Determining3Dmolecularorientationfrompolarization-IRspectra:tutorial

YoungJongLee*

BiosystemsandBiomaterialsDivision,NationalInstituteofStandardsandTechnology,Gaithersburg,MD20899,USA

*youngjong.lee@

Abstract:Despitetheubiquityofmolecularalignmentinnaturalandsynthesizedmaterials,accuratemappingofthree-dimensional(3D)molecularorientationswithsufficientspatialresolvingpowerhasremainedchallenging.Conventionalanalysisapproachesofpolarization-basedopticalimagingcannotmeasuretheout-of-planeangleofmolecularorientationswithouttiltingasample.Recently,weproposedanewtheoreticalalgorithmtodeterminethe3Dorientationangleofalignedmoleculesusingpolarization-controlledinfrared(IR)spectraldatawithouttiltingasample.Thealgorithmwasbasedonaconcurrentanalysisoftheabsorptancesoftwonon-parallelIRtransitiondipolemomentsusingasingleorientationaldescriptor.Theoutcomeofthenon-iterativecalculationisthe3Dangleofthemeanorientationandtheorderparameterofthelocalorientationaldistributionfunction.Thealgorithmwassuccessfullyappliedtotheanalysisofpolarization-controlledhyperspectralIRimagingofsemi-crystallinepolymerfilms.Thistutorialderivesandsummarizesvarioustheoreticaldescriptionspresentedinourrecentpapersanddiscussesassumptionsandlimitationsassociatedwiththeanalysismethod.

1.INTRODUCTION

Microscopicmolecularalignmentoccursatvariouslengthscalesofsolid-statematerials,affectingthestructuralandfunctionalpropertiesofmacroscopicproducts.Thus,thereisacriticalneedtoimagemolecularalignmentwithasufficientspatialresolutiontounderstandtheseanisotropicstructure-propertyrelations.DiffractionofX-rayandelectronbeamshavebeenusedasstandardmethodstovisualizethe3Datomiclocationsandmolecularorientationsincrystallinematerials.However,thoseensemble-baseddiffractionmethodsareunsuitableforsampleswithspatiallyheterogeneousmolecularalignment.Later,imaging-capableX-ray

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[1,2]

andelectron

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[3-5]

diffractiontechniqueshavebeendeveloped,buttheX-rayandelectrontomographicimagingapproachesarestilllimitedtohighlycrystallinesamplesandunsuitablefororganicmaterialsduetoeitherlowcross-sectionorsampledamage.

Alternatively,noninvasiveopticaltechniquesbasedonlinearlypolarizedlightshavebeenwidelyusedtoimagetheopticalanisotropyofasample.Theimagingmethodsincludefluorescence

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[6,7]

,second-harmonicandsum-frequencygeneration

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[8,9]

,infrared(IR)absorption

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[10-14]

,spontaneousRamanscattering

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[15-19]

,andcoherentRamanscattering

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[20-24]

.However,thosepolarization-controlledopticalimagingtechniquescanprovideonlyanisotropysignals2D-projectedontothepolarizationplaneand,thus,cannotdeterminetheout-of-planeangleofmolecularorientations.Someadvancedopticalimagingtechniquescanovercomethislimitationofpolarizationimaging.Forexample,radiallypolarizedexcitationdetectedthesignaldifferencebetweenlongitudinallyandtransverselyalignedmoleculesinspontaneousandcoherentRamanmicroscopies

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[25,26]

.However,thesimpleratioofradialandtransverseRamanintensitiescouldnotbeconvertedintotheout-of-planeanglewithoutknowingthemoleculartwistingangleandtheorientationalbroadening

ADDINEN.CITE<EndNote><Cite><Author>Mino</Author><Year>2012</Year><RecNum>47</RecNum><DisplayText>[27]</DisplayText><record><rec-number>47</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723584356"guid="1bbe0b78-98b7-4bb1-b585-06e9baab431d">47</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Mino,Toshihiro</author><author>Saito,Yuika</author><author>Yoshida,Hiroyuki</author><author>Kawata,Satoshi</author><author>Verma,Prabhat</author></authors></contributors><titles><title>Molecularorientationanalysisoforganicthinfilmsbyz-polarizationRamanmicroscope</title><secondary-title>JournalofRamanSpectroscopy</secondary-title></titles><periodical><full-title>JournalofRamanSpectroscopy</full-title><abbr-1>J.RamanSpectrosc.</abbr-1><abbr-2>JRamanSpectrosc</abbr-2></periodical><pages>2029-2034</pages><volume>43</volume><number>12</number><section>2029</section><dates><year>2012</year></dates><isbn>03770486</isbn><urls></urls><electronic-resource-num>10.1002/jrs.4118</electronic-resource-num></record></Cite></EndNote>

