Download - PHY 208 - wave-optics

Wave Optics

1. Lightasawave1. Huygens'Princinple2. Refraction3. LawofRefraction4. Thelightfromthesun5. ChromaticDispersion

2. Diffraction3. Thedoubleslitexperiment4. WavesandReflections5. Interferometer6. Thediffractionoflight

1. SingleSlit2. MultipleSlits3. CircularApertureDiffraction

1. Light as a wave

We'vespentsometimedevelopingapowerfulsetoftoolsforanalyzingthephysicsofwaves.Sofar,we'vetreatedcaseslikewavesonastring,orsoundwavesastheypropagatethroughair.But,therearemanymorephenomenainthenaturalworldthatcanbeexplainedandanalyzedusingwavephysics.Amajoronethatpersiststhroughoutthedevelopmentofscienceandhumanhistoryhasbeenlight.Fromancienttextstomoderncommunicationtechnology,thequesttounderstandlighthasbeenadrivingforceinscience.We'lltakealooknowathowlightcanbeunderstandasawave.

Intheearlyyearsofscientificthought,thestudyoflightwasveryproblematics.Severaloftheconvenienceswehavenowdidn'texists,andthismadelightaratherdifficultsubjecttostudy.Forexample,weknownowthespeedoflighttobe186,000milespersecond.(Or300,000,000m/s)So,imaginetryingtomeasuresomethingthatmovedthisfastusingonlypendulumclocksorsundials.Itwouldbehard.Also,whenwestudylightnow,weuselensesandothertoolstoshapeitandmakeitbehave.Suchthingswerenotavailableinthebeginningofnaturalscience.So,mostoftheworkonlightwasdoneusingreasoninganddeduction,ratherthenmeasurements.Ofcourse,someexperimentsweredone,buttheywereoftenhardtointerpretcorrectly.Amajorargumentthatpersistedinthescientificcommunitywasweatherlightwasaparticle,orawave.Attimes,itseemstoactlikeawavesfromapebbleonthesurfaceofapond,othertimes,likerubberballsmovinginstraightlines.We'llstudybothapproachesandseehowtheyofferdifferentadvantagesforunderstandingthenaturalworld.

Huygens' Princinple

ChristianHuygens[1629-1695]

1.Eachpointonawavefrontisthesourceofasphericalwaveletthatspreadsoutatthespeedofthewave.

2.Atalatertime,theshapeofthewaveisthecurvethatistangenttoallthewavelets.

Huygens'Sim

PHY 208 - wave-optics

updated on 2018-02-19 J. Hedberg | © 2018

http://localhost/~james/hedberg.ccnysites/sims/huygensprinciple

plane wave

Oneofthefirstscientiststomakeagoodcasefor'lightasawave'wastheDutchphysicistChristianHuygens.Around1678,hewroteatreatiseaboutlightinwhichhedescribedawavebasedunderstandingaboutthepropagationoflight.Althoughthedetailsofthemechanismheproposedwerenotexactlycorrect,thebasicconceptualunderstandingprovidedbythemodelremainsuseful.Atthispointintime,peopleweren'tsureiflightmovedatinfinitespeed(instaneously)orifithadafinitevelocity.HuygensfirmlybelievedhadafinitespeedandwasconvincedofthisbyastronomicalobservationsprovidedbyOlausRoemer.Healsocouldexplaincertainaspectsbytreatinglightasawave.

Essentially,Huygen'sprinciplestatesthatawavefrontcanbecreatedbymanycloselyspaced,coherentpointsourcesthatareallspreadingoutinasphericalmanner.(Coherentmeanstheyhavethesameinitialphase)

1.Eachpointonawavefrontisthesourceofasphericalwaveletthatspreadsoutatthespeedofthewave.




t = 0 t > 01.Eachpointonawavefrontisthesourceofasphericalwaveletthatspreadsoutatthespeedofthewave.


Index of Refraction

Theindexofrefraction(orrefractiveindex)ofamaterialisadimensionlessparameter, ,usedtodescribehowlightmovesinaparticularmedium.

n

n = =speedoflightinvacuum



Theindexofrefractionofvacuumis andisexactly1.

