Optical Instrumentation 9. Interferometer

This article discusses the principle of interferometry. The definition of the term along with its applications are stated in this article. Five most common type of interferometers viz. Michelson Interferometer, Mach-Zahnder Interferometer, Fabry Perot Interferometer, Sagnac Interferometer and Fiber Interferometer are discussed in detial in this article.

1. OPTOMETRY –Part IX
INTERFEROMETER
ER.FARUKBINPOYEN
DEPT.OFAEIE,UIT,BU,BURDWAN,WB,INDIA
FARUK.POYEN@GMAIL .COM

2. Contents:
Definition
PhysicalPrinciplesofInterferometers
Mach–ZehnderInterferometer
MichelsonInterferometer
Fabry–PérotInterferometer
SagnacInterferometer
FiberInterferometers
ApplicationsofInterferometer
2

3. Definition:
Interferometryisafamilyoftechniquesinwhichwaves,usually
electromagneticwaves,aresuperimposedcausingthephenomenonof
interferenceinordertoextractinformation.
Aninterferometerisanopticaldevicewhichutilizestheeffectofinterference.
3

4. Definition:
Typically,itstartswithsomeinputbeam,splitsitintotwoseparatebeamswith
somekindofbeamsplitter(apartiallytransmissivemirror).
Possiblyexposessomeofthesebeamstosomeexternalinfluences(e.g.some
lengthchangesorrefractiveindexchangesinatransparentmedium).
Recombinesthebeamsonanotherbeamsplitter.
Thepowerorthespatialshapeoftheresultingbeamcanthenbeusede.g.fora
measurement.
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5. Physical Principles of Interferometers:
Therearealsosubstantiallydifferentprinciplesofusinginterferometers.
Forexample,Michelsoninterferometersareusedinverydifferentways,using
differenttypesoflightsourcesandphoto-detectors.
Whenalightsourcewithlowopticalbandwidthisused(perhapsevenasingle-
frequencylaser),thedetectorsignalvariesperiodicallywhenthedifferencein
armlengths(opticalpathlength)ischanged.
Suchasignalmakesitpossibletodomeasurementswithadepthresolutionwell
belowthewavelength,butthereisanambiguity.
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6. Physical Principles of Interferometers:
Forexample,amonotonicincreaseordecreaseofthearmlengthdifference
leadstothesamevariationofthedetectedsignal.
Thisproblemmaybesolvedbymodulatingthearmlengthdifferencee.g.with
avibratingmirror(orwithanopticalmodulator)andbymonitoringthe
resultingmodulationonthedetectorinadditiontotheaveragesignalpower.
Simultaneousoperationofaninterferometerwithtwowavelengthsisanother
wayofremovingtheambiguity.
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7. Physical Principles of Interferometers:
Ifthedetectorisakindofcamera(e.g.aCCDchip)andthesurfacesmonitored
arefairlysmooth,thephaseprofile(andthustheprofileofopticalpathlength)
canbereconstructedbyrecordingseveralimageswithdifferentoverallphase
shifts(phase-shiftinginterferometry).
Aphase-unwrappingalgorithmcanbeusedtoretrieveunambiguouslysurface
mapsextendingovermorethanawavelength.
However,suchmethodsmaynotworkforroughsurfacesorforsurfaceswith
steepsteps.
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8. Physical Principles of Interferometers:
Awhitelightinterferometerusesabroadbandlightsource(e.g.,asuper-
luminescentdiode),sothatinterferencefringesareobservedonlyinanarrow
rangearoundthepointofzeroarmlengthdifference.
Inthatway,theabove-mentionedambiguityiseffectivelyremoved.
Awavelength-tunablelasercanbeusedtorecordthedetectorsignalfor
differentopticalfrequencies.
Fromsuchsignals,thearmlengthdifferencecanbeunambiguouslyretrieved.
Thisworksalsowithtwo-dimensionaldetectors(e.g.CCDcameras).
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9. Physical Principles of Interferometers:
Ifoneofthemirrorsisintentionallytilted,aninterferencefringepatternis
obtained.
Anychangeinarmlengthdifferencewillthenmovethefringepattern.
Thismethodmakesitpossibletomeasurephasechangessensitivelyandalsoto
measureposition-dependentphasechanges,e.g.insomeopticalelements.
Anotherclassofinterferometricmethodisnamedasspectralinterferometry.
Here,interferenceinthespectraldomainisexploited.
Thespectralmodulationperiodisessentiallydeterminedbyatimedelay.
9

