The Origin of the Refraction Index文献及翻译.docx

The Origin of the Refraction Index文献及翻译.docx

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TheOriginoftheRefractionIndex文献及翻译

外文文献：

TheOriginoftheRefractionIndex

Wehavesaidbeforethatlightgoesslowerinwaterthaninair,andslower,slightly,inairthaninvacuum.Thiseffectisdescribedbytheindexofrefractionn.Nowwewouldliketounderstandhowsuchaslowervelocitycouldcomeabout.Inparticular,weshouldtrytoseewhattherelationistosomephysicalassumptions,orstatements,wemadeearlier,whichwerethefollowing:

（a）Thatthetotalelectricfieldinanyphysicalcircumstancecanalwaysberepresentedbythesumofthefieldsfromallthechargesintheuniverse.

（b）Thatthefieldfromasinglechargeisgivenbyitsaccelerationevaluatedwitharetardationatthespeedc,always（fortheradiationfield）.

But,forapieceofglass,youmightthink:

"Oh,no,youshouldmodifyallthis.Youshouldsayitisretardedatthespeedc/n."That,however,isnotright,andwehavetounderstandwhyitisnot.

Itisapproximatelytruethatlightoranyelectricalwavedoesappeartotravelatthespeedc/nthroughamaterialwhoseindexofrefractionisn,butthefieldsarestillproducedbythemotionsofallthecharges-includingthechargesmovinginthematerial-andwiththesebasiccontributionsofthefieldtravellingattheultimatevelocityc.Ourproblemistounderstandhowtheapparentlyslowervelocitycomesabout.

Weshalltrytounderstandtheeffectinaverysimplecase.Asourcewhichweshallcall"theexternalsource"isplacedalargedistanceawayfromathinplateoftransparentmaterial,sayglass.Weinquireaboutthefieldatalargedistanceontheoppositesideoftheplate.ThesituationisillustratedbythediagramofFig.31-1,whereSandPareimaginedtobeveryfarawayfromtheplate.Accordingtotheprincipleswehavestatedearlier,anelectricfieldanywherethatisfarfromallmovingchargesisthe（vector）sumofthefieldsproducedbytheexternalsource（atS）andthefieldsproducedbyeachofthechargesintheplateofglass,everyonewithitsproperretardationatthevelocityc.Rememberthatthecontributionofeachchargeisnotchangedbythepresenceoftheothercharges.Theseareourbasicprinciples.ThefieldatPcanbewrittenthus:

（31.1）

or

，（31.2）

whereEsisthefieldduetothesourcealoneandwouldbepreciselythefieldatPiftherewerenomaterialpresent.WeexpectthefieldatPtobedifferentfromEsifthereareanyothermovingcharges.

Whyshouldtherebechargesmovingintheglass?

Weknowthatallmaterialconsistsofatomswhichcontainelectrons.Whentheelectricfieldofthesourceactsontheseatomsitdrivestheelectronsupanddown,becauseitexertsaforceontheelectrons.Andmovingelectronsgenerateafield-theyconstitutenewradiators.ThesenewradiatorsarerelatedtothesourceS,becausetheyaredrivenbythefieldofthesource.ThetotalfieldisnotjustthefieldofthesourceS,butitismodifiedbytheadditionalcontributionfromtheothermovingcharges.Thismeansthatthefieldisnotthesameastheonewhichwastherebeforetheglasswasthere,butismodified,anditturnsoutthatitismodifiedinsuchawaythatthefieldinsidetheglassappearstobemovingatadifferentspeed.Thatistheideawhichwewouldliketoworkoutquantitatively.

Nowthisis,intheexactcase,prettycomplicated,becausealthoughwehavesaidthatalltheothermovingchargesaredrivenbythesourcefield,thatisnotquitetrue.Ifwethinkofaparticularcharge,itfeelsnotonlythesource,butlikeanythingelseintheworld,itfeelsallofthechargesthataremoving.Itfeels,inparticular,thechargesthataremovingsomewhereelseintheglass.Sothetotalfieldwhichisactingonaparticularchargeisacombinationofthefieldsfromtheothercharges,whosemotionsdependonwhatthisparticularchargeisdoing!

Youcanseethatitwouldtakeacomplicatedsetofequationstogetthecompleteandexactformula.Itissocomplicatedthatwepostponethisproblemuntilnextyear.

Insteadweshallworkoutaverysimplecaseinordertounderstandallthephysicalprinciplesveryclearly.Wetakeacircumstanceinwhichtheeffectsfromtheotheratomsisverysmallrelativetotheeffectsfromthesource.Inotherwords,wetakeamaterialinwhichthetotalfieldisnotmodifiedverymuchbythemotionoftheothercharges.Thatcorrespondstoamaterialwhichtheindexofrefractionisverycloseto1,whichwillhappen,forexample,ifthedensityoftheatomsisverylow.Ourcalculationwillbevalidforanycaseinwhichtheindexisforanyreasonverycloseto1.Inthiswayweshallavoidthecomplicationsofthemostgeneral,completesolution.