Medium IndexVacuum 1(exactly)Air(0ºC,1atm) 1.00029Water 1.33Glass 1.52Saphire 1.77Diamond 2.42

[*Noteforlabs:Theindexofrefractioncanchangebasedonthewavelengthofthelight]Waves passing through media

Index of Refraction

Thevelocityofthewaveinamediumisgivenby: .

Fromthepreviousvisualization,wecanseethatthewavelengthwillalsochange.

Thuswecanwrite

Whataboutthefrequency?

Refraction

Refractioninvolvesthebendingoflightataninterfacebetweentwomediumwithdifferentindicesofrefraction.

We'llseethatthefollowingrelationholds:

n = =c

v

speedoflightinvacuumspeedoflightinmaterial

n = 1.000

v = c

n

=λnλ

n

=sinθ1sinθ2

n2

n1



light

n1

n2vacuum (n = 1)

light

n1

n2vacuum (n = 1)

Refractioncanbeunderstoodasadirectconsequenceofthechangeinvelocityofawaveasitentersanewmediumatanangle.

Law of Refraction

Derive the law of refraction

Implicationsofrefraction

Twolightwavesentertwodifferentmaterials.



out of phase!

Upon exiting the material:

Thewavesarenolongerinphase!

Yes,butbyhowmuch?

Quick Question 1

Asinglelightbeamissplitintotwoequalbeams,denotedAandB.BeamAtravelsthroughamediumwithahigherindexofrefractionthanthemediumthatbeamBtravelsthrough.Afterbothbeamsexittheirmediaandarebackintotheair,howdotheirwavelengthscompare?

1. BeamAhasalongerwavelength2. BeamBhasalongerwavelength3. Bothbeamshavethesamewavelength

Basedonthelengthofthematerials,theindicesofrefraction,andthewavelengthofthelightinvacuum,wecandeterminethephasedifference(actuallythedifferenceinnumberofwavelengths)betweenthetwolightwaves.

Wecangetthisrelationinthefollowingway.

Thewavelengthinthemediumwillbedifferentbywavelengthinvacuum:

Also,ifthemediumhasalength ,thentherewillbe wavelengthspresentinthemedium:

Similary,forthewavesinmedium2:

Subtracting - givesusthedesiredrelation.

Color

Thecoloroflightthatweperceiveisbasedonthefrequency(orwavelength)ofthelight.

− = ( − )N2 N1L

λn2 n1

=λn1

λ

n1

L N1

= =N1L

λn1

Ln1

λ

= =N2L

λn2

Ln2

λ

N2 N1



10141018

10-10 10-610-7 10-2 1

1081010

radi

o/tv

mic

row

ave

x ra

ysfrequency of wave [Hz]

wavelength of wave [m]

780 Terahertz 400 Terahertz

~400 nm ~700 nm~500 nm

Thehumaneyehasthreetypesofcellsthataresensitivetodifferentfrequenciesoflight.Thetriggeringofthesecellsbythedifferentfrequenciesresultsincolorperceptions.Theyareresponsivetoonlyaverynarrowsetoffrequencieshowever.Thecompletespectrumoflightcontainfrequenciesandwavelengthsthatdonottriggerthecellsoureyes.We'llseethisspectrumagain,afterwegothroughelectricityandmagnetism,sincethosetopicswillcontainthematerialtoreallyunderstandwhatlightis.

The light from the sun

ThephysicalprocesseshappeningintheSuncreatelightofmanydifferent



n > 1

Theimplicationsarethattheangleofrefractionisnowdependentonthewavelengthofthelightbecause carriesthisdependence: .

Fig.1

frequencies.Thischartshowstheintensitiesemittedfrequenciesasafunctionofwavelength.Itisanotherexampleofaspectrum.Wesawaspectrumgraphwhenwecoveredsoundwaves.Thereweplottedtheintensityofthesoundwavesasafunctionofthefrequency(pitch)ofthesound.Thisgraphcontainssimilarinformation.Fromit,wecanseethemostofthelightthatthesunproducesisaround500nmwavelength.Fortunately,thisisaroundthefrequencythatoureyesareverysensitivetoo.