[27]

.Otheradvancedhigh-resolutionimagingmethodsincludeanalyzingfinelyz-scanneddefocusedimages

ADDINEN.CITE<EndNote><Cite><Author>Toprak</Author><Year>2006</Year><RecNum>48</RecNum><DisplayText>[28]</DisplayText><record><rec-number>48</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723584356"guid="2ff0a1cc-f0f7-483e-b926-8e2a16931f4e">48</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Toprak,Erdal</author><author>Enderlein,Joerg</author><author>Syed,Sheyum</author><author>McKinney,SeanA</author><author>Petschek,RolfeG</author><author>Ha,Taekjip</author><author>Goldman,YaleE</author><author>Selvin,PaulR</author></authors></contributors><titles><title>Defocusedorientationandpositionimaging(DOPI)ofmyosinV</title><secondary-title>ProceedingsoftheNationalAcademyofSciences</secondary-title></titles><periodical><full-title>ProceedingsoftheNationalAcademyofSciences</full-title><abbr-1>Proc.Natl.Acad.Sci.USA</abbr-1><abbr-2>Proc.Natl.Acad.Sci.USA</abbr-2></periodical><pages>6495-6499</pages><volume>103</volume><keywords><keyword>3Dorientation</keyword><keyword>fluorescenceimagingwithone-nanometeraccuracy</keyword><keyword>leverarm</keyword><keyword>singlemolecule</keyword></keywords><dates><year>2006</year></dates><isbn>0027-8424,1091-6490</isbn><urls></urls><electronic-resource-num>10.1073/pnas.0507134103</electronic-resource-num><language>en</language></record></Cite></EndNote>

[28]

,comparingwithpointspreadfunctions

ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2019</Year><RecNum>49</RecNum><DisplayText>[29]</DisplayText><record><rec-number>49</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723584356"guid="4891dc41-8931-4253-8944-d4bbd95cd99a">49</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Muzhou</author><author>Marr,JamesM.</author><author>Davanco,Marcelo</author><author>Gilman,JeffreyW.</author><author>Liddle,J.Alexander</author></authors></contributors><titles><title>Nanoscaledeformationinpolymersrevealedbysingle-moleculesuper-resolutionlocalization-orientationmicroscopy</title><secondary-title>MaterialsHorizons</secondary-title></titles><periodical><full-title>MaterialsHorizons</full-title><abbr-1>Mater.Horiz.</abbr-1><abbr-2>MaterHoriz</abbr-2></periodical><pages>817-825</pages><volume>6</volume><dates><year>2019</year></dates><publisher>RoyalSocietyofChemistry</publisher><urls></urls><electronic-resource-num>10.1039/c8mh01187g</electronic-resource-num></record></Cite></EndNote>

[29]

,ordetectingemissionsfrom2D-degeneratefluorophores

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[30,31]

.However,thoseexquisitemethodsaresuitableonlyforsinglechromophoremoleculesorisolateddefectsbutunsuitableforcontinuouslydistributedmaterials,suchaspolymersandbiologicaltissues.

Recently,weintroducedasimpletheoreticalalgorithmthatcandeterminethe3Dorientationofcontinuouslydistributedmoleculesfromconventionalpolarization-controlledIRandRamanspectra

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[32,33]

.Inthisanalysismethod,twoIRpeaksoriginatingfromnon-paralleltransitiondipolemomentsaredescribedwiththeorientationanglesofasingledirector,themolecularsymmetryaxis.Then,thepolarizationprofilesoftwoIRabsorptionpeaksareanalyzedconcurrently.Thephaseangles,amplitudes,andoffsetsofthepolarizationprofilesareusedtodeterminethe3Dorientationangleofthemolecularsymmetryaxis.Usingthisconcurrentpolarizationanalysismethod,Wrobeletal.andmyteamseparatelypublishedexperimentalresultstodeterminethe3Dorientationangleofalignedmoleculesinsemi-crystallinepolymerfilms

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[34-36]

.ThistutorialsummarizestheoreticalderivationsscatteredinseveralpapersoftheconcurrentanalysisalgorithmforIR

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[33-36]

andsomefromcoherentRamanmicroscopy

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[32,37,38]

.

