The Mathematics of Geometrical and Physical Optics. The k-function and its Ramification

225 Pages·1.95 MB·english

by  Stavroudis O.N.

Orestes N. Stavroudis
The Mathematics of Geometrical and
Physical Optics
k
The -function and its Ramifications
TheAuthor AllbookspublishedbyWiley-VCHarecarefully
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[e-mail] areadvisedtokeepinmindthatstatements,data,
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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.