Chromatic Dispersion

n = 1.55

n = 1.45

200 nm 400 nm 600 nm 800 nm 1000 nm 1200 nm 1400 nm wavelength

inde

x of

refra

ctio

n

Hereisaqualitativeplotoftheindexofrefractionforahypotheticalmaterialasafunctionofthewavelengthoftheincidentlight.

Whenlightentersamedium,theindexofrefractionitencounterswilldependslightlyonthewavelengthofthelight.ThiseffectiscalledChromaticDispersion.

Dispersion

2. Diffraction

Diffraction

Diffraction

n n → n(λ)

sin (λ) = sin (λ)θ1n1 θ2n2



point 1

point 2

crest

trough

L1L2

L2L1

Thespreadingoutofthewavesafterpassingthroughtheopeningiscalleddiffraction.

Diffractionisageneralpropertiesofallwavephenomenon.Waterwavesareparticularlyeasytoseediffractioneffectsbecausethewavelengthsareaboutasbigastheobstacleswhichcausethediffraction.

CheckoutthePanamacanalentrance:LinktoMap

Thereweretwocompetingtheoriesaboutwhatexactlylightwas.Newton'sexperimentsledhimtopostulatethatlightwasaparticle(hecalledthem'corpuscles'.)

WhileHuygens'andothershaddoneworkwhichseemedtoshowthatlightactedasawave.

ThomasYoungwasanEnglishScientist.Around1801hedidsomeexperimentswithlightwiththegoalofdemonstratingitswave-likenature.

Wave interference

Twospeakers?

Thephysicsoflightborrowsmuchfromthetreatmentofsoundwaves.Wefiguredouthowtocalculatewhereloadandquitespotsshouldbeinthecaseoftwointerferingsoundwaves.Asimilarapproachwillgiveusinformationabouttheinterferenceoflight.

Thomas Young

3. The double slit experiment



https://maps.google.com/maps?q=Colon,+Panama&hl=en&ll=9.370652,-79.912748&spn=0.05784,0.065832&sll=52.788217,1.606772&sspn=0.008863,0.016458&oq=colon+pana&t=h&hnear=Colon,+Panama&z=14

single slit (hole)

incident light

diffracted waves

Fig.2

double slit (hole)



double slit (hole)

max

max

max

max

max

max

min

min

min

min

min

min

screen

Thebasicsetupisasfollows.Light(monochromaticsandcoherent)passesthroughtwosmallslits.Fromeachslightemergesadiffractedbeam.Thesebeamsundergowaveinterferenceandtheresultinglightthatappearsonascreenawayfromtheslitsshowstheinterferencepattern.



r1

r2 Lslits

screen

P

m = 0

m = 1

m = 1

m = 2

m = 2

m = 3

m = 3

y0

y1 y’0

y’1

y’2y2

y3+y

r2

r1

path length r1

path length r2

L

slits

screen

P

y = 0

y = L tan

d

∆r = path length difference

Whenconstructive:

andfordestructive:

Quick Question 2

Abeamofmonochromaticlightwithawavelengthof660nmisdirectedatadoubleslit.Considerthefivelocationslabeledinthedrawing,(thecentralmaximumislabeled“B.”)Whichoneofthesefringesisproducedwhenthepathdifferenceis1320nm?

LightfromaLaserhasawavelengthof633nm.Thisispassedthroughtwosmallslitsspaced.4mmapart.Aviewingscreenis2.0mbehindtheslits.Whatarethedistancesbetweenthetwom=2brightfringesandbetweenthetwom=2darkfringes,onthescreen?

ΔL = d sinθ

d sinθ = mλform = 0,1,2

d sinθ = (m + 1/2)λform = 0,1,2

Example Problem #1:



Quick Question 3

Alaboratoryexperimentproducesadouble-slitinterferencepatternonascreen.Ifthescreenismovedfartherawayfromtheslits,thefringesmarkedBandCwillbe:

1. Closertogether.2. Inthesamepositions.3. Fartherapart.4. Fuzzyandoutoffocus.

Adoubleslitinterferencepatternisobservedonascreen1.0mbehindtwoslitsspaced0.30mmapart.Tenbrightfringesspanadistanceof1.7cm.Whatisthewavelengthofthelight?

