . . . . . . . . . . 68 5.2 TheHomogeneousFunction . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3 TheBilinearConcomitant. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4 Non-LinearEquation: TheMethodofLagrangeandCharpit . . . . . . . . 72 5.5 TheGeneralSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.6 TheExtensiontoThreeIndependentVariables . . . . . . . . . . . . . . . . 76 5.7 TheEikonalEquation. TheCompleteIntegral . . . . . . . . . . . . . . . . 77 5.8 TheEikonalEquation. TheGeneralSolution . . . . . . . . . . . . . . . . 79 5.9 TheEikonalEquation. ProofofthePudding . . . . . . . . . . . . . . . . . 81 II Thek-function 83 6 TheGeometryofWaveFronts 85 6.1 PreliminaryCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2 TheCausticSurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.3 SpecialSurfacesI:PlaneandSphericalWavefronts . . . . . . . . . . . . . 90 6.4 ParameterTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.5 AsymptoticCurvesandIsotropicDirections . . . . . . . . . . . . . . . . . 94 7 RayTracing: GeneralizedandOtherwise 97 7.1 TheTransferEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.2 TheAncillaryQuantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.3 TheRefractionEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.4 RotationalSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.5 TheParaxialApproximation . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.6 GeneralizedRayTracing–Transfer . . . . . . . . . . . . . . . . . . . . . 104 7.7 GeneralizedRayTracing–PreliminaryCalculations . . . . . . . . . . . . 105 7.8 GeneralizedRayTracing–Refraction . . . . . . . . . . . . . . . . . . . . 109 7.9 TheCaustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.10 TheProlateSpheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.11 RaysintheSpheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8 AberrationsinFiniteTerms 121 8.1 Herzberger’sDiapoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.2 Herzberger’sFundamentalOpticalInvariant . . . . . . . . . . . . . . . . . 122 8.3 TheLensEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.4 AberrationsinFiniteTerms . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.5 Half-Symmetric,SymmetricandSharpImages . . . . . . . . . . . . . . . 127 Contents XIII 9 Refractingthek-Function 131 9.1 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.2 TheRefractingSurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.3 ThePartialDerivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.4 TheFiniteObjectPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.5 TheQuestforC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.6 DevelopingtheSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 10 MaxwellEquationsandthek-Function 147 10.1 TheWavefront. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.2 TheMaxwellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.3 GeneralizedCoordinatesandtheNablaOperator . . . . . . . . . . . . . . 149 10.4 ApplicationtotheMaxwellEquations . . . . . . . . . . . . . . . . . . . . 150 10.5 ConditionsonV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10.6 ConditionsontheVectorV . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10.7 SphericalWavefronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 III Ramifications 163 11 TheModernSchiefspiegler 165 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.2 TheSingleProlateSpheroid . . . . . . . . . . . . . . . . . . . . . . . . . 167 11.3 CoupledSpheroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 11.4 TheConditionforthePseudoAxis . . . . . . . . . . . . . . . . . . . . . . 172 11.5 MagnificationandDistortion . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 12 TheCartesianOvalanditsKin 179 12.1 TheAlgebraicMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 12.2 TheObjectatInfinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 12.3 TheProlateSpheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 12.4 TheHyperboloidofTwoSheets . . . . . . . . . . . . . . . . . . . . . . . 183 12.5 OtherSurfacesthatMakePerfectImages. . . . . . . . . . . . . . . . . . . 184 13 ThePseudoMaxwellEquations 187 13.1 MaxwellEquationsforInhomogeneousMedia. . . . . . . . . . . . . . . . 187 13.2 TheFrenet-SerretEquations . . . . . . . . . . . . . . . . . . . . . . . . . 188 13.3 InitialCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 13.4 DivergenceandCurl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 13.5 EstablishingtheRelationship . . . . . . . . . . . . . . . . . . . . . . . . . 192 14 ThePerfectLensesofGaussandMaxwell 197 14.1 Gauss’Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 XIV Contents 14.2 Maxwell’sApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 A Appendix. VectorIdentities 205 A.1 AlgebraicIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.2 IdentitiesInvolvingFirstDerivatives . . . . . . . . . . . . . . . . . . . . . 207 A.3 IdentitiesInvolvingSecondDerivatives . . . . . . . . . . . . . . . . . . . 207 A.4 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.5 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A.6 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.7 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.8 DirectionalDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.9 OperationsonWanditsDerivatives . . . . . . . . . . . . . . . . . . . . . 