10. Mach–ZehnderInterferometer:
TheMach–ZehnderinterferometerwasdevelopedbythephysicistsLudwigMachand
LudwigZehnder.
TheMach–Zehnderinterferometerisaparticularlysimpledevicefordemonstrating
interferencebydivisionofamplitude.Alightbeamisfirstsplitintotwopartsbyabeam
splitterandthenrecombinedbyasecondbeamsplitter.Dependingontherelativephase
acquiredbythebeamalongthetwopathsthesecondbeamsplitterwillreflectthebeam
withefficiencybetween0and100%.
AsshowninFigure,itusestwoseparatebeamsplitters(BS)tosplitandrecombinethe
beams,andhastwooutputs,whichcane.g.besenttophoto-detectors.
Theopticalpathlengthsinthetwoarmsmaybenearlyidentical(asinthefigure),or
maybedifferent(e.g.withanextradelayline).
Thedistributionofopticalpowersatthetwooutputsdependsontheprecisedifference
inopticalarmlengthsandonthewavelength(oropticalfrequency).
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11. Mach–ZehnderInterferometer:
Iftheinterferometeriswellaligned,thepathlengthdifferencecanbeadjusted
(e.g.byslightlymovingoneofthemirrors)sothatforaparticularoptical
frequencythetotalpowergoesintooneoftheoutputs.
Formisalignedbeams(e.g.withonemirrorbeingslightlytilted),therewillbe
somefringepatternsinbothoutputs,andvariationsofthepathlengthdifference
affectmainlytheshapesoftheseinterferencepatterns,whereasthedistribution
oftotalpowersontheoutputsmaynotchangeverymuch.
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12. Mach–ZehnderInterferometer: Set -Up
Acollimatedbeamissplitbyahalf-silveredmirror(beamsplitters).
Thetworesultingbeams(the"samplebeam"andthe"referencebeam")areeach
reflectedbyamirror.
Thetwobeamsthenpassasecondhalf-silveredmirrorandentertwodetectors.
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13. Mach–ZehnderInterferometer: Working
The"half-silvered"mirrorisjustacrummymirrorreflectinghalfthelightincidentonit,
refractingtheotherhalfthroughit.Suchmirrorsaresometimescalledonewayglass.
Sometimesweshallcallitabeamsplitter.
Thespeedoflightinglassissignificantlylessthanc.Formostglasses,theindexof
refractionisontheorderof1.5orso.
Whenalightrayisincidentonasurfaceandthematerialontheothersideofthesurfacehas
ahigherindexofrefraction(i.e.alowerspeedoflightthanthemediumthatthelightis
travellingin),thenthereflectedlightrayisshiftedinitsphasebyexactlyonehalfa
wavelength.
Theindexofrefractionofaperfectmirrorcanbethoughtofasinfinite.Thuslightreflected
byamirrorhasitsphasechangedbyonehalfawavelength.
Whenalightrayisincidentonasurfaceandthematerialontheothersideofthesurfacehas
alowerindexofrefraction,thereflectedlightraydoesnothaveitsphasechanged.
Whenalightraygoesfromonemediumintoanother,itsdirectionchangesduetorefraction
butnophasechangeoccursatthesurfacesofthetwomediums.
Whenalightraytravelsthroughamedium,suchasaglassplate,itsphasewillbeshiftedby
anamountthatdependsontheindexofrefractionofthemediumandthepathlengthofthe
lightraythroughthemedium.
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14. Mach–ZehnderInterferometer: Working
Weconsiderthetwopathsforlightarrivingatdetector1.
Path"U":
1.Reflectedbythefrontofthefirstbeamsplitter,givingaphasechangeofone-halfa
wavelength.
2.Reflectedbytheupper-leftmirror,givingafurtherphasechangeofone-halfa
wavelength.
3.Transmittedthroughtheupper-rightbeamsplitter,givingsomeconstantphasechange.
Path"D":
1.Transmittedthroughthelower-leftbeamsplitter,givingsomeconstantphasechange.
2.Reflectedbythefrontofthelower-rightmirror,givingaphasechangeofone-halfa
wavelength.
3.Reflectedbythefrontofthesecondbeamsplitter,givingaphasechangeofone-halfa
wavelength.
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15. Mach–ZehnderInterferometer: Working
Nowweconsiderlightenteringdetector2:
Path"U":
1.Reflectedbythefrontofthefirstbeamsplitter,givingaphasechangeofone-halfa
wavelength.
2.Reflectedbytheupper-leftmirror,givingafurtherphasechangeofone-halfawavelength.
3.Transmittedthroughthesecondbeamsplitter,givingsomeconstantphasechange.
4.Reflectedbytheinnersurfaceofthesecondbeamsplitter,givingnophasechange.
5.Transmittedthroughthebeamsplitterasecondtime,givinganadditionalconstantphase
change.
Path"D":
1.Transmittedthroughthelower-leftbeamsplitter,givingsomeconstantphasechange.
2.Reflectedbythefrontofthelower-rightmirror,givingaphasechangeofone-halfa
wavelength.
3.Transmittedthroughthesecondbeamsplitter,givingsomeconstantphasechange.
15