I

image data

Intensity plot

y

Ofcourse,asscientists,wewouldratherhaveourdatainamoreuseableform.Tothisend,wecanconvertour'screen'imagesofdarkandlightspotstoanintensityplot.Thisletsusputunits,like forintensityasafunctionofdistance,andusefunctionstodesribetheplot.

Thesameinformationispresent--it'sjusteasiertoworkwith.

Quick Question 4

Alaboratoryexperimentproducesadouble-slitinterferencepatternonascreenusingredlight.Ifgreenlightisused,witheverythingelsethesame,thebrightfringeswillbe:

1. Closertogether.2. Inthesamepositions.3. Fartherapart.4. Fuzzyandoutoffocus.

4. Waves and Reflections

Wesawwhathappenedwhenawaveonastringreflectedoffaboundary.

Boundary is not perfect

Example Problem #2:

W/m2



incident wave

reflected wave

transmitted wave

Dependingonifthelightisreflectingoffahigherindexoralowerindex,wewillobtainthefollowingphaseshiftsforthereflectedwaves.

Reflection PhaseshiftReflectsoffalowerindex 0Reflectsoffahigherindex 0.5

incident wave

reflected wave

transmitted wave

incident wave

reflected wave

transmitted wave

n1 > n2

n1 < n2

Herewehaveawaveonaropetravelingtowardsapartiallyreflectingboundary.The'boundary'isreallyaconnectionbetweenalowdensityandhighdensitysectionofrope.

Wecanseethatmostofthewaveisreflected,andinverted,attheboundary,however,someoftheenergyofthewaveistransmittedacrosstheboundaryandcontinuesintothehigherdensityregion.Additionally,since ,thewavespeedisdifferentforthereflectedandthe

transmittedwaves.

Boundary is not perfect

Likewise,ifthewavetravelsfromahigherdensityregiontoalowerdensityregion,mostofthewave'senergywillbetransmitted,whilesomewillbereflected.

Again,thelowerdensitysectionofropewillhaveahigherwavespeedthanthehighdensityregion.Thistime,thereisnoamplitudeinversion.vid

Phase change due to reflection

Sincetheindexofrefractionofamaterialissomewhatanalogoustothedensityofastring,asfaraswavesgo,(theybothareparametersthataffectthespeedofthewaves),wewouldexpectasimilarphenomenawithlightasitreflectsfromboundaries.

v = τ

μ

−√



http://www.animations.physics.unsw.edu.au/jw/waves_superposition_reflection.htm

n = 1 n = 1.5 n = 1.3

incident ray

reflected ray

1 2

incident wave

reflected wave (from first surface)

reflected wave (from glass surface)

transmitted wave

air thin film glass

t

nair= 1 nglass= 1.5nx= 1.33

incident ray

r1

n1

r2

n2 n3

12

3

Quick Question 5

Bluelaserlightistravelingthroughasandwichofmaterialsasshown.Ateachinterface,somelightwillbereflected.Comparingtheoriginalincidentraywiththefinalreflectedray,whatisthephaseshiftsduetoreflection?

1. 0rad2. rad

3. rad4. rad5. rad

Phase Shift Sources

Wenowhave3sourcesofphaseshiftsforalightwave:

1. Reflection:dependingonthe ofthetwomedia,thereflectedwavemayormaynotbephaseshifted

2. PathLength:Theanalysisofthedoubleslitshowedthatpathlengthdifferencecanalsoleadtoaphaseshift.

3. IndexofRefraction:Lighttravelingthroughmediawithdifferent canundergoaphaseshift.

Thin Film Interference

Thin Film Interference

①Theoriginalincidentrayisbothreflectedandtransmittedatthefirstboundary.Thisreflectionbecomes .Thetransmittedwavegoesuntilthenextboundary,②,whereitisalsotransmittedandreflected.Thereflectedrayproceedsbackthroughmaterial2,anditthentransmittedthroughtheboundaryat③.Thislasttransmittedrayis .

Byfiguringoutallthephaseshiftsalongtheway,wecantellifrays and willbeinphaseoroutofphase.Thatis,willtheyconstructivelyinterfereordestructivelyinterfere?