Incidentally,youshouldnoticethatthereisanothereffectcausedbythemotionofthechargesintheplate.ThesechargeswillalsoradiatewavesbacktowardthesourceS.Thisbackward-goingfieldisthelightweseereflectedfromthesurfacesoftransparentmaterials.Itdoesnotcomefromjustthesurface.Thebackwardradiationcomesfromeverywhereintheinterior,butitturnsoutthatthetotaleffectisequivalenttoareflectionfromthesurfaces.Thesereflectioneffectsarebeyondourapproximationatthemomentbecauseweshallbelimitedtoacalculationforamaterialwithanindexsocloseto1thatverylittlelightisreflected.

Beforeweproceedwithourstudyofhowtheindexofrefractioncomesabout,weshouldunderstandthatallthatisrequiredtounderstandrefractionistounderstandwhytheapparentwavevelocityisdifferentindifferentmaterials.Thebendingoflightrayscomesaboutjustbecausetheeffectivespeedofthewavesisdifferentinthematerials.ToremindyouhowthatcomesaboutwehavedrawninFig.31-2severalsuccessivecrestsofanelectricwavewhicharrivesfromavacuumontothesurfaceofablockofglass.Thearrowperpendiculartothewavecrestsindicatesthedirectionoftravelofthewave.Nowalloscillationsinthewavemusthavethesamefrequency.（Wehaveseenthatdrivenoscillationshavethesamefrequencyasthedrivingsource.）Thismeans,also,thatthewavecrestsforthewavesonbothsidesofthesurfacemusthavethesamespacingalongthesurfacebecausetheymusttraveltogether,sothatachargesittingattheboundarywillfeelonlyonefrequency.Theshortestdistancebetweencrestsofthewave,however,isthewavelengthwhichisthevelocitydividedbythefrequency.Fromthefigurewecanseethattheonlywayforthewavesto"fit"properlyattheboundaryisforthewavesinthematerialtobetravellingatadifferentanglewithrespecttothesurface,fromthegeometryofthefigureyoucanseethatfora"fit"wemusthave

or

whichisSnell'slaw.Weshall,fortherestofourdiscussion,consideronlywhylighthasaneffectivespeedofc/ninmaterialofindexn,andnolongerworry,inthischapter,aboutthebendingofthelightdirection.

WegobacknowtothesituationshowninFig.31-1.WeseethatwhatwehavetodoistocalculatethefieldproducedatPbyalltheoscillatingchargesintheglassplate.WeshallcallthispartofthefieldEa,anditisjustthesumwrittenasthesecondterminEq.（31.2）.WhenweaddittothetermEs,duetothesource,wewillhavethetotalfieldatP.31-2Thisisprobablythemostcomplicatedthingthatwearegoingtodothisyear,butitiscomplicatedonlyinthattherearemanypiecesthathavetobeputtogether;eachpiece,however,isverysimple.Unlikeotherderivationswherewesay,"Forgetthederivation,justlookattheanswer!

"inthiscasewedonot

needtheanswersomuchasthederivation.Inotherwords,thethingtounderstandnowisthephysicalmachineryfortheproductionoftheindex.

Toseewherewearegoing,letusfirstfindoutwhatthecorrectionfield"EawouldhavetobeifthetotalfieldatPisgoingtolooklikeradiationfromthesourcethatissloweddownwhilepassingthroughthethinplate.Iftheplatehadnoeffectonit,thefieldofawavetravellingtotheright（alongthez-axis）wouldbe

or,usingtheexponentialnotation,

Nowwhatwouldhappenifthewavetravelledmoreslowlyingoingthroughtheplate?

Letuscallthethicknessoftheplate

.Iftheplatewerenottherethewavewouldtravelthedistance

inthetime

Butifitappearstotravelatthespeedc/nthenitshouldtakethelongertime

ortheadditionaltime

.Afterthatitwouldcontinuetotravelatthespeedcagain.WecantakeintoaccounttheextradelayingettingthroughtheplatebyreplacinginEq.（31.4）by

orby

.Sothewaveafterinsertionoftheplateshouldbewritten

（31.5）

Wecanalsowritethisequationas

（31.6）

31-3Dispersion

Noticethatintheaboveprocesswehaveobtainedsomethingveryinteresting.Forwehavenotonlyanumberfortheindexofrefractionwhichcanbecomputedfromthebasicatomicquantities,butwehavealsolearnedhowtheindexofrefractionshouldvarywiththefrequencycofthelight.Thisissomethingwewouldneverunderstandfromthesimplestatementthat"lighttravelsslowerinatransparentmaterial."Westillhavetheproblem,ofcourse,ofknowinghowmanyatomsperunitvolumethereare,andwhatistheirnaturalfrequencyw0.Wedonotknowthisjustyet,becauseitisdifferentforeverydifferentmaterial,andwecannotgetageneraltheoryofthatnow.Formulationofageneraltheoryofthepropertiesofdifferentsubstances-theirnaturalfrequencies,andsoon-ispossibleonlywithquantumatomicmechanics.Also,differentmaterialshavedifferentpropertiesanddifferentindexes,sowecannotexpect,anyway,togetageneralformulafortheindexwhichwillapplytoallsubstances.

However,weshalldiscusstheformulawehaveo