16. Mach–ZehnderInterferometer: Working
Addingupallthecontributionsforthetwopaths,weseethattheyarethesame.Thus
lightenteringdetector1viathetwopathsisinphase.Thuswegetconstructive
interferenceforthelightenteringdetector1.
Addingupallthese,weseethatthetotaldifferencebetweenthetwopathsisthattheU
pathhasgonethroughoneadditionalphasechangeofone-halfawavelength.Therefore,
therewillbecompletedestructiveinterference,andnolightwillreachdetector2.
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17. Michelson Interferometer:
AMichelsoninterferometer,asinventedbyAlbertAbrahamMichelson,usesasingle
beamsplitterforseparatingandrecombiningthebeams.
Ifthetwomirrorsarealignedforexactperpendicularincidence,onlyoneoutputis
accessible,andthelightoftheotheroutputgoesbacktothelightsource.
Ifthatopticalfeedbackisunwanted(asisoftenthecasewithalaser,whichmightbe
destabilized),and/oraccesstothesecondoutputisrequired,therecombinationofbeams
canoccuratasomewhatdifferentlocationonthebeamsplitter.
Onepossibilityistouseretroreflectors,asshowninthelowerfigure;thisalsohasthe
advantagethattheinterferometerisfairlyinsensitivetoslightmisalignmentofthe
retroreflectors.Alternatively,simplemirrorsatslightlynon-normalincidencecanbe
used.
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18. Michelson Interferometer:
Ifthepathlengthdifferenceisnon-zero,asshowninbothpartsofthefigure,
constructiveordestructiveinterferencee.g.forthedownward-directedoutputcanbe
achievedonlywithinafiniteopticalbandwidth.
MichelsonoriginallyusedabroadbandlightsourceinthefamousMichelson–Morley
experiment,sothathehadtobuildaninterferometerwithclosetozeroarmlength
difference.
TherearemanyvariationsoftheMichelsoninterferometer.Forexample,aTwyman–
GreeninterferometerisessentiallyaMichelsoninterferometerilluminatedwitha
monochromaticpointsource.Itisusedforcharacterizingopticalelements.
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19. Michelson Interferometer: Working
TheMichelsoninterferometerproducesinterferencefringesbysplittingabeamof
monochromaticlightsothatonebeamstrikesafixedmirrorandtheotheramovable
mirror.
Whenthereflectedbeamsarebroughtbacktogether,aninterferencepatternresults.
19