Constructive interference

Inthissituation,wehave .

π/2π

3π/22π

n

n

r1

r2

r1 r2

< >n1 n2 n3



incident ray

r1

air

r2

air

12

3

n2

incident ray

r1

air

r2

air

12

3

n2

incident ray

r1

r2

12

3

n2n1 n3

t < .1 l

Sinceattheinterface①therayisreflectingoffalarge ,thereflectedray willhaveaphaseshiftof

.

willhaveapathlengthdifferencecomparedtoof2L.Thustogetconstructiveinterferencebetweenand ,weneed

Destructive Interference

Ifwewherelookingtohavedestructiveinterferencebetweenthetworeflectedrays,wewouldneed:

.

Thus,withthesamesubstitutions,wecanarriveat:

WhatisthethinnestfilmofMgF2(n=1.38)onglassthatproducesastrongreflectionforlightwithawavelengthof600nm.

Two15-cmlongflatglassplatesareseparatedbya10 mthickspaceratoneend,leavingathinwedgeofairbetweentheplates.Amonochromaticlight( nm)fromaboveilluminatestheplates.Alternatingbrightanddarkfringesareobserved.Whatisthespacingbetweentwobrightfringes.

Ifthefilmismuchsmallerthanthewavelengthofthelightpassingthrough,(i.e ),thenwecanignorethepathlengthcreatedbythefilmandonlyconcernouranalysiswiththereflectionsofthelightattheboundaries.

n r1λ/2

r2 r1

r1 r2

2L = ( ) = (m + ) form = 0,1,2,…oddnumber

2λn2

12

λ

n2

2L = integer × λ

2L = (integer) = m form = 0,1,2,…λn2

λ

n2

Example Problem #3:

Example Problem #4:

μ

λ = 589

t < .1λ



incident

reflected

reflected

transmitted

transmitted

incident

soap filmSincethecriteriaforconstructiveanddestructiveinterferencedependsonwavelength,lightofdifferentcolorswillresponddifferentlytoathinfilm,likeasoapbubbleoroilslick.Somecolorswillexperienceconstructiveinterference,whileotherswillhavedestructiveinterferenceafterthetworeflections.

5. Interferometer

We’veseenthattheamountoflightvisibleisverysensitivetothepathdifferencebetweentworays.Thismeanswecanmakeadevicetomeasureverysmalldistances.

emitters detectorThesmallestlengthwecanmeasureusingthisdevicewillbegivenbythewavelengthofthelightused.

Interferometer



L1

L2source splittermirror

detector

mirrorWeknowthepathlengthdifferenceis:

When isequaltoawholenumberofwavelengths,ie: ,thentheinterferencebetweenthetwopathswillbeperfectlyconstructive.Anychangeintheamplitudeofthefinal,detectedlight,impliesachangeineitherthewavelengthofthelightduringonearmoftravel,orachangeinthelengthofonearm.

L1

L2source

splittermirror

detector

mirror

Interferometer

Usesofinterferometry:

1. DisprovedtheLuminiferousAether2. Surfacetopographyofsamples.3. MeasuredNewton’sGravitationalConstant,G.4. Maybegravitationalwaves???

(lookupLIGO-http://www.ligo.caltech.edu/)

6. The diffraction of light

Ouruseoftheraymodelreliedontheassumptionthatthelightraysdidn'tinteractwithobjectsthathad

Δr = 2 − 2L2 L1

Δr

Δr = mλ



ⒶThewaveletsfromeachpointontheinitialwavefrontoverlapandcreateaninterferencepattern

ⒷThesealltravelthesamedistanceandcreatethecentralmaximum

ⒸThesetraveldifferentdistancesandcreatethefringes.

featuresinthesamelengthscaleasthewavelengthofthelight.

Clearly,thislimitstherangeoftopicswecandiscuss,anditlimitsthedepthofphenomenawecanunderstand.

Let’sgobacktothis:Amonochromaticlightsourcepassingthroughasmallslit.

Thewavelengthofthislightisroughlythesamesizeastheslitthroughwhichitispassing.

