35-5 Michelson’s Interferometer An interferometer is a device that can be used to measure lengths or changes in length with great accuracy by means of interference fringes. In Michelson’s interferometer, a light wave is split into two beams that then recombine a=er traveling along different paths. The interference pa@ern they produce depends on the difference in the lengths of those paths and the indexes of refracBon along the paths. If a transparent material of index n and thickness L is in one path, the phase difference (in terms of wavelength) in the recombining beams is equal to where λ is the wavelength of the light. Michelson’s interferometer, showing the path of light originaBng at point P of an extended source S. Mirror M splits the light into two beams, which reflect from mirrors M 1 and M 2 back to M and then to telescope T. In the telescope an observer sees a pa@ern of interference fringes. © 2014 John Wiley & Sons, Inc. All rights reserved. 35-5 Michelson’s Interferometer
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35-5Michelson’sInterferometer
Aninterferometerisadevicethatcanbeusedtomeasurelengthsorchangesinlengthwithgreataccuracybymeansofinterferencefringes.InMichelson’sinterferometer,alightwaveissplitintotwobeamsthatthenrecombinea=ertravelingalongdifferentpaths.Theinterferencepa@erntheyproducedependsonthedifferenceinthelengthsofthosepathsandtheindexesofrefracBonalongthepaths.IfatransparentmaterialofindexnandthicknessLisinonepath,thephasedifference(intermsofwavelength)intherecombiningbeamsisequaltowhereλisthewavelengthofthelight.
Michelson’sinterferometer,showingthepathoflightoriginaBngatpointPofanextendedsourceS.MirrorMsplitsthelightintotwobeams,whichreflectfrommirrorsM1andM2backtoMandthentotelescopeT.Inthetelescopeanobserverseesapa@ernofinterferencefringes.©2014JohnWiley&Sons,Inc.Allrights
reserved.
35-5Michelson’sInterferometer
35Summary
Huygen’sPrinciple• Thethree-dimensionaltransmissionofwaves,includinglight,mayo=enbepredictedbyHuygens’principle,whichstatesthatallpointsonawavefrontserveaspointsourcesofsphericalsecondarywavelets.
Young’sExperiment• InYoung’sinterferenceexperiment,lightpassingthroughasingleslitfallsontwoslitsinascreen.Thelightleavingtheseslitsflaresout(bydiffracBon),andinterferenceoccursintheregionbeyondthescreen.Afringepa@ern,duetotheinterference,formsonaviewingscreen.
• ThecondiBonsformaximumandminimumintensityare
WavelengthandIndexofRefracAon• ThewavelengthλnoflightinamediumdependsontheindexofrefracBonnofthemedium:
inwhichλisthewavelengthinvacuum.
Eq.35-6
Eq.35-14
Eq.35-16
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35Summary
35Summary
Coherence• IftwolightwavesthatmeetatapointaretointerferepercepBbly,bothmusthavethesamewavelengthandthephasedifferencebetweenthemmustremainconstantwithBme;thatis,thewavesmustbecoherent.
Thin-FilmInterference• Whenlightisincidentonathintransparentfilm,thelightwavesreflectedfromthefrontandbacksurfacesinterfere.Fornear-normalincidence,thewavelengthcondiBonsformaximumandminimumintensityofthelightreflectedfromafilmofindexn2inairare
IntensityinTwo-SlitInterference• InYoung’sinterferenceexperiment,twowaves,eachwithintensityI0,yieldaresultantwaveofintensityIattheviewingscreen,with
Eqs.35-22&23
Eq.35-36
Eq.35-37
Michelson’sInterferometer• InMichelson’sinterferometeralightwaveissplitintotwobeamsthat,a=ertraversingpathsofdifferentlengths,arerecombinedsotheyinterfereandformafringepa@ern.©2014JohnWiley&Sons,Inc.Allrights
reserved.
35Summary
DiffracAon
Chapter36
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36-1Single-SlitDiffracBon
Whenwavesencounteranedge,anobstacle,oranaperturethesizeofwhichiscomparabletothewavelengthofthewaves,thosewavesspreadoutastheytraveland,asaresult,undergointerference.ThistypeofinterferenceiscalleddiffracBon.
Wavespassingthroughalongnarrowslitofwidthaproduce,onaviewingscreen,asingle-slitdiffracBonpa@ernthatincludesacentralmaximum(brightfringe)andothermaxima.TheyareseparatedbyminimathatarelocatedrelaBvetothecentralaxisbyanglesθ:Themaximaarelocatedapproximatelyhalfwaybetweenminima.