20. Michelson Interferometer: Working
LightfromamonochromaticsourceSisdividedbyabeamsplitter(BS),whichis
orientedatanangle45°tothebeam,producingtwobeamsofequalintensity.
Thetransmittedbeam(T)travelstomirrorM1anditisreflectedbacktoBS.
50%ofthereturningbeamisthenreflectedbythebeamsplitterandstrikesthescreen,
E.Thereflectedbeam(R)travelstomirrorM2,whereitisreflected.
50%ofthisbeampassesstraightthroughbeamsplitterandreachesthescreen.
20
Fig 2
Fig 3

21. Michelson Interferometer: Working
SincethereflectingsurfaceofthebeamsplitterBSisthesurfaceonthelowerright,the
lightraystartingfromthesourceSandundergoingreflectionatthemirrorM2passes
throughthebeamsplitterthreetimes,whiletherayreflectedatM1travelsthroughBS
onlyonce.
Theopticalpathlengththroughtheglassplatedependsonitsindexofrefraction,which
causesanopticalpathdifferencebetweenthetwobeams.
Tocompensateforthis,aglassplateCPofthesamethicknessandindexofrefractionas
thatofBSisintroducedbetweenM1andBS.
TherecombinedbeamsinterfereandproducefringesatthescreenE.Therelativephase
ofthetwobeamsdetermineswhethertheinterferencewillbeconstructiveor
destructive.
ByadjustingtheinclinationofM1andM2,onecanproducecircularfringes,straight-
linefringes,orcurvedfringes.Fig2showscircularfringes.
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22. Michelson Interferometer: Working
Fromthescreen,anobserverseesM2directlyandthevirtualimageM1'ofthemirror
M1,formedbyreflectioninthebeamsplitter,asshowninFig.3.Thismeansthatoneof
theinterferingbeamscomesfromM2andtheotherbeamappearstocomefromthe
virtualimageM1'.
Ifthetwoarmsoftheinterferometerareequalinlength,M1'coincideswithM2.Ifthey
donotcoincide,letthedistancebetweenthembed,andconsideralightrayfroma
pointS.
ItwillbereflectedbybothM1'andM2,andtheobserverwillseetwovirtualimages,
S1duetoreflectionatM1',andS2duetoreflectionatM2.
Thesevirtualimageswillbeseparatedbyadistance2d.Ifθistheanglewithwhichthe
observerlooksintothesystem,thepathdifferencebetweenthetwobeamsis2��??????????????????.
WhenthelightthatcomesfromM1undergoesreflectionatBS,aphasechangeofπ
occurs,whichcorrespondstoapathdifferenceofλ/2.
22

23. Michelson Interferometer:
Withanopticalinterferometer,onecanmeasuredistancesdirectlyintermsof
wavelengthoflightused,bycountingtheinterferencefringesthatmovewhenoneor
theotheroftwomirrorsaremoved.
PrecisedistancemeasurementscanbemadewiththeMichelsoninterferometerby
movingthemirrorandcountingtheinterferencefringeswhichmovebyareference
point.
ThedistanceΔassociatedwithmfringesisΔ=????????????/2.
Thetotalpathdifferencebetweenthetwobeamsis
Theconditionforconstructiveinterferenceisthen
23