Single Slit

Thesetupisthesameasthedoubleslit,exceptthereisonlyoneslit.

Again,weseeadiffractionpatternemergeonthescreen.Thisoneisabitdifferenthowever.

Huygens-like analysis of a single slit

A B C



aa/2

1

2

3

4

5

6∆r

Thepathlengthdifferencebetweenwavelet1and2is:

Whichisanotherwayofsayingthatwavelet2travels moretogetthescreenattheangle .

If ,thensoarealltheother∆rvalues,andtheinterferenceatthescreenwillbedestructive,meaningtherewillbeadarkspotthere.Thelocationsofthedarkspotsarethen:

or,

Theotherminimumscanbefiguredoutinasimilarway.Wegetthefollowingfortheangularlocationsofthedarkregions.

Thewidthofthecentralmaximumiseasilyfiguredbylookingatthedistancebetweenthefirstminimums.

thus:

a

centralmaximum

m = 1

m = 1m = 2

m = 2

w

single slit screen y

LQuick Question 6

Lightofwavelength illuminatesasingleslitsothatthewidthofthecentraldiffractionmaximais .Theslitwidthisthendecreasedto .Whatisthewidthofthecentraldiffractionmaxima?

1.2.3.4.5.

Δ = sinθr12a

2

Δr θ

Δ = λ/2r12

sinθ =a

2λ

2

a sinθ = λ

a sinθ = mλ

y =mλL

a

w = + =1λL

a

1λL

a

2λL

a

λ

l a/2

l/22l

l/44l

l



d

d towards screen

plane wavesapproaching from left

d

y1

y2

y1

0

y2

screen

5 slit grating

m = 2

m = 2

m = 1

m = 1

m = 0

Quick Question 7

Thistimeyoukeepthesameslitwidth,butuseanothermonochromaticlightofwavelength500nm.Howdoesthebroadness(width)ofthecentralbrightfringechangecomparedtothatproducedbythe600nmwavelength?

1. Increases2. Decreases3. Staysthesame4. Dependsontheexactvalueofslits

Multiple Slits

Whathappenswhenweincreasethenumberofslits?

Thissystemiscalledadiffractiongrating.Wecanthinkofitasamulti-slitdevice.

Wecanincreasethenumberofslitsperinch.Here's5slitsandasampleofwhatthediffractionpatternwouldlooklike.

Usingthesameanalysisasbefore,wecanfindthelocationsonthescreenwherethepathlengthdifferenceleadstoconstructiveinterference.

Again,notethatthisvalueisdependentonthewavelength.

Thequality(sharpnessandintensity)ofthediffractionpatternisdeterminedbythenumberofslitsinthegrating.Asthenumberofslits,n,increases,thewidthofthefringesdecreases,buttheirintensityincreases.

Δr = d sinθ = mλ

(form = 1,2,3)



Differentwavelengthswillhavemaximumsatdifferentpositionsalongthescreen.

Awesome Tool: The spectrometer

Buildyourown:DIYspectrometerkithereCircular Aperture Diffraction



http://publiclab.org/wiki/foldable-spec

D

circularaperture

screen

light intensity

incident light

L

Monochromaticlightwhichpassesthroughaverysmallcircularaperturewillcreateacirculardiffractionpatternlikethis.

Thelocationofthefirstminimumcanbefoundby:

The lens as an aperture

Alenscanbeconsideredatypeofaperture.Therewillbediffractionofthelightwavesasitpassesthrough.

Two distant stars

D

= 1.22θ1λ

D



Rayleigh's Criterion

Twopointsourcescanonlyberesolvediftheirangularseparation, isgreaterthan:

Twolightbulbsare1.0mapart.Fromwhatdistancecantheselightbulbsberesolvedbyasmalltelescopewitha4.0cmdiameterobjectivelens.Assumethatthelensisdiffractionlimitedand =600nm.

ResolvingPower:Ingeneral,wecansaytheresolutionforanyopticalinstrumentislimitedbythewavelengthsofthelightused.

GuesswhatcolorthelightfromaBlu-rayplayerlaseris?Why?

α

= ≈θR sin−1 1.22λ

D

1.22λ

D

Example Problem #5:

λ