(a)Wavesfromthetoppointsoffourzonesofwidtha/4undergofullydestrucBveinterferenceatpointP2.(b)ForD>>a,wecanapproximateraysr1,r2,r3,andr4asbeingparallel,atangleθtothecentralaxis.
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36-1Single-SlitDiffracBon
36-2IntensityinSingle-SlitDiffracBon
TheintensityofthediffracBonpa@ernatanygivenangleθiswhere,Imistheintensityatthecenterofthepa@ernand
TheplotsshowtherelaBveintensityinsingle-slitdiffracBonforthreevaluesoftheraBoa/λ.Thewidertheslitis,thenarroweristhecentraldiffracBonmaximum.
Answer(a) 650nm(b) 430nm
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36-2IntensityinSingle-SlitDiffracBon
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36-3DiffracBonbyaCircularAperture
ThediffracBonpa@ernofacircularaperture.Notethecentralmaximumandthecircularsecondarymaxima.Thefigurehasbeenoverexposedtobringoutthesesecondarymaxima,whicharemuchlessintensethanthecentralmaximum.
DiffracBonbyacircularapertureoralenswithdiameterdproducesacentralmaximumandconcentricmaximaandminima,givenby
Theangleθhereistheanglefromthecentralaxistoanypointonthat(circular)minimum.
whichlocatesthefirstminimumforalongnarrowslitofwidtha.Themaindifferenceisthefactor1.22,whichentersbecauseofthecircularshapeoftheaperture.
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36-3DiffracBonbyaCircularAperture
36-3DiffracBonbyaCircularAperture
Theimagesoftwopointsources(stars)formedbyaconverginglens.Atthebo@om,representaBonsoftheimageintensiBes.In(a)theangularseparaBonofthesourcesistoosmallforthemtobedisBnguished,in(b)theycanbemarginallydisBnguished,andin(c)theyareclearlydisBnguished.Rayleigh’scriterionissaBsfiedin(b),withthecentralmaximumofonediffracBonpa@erncoincidingwiththefirstminimumoftheother.
Resolvability
Rayleigh’scriterionsuggeststhattwoobjectsareonthevergeofresolvabilityifthecentraldiffracBonmaximumofoneisatthefirstminimumoftheother.TheirangularseparaBoncanthenbenolessthaninwhichdisthediameteroftheaperturethroughwhichthelightpasses.©2014JohnWiley&Sons,Inc.Allrights
reserved.
36-3DiffracBonbyaCircularAperture
36-3DiffracBonbyaCircularAperture
PoinAllism Rayleigh’scriterioncanexplainthearresBngillusionsofcolorinthestyleofpainBngknownaspoinBllism.Inthisstyle,apainBngismadenotwithbrushstrokesintheusualsensebutratherwithamyriadofsmallcoloreddots.OnefascinaBngaspectofapoinBllisBcpainBngisthatwhenyouchangeyourdistancefromit,thecolorsshi=insubtle,almostsubconsciousways.Thiscolorshi=inghastodowithwhetheryoucanresolvethecoloreddots.
Answer:Resolvabilityimproves.
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36-3DiffracBonbyaCircularAperture
36-4DiffracBonbyaDoubleSlit
(a)Theintensityplottobeexpectedinadouble-slitinterferenceexperimentwithvanishinglynarrowslits.(b)TheintensityplotfordiffracBonbyatypicalslitofwidtha(notvanishinglynarrow).(c)Theintensityplottobeexpectedfortwoslitsofwidtha.Thecurveof(b)actsasanenvelope,limiBngtheintensityofthedouble-slitfringesin(a).NotethatthefirstminimaofthediffracBonpa@ernof(b)eliminatethedouble-slitfringesthatwouldoccurnear12°in(c).
WavespassingthroughtwoslitsproduceacombinaBonofdouble-slitinterferenceanddiffracBonbyeachslit.
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36-4DiffracBonbyaDoubleSlit
36-4DiffracBonbyaDoubleSlit
WavespassingthroughtwoslitsproduceacombinaBonofdouble-slitinterferenceanddiffracBonbyeachslit.
ForidenBcalslitswithwidthaandcenter-to-centerseparaBond,theintensityinthepa@ernvarieswiththeangleθfromthecentralaxisasinwhichand
NotecarefullythattherightsideofdoubleslitequaBonistheproductofImandtwofactors.(1)Theinterferencefactorcos2βisduetotheinterferencebetweentwoslitswithslitseparaBond.(2)ThediffracBonfactor[(sina)/a]2isduetodiffracBonbyasingleslitofwidtha.
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36-4DiffracBonbyaDoubleSlit
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