24. Fabry–PérotInterferometer:
AFabry–Pérotinterferometerconsistsoftwoparallelmirrors,allowingformultiple
roundtripsoflight.(Amonolithicversionofthiscanbeaglassplatewithreflective
coatingsonbothsides.)
Forhighmirrorreflectivities,suchadevicecanhaveverysharpresonances(ahigh
finesse),i.e.exhibitahightransmissiononlyforopticalfrequencieswhichclosely
matchcertainvalues.
Basedonthesesharpfeatures,distances(orchangesofdistances)canbemeasuredwith
aresolutionfarbetterthanthewavelength.Similarly,resonancefrequenciescanbe
definedveryprecisely.
AmodifiedversionistheFizeauinterferometer,wherethesecondmirroristotally
reflective,andslightlytilted.Thereflectedlightisused(e.g.withanangledbeam
splitter)e.g.forcharacterizingopticalcomponents.
AnotherspecialkindofFabry–Pérotinterferometer,usedfordispersioncompensation,
istheGires–Tournoisinterferometer.
24

25. Fabry–PérotInterferometer:
Thisinterferometermakesuseofmultiplereflectionsbetweentwocloselyspaced
partiallysilveredsurfaces.
Partofthelightistransmittedeachtimethelightreachesthesecondsurface,resulting
inmultipleoffsetbeamswhichcaninterferewitheachother.
Thelargenumberofinterferingraysproduceaninterferometerwithextremelyhigh
resolution,somewhatlikethemultipleslitsofadiffractiongratingincreaseits
resolution.
25

26. Fabry–PérotInterferometer:
TheFabry-PerotInterferometermakesuseofmultiplereflectionswhichfollowthe
interferenceconditionforthinfilms.
Thenetphasechangeiszerofortwoadjacentrays,sothecondition2�cos??????=??????λ
representsanintensitymaximum.
26

27. Fabry–PérotInterferometer: Resolution
Ahigh-resolutioninterferometer,theFabry-PerotInterferometerhasaresolvanceof
Whichmeansthattheleastseparationoftwospectrallinesisgivenby
ThisseparationmeansthatthetwowavelengthssatisfytheRayleighcriterion.
Theinterferometercanalsobecharacterizedbyitsfreespectralrange,thechangein
wavelengthnecessarytoshiftthefringesystembyonefringe:
27

28. SagnacInterferometer:
Thesagnacinterferometerisatypeofcommonpathinterferometer.
ItusestheconceptofSagnaceffecttomeasurerotationusingopticalinterferometry.
ASagnacinterferometer(namedaftertheFrenchphysicistGeorgesSagnac)uses
counter-propagatingbeamsinaringpath,realizede.g.withmultiplemirrorsorwithan
opticalfiber.
Ifthewholeinterferometerisrotatede.g.aroundanaxiswhichisperpendiculartothe
drawingplane,thisintroducesarelativephaseshiftofthecounter-propagatingbeams
(Sagnaceffect).
Thesensitivityforrotationsdependsontheareacoveredbythering,multipliedbythe
numberofroundtrips(whichcanbelargee.g.whenusingmanyturnsinanoptical
fiber).
Itispossiblee.g.toobtainasensitivitywhichissufficientformeasuringtherotationof
theEartharounditsaxis.
28

29. SagnacInterferometer: SagnacEffect
TheSagnacEffectmanifestsitselfinasetupcalledaringinterferometer.
Abeamoflightissplitandthetwobeamsaremadetofollowthesamepathbutin
oppositedirections.
Onreturntothepointofentrythetwolightbeamsareallowedtoexittheringand
undergointerference.
Therelativephasesofthetwoexitingbeams,andthusthepositionoftheinterference
fringes,areshiftedaccordingtotheangularvelocityoftheapparatus.
Inotherwords,whentheinterferometerisatrestwithrespecttotheearth,thelight
travelsataconstantspeed.
However,whentheinterferometersystemisspun,onebeamoflightwillslowwith
respecttotheotherbeamoflight.
ThisarrangementisalsocalledaSagnacinterferometer.
29

30. SagnacInterferometer: Working
Theshiftoftheinterferencefringesisproportionaltotheplatform'sangularvelocity.
Thesensitivityforrotationsisbasedontheareacoveredbythering,multipliedbythe
numberofroundtrips.
Sagnacinterferometerismadeofaringarchitectureasitcanbeobserved
30

