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Chapter4Duality
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&wodifferentviewsoflight，asapartidasawave，bothsightaheyhaveeaabledbothandiuralworldaanddesigeologies.Yettheyappeartobevastlydifferentintheirofwhatlightactuallyis.Oheparticlemodelviewslightasalotity，abuhatmovesalongawell-defiory.Ohewavemhtasadiffuseeingthroughspaotothemotionofsolidthings.Howthesetwopicturespossiblyrefertothesamething?ThisdilemmawasreizedearlyonbyHuygensandhisporaries，butthetwoviewsremaiensioivedessoflight，uhtury.
WhenMaxwelldevelopedhistheoryicfields，hewasabletousethistoexplaiiesoflightaswavemotionofthosefields，aster3.ThistriumphappearedtotheexperimentsofThomasYoungandAugusteFresnel（desChapter3）byprovidiionoftheerferenddiffra，thatdidwithiiclemodel.Yettheceptoftrajeaiillremairaordinarilypowerfuloheanalysisanddesignofopticalsystems.Sothere’saruceofthesetwopictures—adualismhysics—thatrequiressomesideration.Howtheybereciled?
Lookingattrajectain
IhturytheFrenPierredeFermatproposedaningeniousformulatioionthatwasverydifferentfromthatofSSnell’slawdealswiththegeofdirehtataweentareheray，defiioninwhichitistravellingtowardstheihepointatwhichithitstheisdirealteredbyanamountproportioiooftherefradicesofthetwomaterials.Itisonlythelocalpropertiesoftherayahatareimportant.Snell’slaliesateataloory，asiftherayis‘feeling’itswayalong，adjustiioentersaerface.
&’swasradicallydifferehatoneshoulddefioryiartingandendingpoints，asshowninFigure21.Hesuggestedthatthequestiontoaskis:whatisthepaththatthelighttakestotraversethespathetwopoints?Heproposedthatitshouldtakethepaththatmiimeofflightbetweewopoints.ThatthisgivesthesameanswerasSnellisremarkableandprofou’s‘pritime’suggeststhatthelightsiderstheoverallpictureofthesituation，andthatthenotionofarayisoakesintoatboththeinitialandfinalpositionsaionsaswellaseverythihetrastwiththelocalmodelofaparticlereaediateeelling.
21.Fermat’sofalightrayasapathofleasttimegthestartahetrajectory.Therayiweentwoopticalmediainwhichlightmovesatdiferentspeeds.
ThisideabytheGermannaturalphilottfriedWilhelmvonLeibniz，ton’sporaryandantagonist.LeibnizressedbytheholisticpictureoftheprocessdescribedbyFermat，aof‘optimization’thatitimplied:arayexploresthewholeofspadpicksjustthatpaththatwillmiraweenthespecifiedbegins.Hedevelopedthemathematicaltoolsforanalysingthisidea—thecalculusofvariations—bywhichtheeffectssmallatrajectorywouldhaveootraversethemodifiedtrajectorycouldbecalculated.LeibnizreizedtheimportahatFermat’sprincipleprovided:themovementoflightfromooanotherdefiimal’trajectory.
&akenwasLeibnizbythisizatioedittoateleologiciple:thattheworlditself，inallitsaspects，wasoimaltrajectorybetointandafinishingpoiradiherentinsuchaposition，liedoutsideoftherealmofsce，ooaireinhisnoveldide，whereLeibniz’sideasareputihloss，whoinsistsdisastersbothnaturalandmahelessevidehisisthe‘bestofallpossibleworlds’.
&ingwavesandrays
&heless，Leibniz’smathematicalideasprovedtobeveryfruitful.TheybytherenownedIrishmathematiWilliamRowanHamiltohtury.Heshowedformallyhowtheideaofawavebealliedtothatofaofparticles.Wavesbedefiheirwavelength，amplitude，andphase（seeFigure15）.Particlesaredefiioionoftravel（seeFigure5），aionofparticlesbytheirdehehematagivenpositieofdireediainwhichthelightmovesarecharacterizedbytheirrefradices.Thisvaryacrossspaple，attheinterfaihereisasteptherefradexacrossthebouhet>
Hamiltoortantidlytherefradexspaparedwiththelengthofanopticalwave.Thatis，iftheiookplaascaleofclosetoawavelehewavecharacteroflightwasevident.Ifitvariedmoresmoothlyandveryslowlyiheparticlepictureprovidedaiohesimplerraypictureemergesfromthemoreplexioeredsituations.Theappearanceofhenomena，suchasdiffradinterference，othesizescalesofthewavelengthoflightauresinwhichitpropagatesaresimilar.Thusyouseediffrapatternsarisiheobjectthatthelighthitsisafewmidiameter，orhasaverysharpedge，suchasthedelicatestruabird’sfeather，orabutterflywing.Otherwise，asintheeraleoryprovidesasuffitdes，siiveindexisunifhouttheglassofthelensitself.
22.Hamilton’sideaofraysasgwavefronts-thusjoitionsoflight.
Further，HamiltoFermat’strajectoriesrelateddirectlytoapropertyofthewave—thewavefrohelotsatwhichthewavehasthesamephaseateatinspastanyouseetheripplesonthesurfaceofapoointoit，thecircularpatterhesewavefroheplathesurfacewheretheeaks’（hs）atagiveninstantoftime.Now，whatHamiltohatrayscouldbesideredasliihewavefrles，asshownihusgadjatwavefrontsbyawell-defiory.
Hamilton’s‘optialogy’
Thisremarkableresultsuggestedanotherprofoundilton’sso-called‘optialogy’.Whathehatthewell-knownformulatiohemotionandpositionofsolidbodiesofmatter—wasbasedorajectories.Theideathatthesemayalsobeiimal’hadbeensideredbyPierreLouisMaupertuisihtury.
&uishadformulatedawaytoevaluatetheoptimalvalueofaquantitycalledthe‘a’—essentiallythevelocityofthebodymultipliedbythedistamoves（asmass）—alorajectory.
&hattheainimalforanactualtrajectorybetweentwopoints，justasi’sargumentthatthetimetakenthtrayshouldbeminimal.Maupertuis’‘pria’isverysimilarioFermat’s‘pritime’.Indeed，LeonhardEuler，aSwissmathemati，showedhowtouseLeibniz’scalculustoderiveon’sfamousequationsofmotionfromMaupertuis’prihusEulerectedadesofatrajetermsofapartigitswaythroughitseooneinwhichthewholeofspathespecifiedstartingandfinishingpoihepath.
WhatHamiltondidwastofiioedthevariationsinasofasimpledesofthespeviroinwhioving.Aurnsouttohaveaverysimilarformtotheonehefoundfthetrajectoriesoflightrays（forwhiviroionisjusthowtherefradexgeswithpositioninthemedium）.Sothereisahintofalatentahetrajectoriesofsolidobjedafictivewavefront:perhapsallbodiesmighthavebothparticle-liketrajedroperties?Ioion，andhiseponymousfun，turnsouttobeveryimportantinthinkingabouttheepiandinglight—quantummeics.
Unsolvedpuzzles
Thiswasbyheonlyhiunityforsce.Aboutthistime，towardstheehtury，lightstillofferedafewpuzzlesthatwereunexplaiermsmodelsofitsproperties，evenwiththerethatHamiltonhadprovided.Twoofthemostimportantofthesewere:thecolourofhotobjegtheSun），andthecolourofdifferentatomsinaflame.
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