213 A.10 AnAdditionalLemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 B Bibliography 217 Index 223 Introduction VII So I pass from a task, which has filled the greater part of many years of my life, which has broadenedinmyviewastheypassed,andwhichhassufferedinterruptionsthatthreatenedto enditbeforeitscompletion. Manyofitsdefectsareknowntome;afterithasgonefromme, otherswillbecomeapparent. Nevertheless, myhopeisthatmyworkwilleasethelabourof thosewho,comingafterme,maydesiretopossessasystematicaccountofthisbranchofpure mathematics. A.R.Forsyth TrinityCollege,Cambridge October,1906. Introduction Thisworkisaboutgeometricalopticsthoughitshallextendintosomefundamentalareasof physicalopticsaswell. Itmakesheavyuseofseveralbranchesofmathematicswhich,perhaps, thereaderwillfinddisturbinglyunfamiliar. TheseIwilldescribewithsomecarebutwithonly lipservicetomathematicalrigorandvigor. Keep in mind that geometrical optics is a peculiar science. Its fundamental artifacts are rays,whichdonotexist,andwavefronts,whichindeeddoexistbutarenotdirectlyobservable. A third item is the caustic, a surface in image space which is certainly observable, defined variouslyastheenvelopeofanarrayofraysassociatedwithsomepointobject, thelocusof theprincipalcentersofwavefrontcurvatures, orasthelocusofpointswherethedifferential elementofareaofawavefrontvanishes. Ofcourse,thesewavefrontsmustbeinawavefront traingeneratedbyalensandassociatedwithsomefixedobjectpoint. The peculiarities of geometrical optics go even further. Rays, which do not exist, are trajectoriesofcorpuscles,whichalsodonotexist. Thesetrajectories,accordingtotheprinciple ofFermat,arethosepathsoverwhichthetimeoftransitofacorpuscle,passingfromonepoint toanother,iseitheramaximumoraminimum. Yet it works. Geometrical optics, anachronistic as it is, remains the basis for modern opticaldesign,thehighlysuccessfulengineeringapplicationbuiltonthesandiestfoundation imaginable. Thereishardlyoneareaofmodernscienceinwhichinstrumentsareusedwhose designdependsultimatelyonFermat’spostulateontheintrinsiclazinessofmothernature. InwhatfollowsIshalluseamethodbestdescribedasaxiomatic,theaxiombeingFermat’s principle. Thiswemustmodify,however. Sincepoint-to-pointtransittimescanbemaximaas wellasminimawemustuse,inthelanguageoftheCalculusofVariations,extrema(singular: extremum)asourcriterioninapplyingFermat’sprinciple. IndeedtheinterpretationoftheprincipleofFermatintermsofthelanguageofthevariational calculuswillleadustoraypathsininhomogeneousmedia;mediainwhichtherefractiveindex TheMathematicsofGeometricalandPhysicalOptics:Thek-functionanditsRamifications.O.N.Stavroudis Copyright(cid:1)c 2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40448-1 VIII Introduction isacontinuousfunctionofposition. Theseraypathswillbeexpressedintheformofasystem ofordinarydifferentialequationsthatcanbeappliedtoanyspecifiedmedia. Theseraypathsarethensubjecttoanalysisusingthetechniquesofthedifferentialgeometry ofspacecurves. Usingthesedifferentialequationsforaraypathwecandeduceitsshapeand itsrelationshiptotherefractingmediumitself. Fromtheseresultswecandetermine,quickly andeasily,thenatureofraysin,say,Maxwell’sfisheye. From here we pass on to the Hilbert integral, developed originally for dealing with the problemofthevariableendpointintheCalculusofVariations. Thisveryrichtheoryleadsus toanumberofveryimportantdeductionsingeometricaloptics; conditionsfortheexistence ofwavefronts,Snell’slaw,TheHamilton-Jacobiequations(thoughbothHamiltonandJacobi preceded Hilbert by as much as a half century), the eikonal equation, among others. In this contextthetheoremofMalusbecomestrivial. FromthiscontextHerzbergerrecognizedthe importanceofthenormalcongruenceortheorthotomicsystemofrays. With the concept of the wavefront in hand we proceed to the differential geometry of surfacesandtopartialdifferentialequationsofthefirstorder. Onesuchistheeikonalequation, mentioned above, obtained from the Hamilton-Jacobi equation, for which we find a general solution descriptive of any wavefront train in a homogeneous optical medium; one with a constantrefractiveindex. In terms of the differential geometry of surfaces we can find, for the general wavefront train, wavefront principal directions and curvatures. This leads to the important concept of the caustic, that surface that is the locus of the principal centers of wavefront curvature. In thecausticresidesallofthemonochromaticaberrationsassociatedwithawavefronttrainand, ultimately, with the lens and object point that give rise to it. The structure of this caustic describes completely the image errors: spherical aberration, coma and astigmatism. Its locationinspaceindicatesthefielderrors;distortionandfieldcurvature. Alongthewaywelookatgeneralizedraytracing,moreproperly,ageneralizationofthe Coddingtonequations,thatdeterminestheprincipaldirectionsandprincipalcurvaturesatany pointonawavefrontthroughwhichatracedraypasses. This we apply to the prolate spheroid, a rotationally symmetric ellipsoid generated by rotatinganellipseaboutitsmajoraxis. Thisleadstoareflectingopticalsystem,consistingof twoconfocalspheroids,thatIhavecalledthemodernschiefspiegler. We also look at Herzberger’s fundamental optical invariant and his diapoint theory and applyittotherepresentationofwavefrontsobtainedformthesolutionoftheeikonalequation. Thisleadstoahierarchialsystemofaberrations. ThecanonthatIhavedescribedhere,basedonFermat’sprinciple,omitsmanyimportant items. Outstandingamongtheseisparaxialtheoryandparaxialraytracing. Althoughisitis oftremendouspracticalimportance,itisbasedonanapproximationthat,inmyopinion,does notbelonghere. AfarmorefundamentalomissionisGaussianoptics,inparticularly,itsmodelasdeveloped byMaxwell. Hebeganwithcertainassumptionsaboutperfectlensesfromwhichherepresented perfectimageformationbyafractional-lineartransformation. Uponassumingthathisperfect lens is rotationally symmetric, he was able to derive its cardinal points; the foci, the nodal pointsandtheprincipalpoints. Introduction IX Other omissions are the Seidel aberrations and their higher order extensions. These are fromasolutionoftheeikonalequationintheformofapowerseriesthathasneverbeenshown toconverge. Huygens’principleisomitted. Itisclearlyindependentofanycorpuscularconceptsandis basedonwavefrontpropagationastheenvelopeofsphericalwavelets,whichalsodonotexist, centeredonapreviouspositionofthewavefront. ItalsoleadstoSnell’slaw. Itwasformany centuriesthemaincompetitortoFermat’scorpuscules. But nowadays the photon incorporates the best of both the corpuscule and the wavelet, a compromise that has resulted in a far more useful theory with applications far beyond the dreamsofFermatandHuygens.