31. SagnacInterferometer: Working
TheinterferometerinputandoutputspotisA.
Bothplanewavespropagateaccordingto(A,B,C,D,A)and(A,D,C,B,A)pathsand
soarecontra-propagative.
Theyfollowidenticalpaths.
Forwave1andwave2,opticalphasebetweentheinputandoutputis
??????
1=??????
2=
2??????
λ
�����
Thephasedifferencebetweenbothwavesiszero
??????
1−??????
2=0
Duetotheinterferometersymmetry,anyswitchofoneofthethreemirrorshasno
influenceonthefringesfigureastheopticalpathsareallthesameonthephasefront.
Thedifferenceofopticalpathbetweenbothwavesisconstantandzero.
31

32. SagnacInterferometer: Working
Asaconsequence,theinterferencesignaliswrittenas
??????�,�,�=
�
0
2
2
1+cos0=�
0
2
Thefringesfigureisuniform:Ithasapaletone.
Thisinterferometerisofinterestinthecasewherethecavityisinrotationaroundan
axisΔperpendiculartothefigureplane.
Wecanobservethatinthiscasebothlightbeamsareoutofphaseattheinterferometer
output.
Thephasedifferencedependsontheangularspeedofrotationω,ofthespeedoflight
‘c’andofthecavitysurface‘S’accordingtotherelationship
??????=
2??????
λ
.
4????????????
�
FindsapplicationinringgyroscopeandGlobalpositioningsystem(GPS).
32

33. FiberInterferometers:
Alltheinterferometertypesdiscussedabovecanalsobeimplementedwithoptical
fibers.Insteadofbeamsplitters,oneusesfibercouplers.
Apotentialdifficultyisthatthepolarizationstateoflightmaychangeduring
propagationinthefiber.
Thisoftenrequiresonetoincludeafiberpolarizationcontroller(whichmay
occasionallyhavetobereadjusted)ortousepolarization-maintainingfibers.
Alsonotethattemperaturechangesinthefibers(aswellasbending)canaffectthe
opticalphaseshifts.
Thiscanbeaproblemifdifferentfibersbelongtodifferentinterferometerarms.
However,therearealsofiberinterferometerswhereonefiberservesforbotharms,e.g.
usingtwodifferentpolarizationdirectionsinthesamefiber.
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34. Applications:
Interferometersareusedtomeasurelength,distance,displacementwithan
accuracy&sensitivityoftheorderofwavelengthoflight.
Formeasuringthewavelengthe.g.ofalaserbeam(→wavemeter),orfor
analyzingabeamintermsofwavelengthcomponents.
Formonitoringslightchangesinanopticalwavelengthorfrequency(typically
usingthetransmissioncurveofaFabry–Pérotinterferometer)(frequency
discriminators).
Formeasuringrotations(withaSagnacinterferometer).
Formeasuringslightdeviationsofanopticalsurfacefromperfectflatness(or
fromsomeothershape).
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35. Applications:
Formeasuringthelinewidthofalaser(→self-heterodynelinewidth
measurement,frequencydiscriminator).
Forrevealingtinyrefractiveindexvariationsorinducedindexchangesina
transparentmedium.
Formodulatingthepowerorphaseofalaserbeam,e.g.withaMach–Zehnder
modulatorinanopticalfibercommunicationsystem.
Formeasurementsofthechromaticdispersionofopticalcomponents.
Asanopticalfilter.
Forthefullcharacterizationofultrashortpulsesviaspectralinterferometry.
35

36. References:
www.rp-photonics.com/interferometers.html
https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer
https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_ze
hnder00.pdf
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/michel.html
http://vlab.amrita.edu/?sub=1&brch=189&sim=1106&cnt=1
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/fabry.html#c1
https://en.wikipedia.org/wiki/Sagnac_effect
http://www.optique-ingenieur.org/en/courses/OPI_ang_M02_C05/co/Contenu_26.html
36