2.FUNDAMENTALS

2.1.TransitionProbabilityofOrientedDipoleMoments

Thetransitionprobabilityofadipolemomentbylinearlypolarizedlightisproportionaltothesquareoftheinnerproductofatransitiondipolemomentandalightpolarizationdirectionvector.Thedirectionalunitvectorofatransitiondipolemomentcanbeexpressedasμψ,θ=cosψsinθ,sinψsinθ,cosθ,whereψistheazimuthalangleonthexy-plane;andθistheaxialanglefromthez-axis.Ifthelightpropagationdirectionisdefinedasthez-axis,thelightpolarizationvectorscanbeexpressedaseη=cosη,sinη,0,whereisthelightpolarizationanglefromthex-axis.Then,thetransitionprobability,fη,canbeexpressedas

fη∝eη?μψ,θ2∝sin2θ?cos2η?ψ (

SEQ(\*ARABIC

1

)

Analyzingthephaseoffηwillleadtoψ.However,theamplitudeoffηcannotbereadilyconvertedtosin2θbecauseitsobservedamplitudeisundeterminableandisaffectedbytheconcentration,thecrosssection,andotherexperimentalconditions.ThisextrinsicnatureofanobservedamplitudeiswhyconventionalpolarizationIRanalysisofasinglevibrationalmodecannotdeterminetheout-of-planeangle,θ.Alternatively,weproposedtoconcurrentlyanalyzeapairoforthogonalIRtransitiondipolemoments,whichiscalledorthogonal-pairpolarizationIR(OPPIR).AnIRmodewhosetransitiondipolemomentisparalleltothemolecularsymmetryaxisisassignedastheprimarymode.AnIRmodewhosetransitiondipolemomentisperpendiculartothemolecularsymmetryaxisiscalledthesecondarymode.Thetransitiondipolemomentsofthetwomodesareillustratedastheredandbluearrowsin

REF_Ref138071518

Fig.1

A.InOPPIR,thepolarizationangle-dependenttransitionprobabilitiesofμ1andμ2areexpressedwithasingledirectororientation,ψm,θm,?m,correspondingtothemolecularsymmetryaxis.ψm,θm,and?maretheazimuthalangle,theaxialangle,andtherotationangleofthemolecularsymmetryaxis,respectively.DescribingthetransitionsofthetwomodeswithasingledirectororientationisthemostsignificantdifferencebetweenOPPIRandotherconventionalpolarizationapproaches,wherethepolarizationangle-dependenttransitionprobabilities,f1ηandf2η,areseparatelydescribedbytheirorientationanglesψ1,θ1,?1andψ2,θ2,?2.

Fig.

SEQFigure\*ARABIC

1

.(A)IllustrationofthetransitiondipolemomentsofaprimaryIRmodeandasecondaryIRmode(μ1andμ2)ofapolymerchainrelativetothemolecularsymmetryaxis.(B)μ1andμ2orientedinthelaboratoryframe.ψmandθmareitsazimuthalandaxialangleofthemolecularsymmetryaxis,respectively.isthepolarizationangleoftheincidentlight.(C)Polarization-controlledIRabsorptionmicroscopysetup,whereahalf-waveplaterotates.(D)Illustrationofthetransitionprobabilityfunctions,f1ηandf2η,ofμ1andμ2,respectively.

First,theoriginalμ1o=0,0,1andμ2o=1,0,0areassumedtobeparalleltothez-andx-axes,respectively,inthemolecularcoordinate.Theyaretransformedintoμ1andμ2inthelaboratorycoordinateusinganEulerrotationmatrix.

Rψm,θm,?m=cosψm?sinψm0sinψmcosψm0001cosθm0sinθm010?sinθm0cosθmcos?m?sin?m0sin?mcos?m0001 (

SEQ(\*ARABIC

2

)

Then,μ1=Rψm,θm,?mμ1o,andthetransitionprobabilityofμ1becomes

f1η=f1osin2θm?cos2η?ψm (

SEQ(\*ARABIC

3

)

=f1maxcos2η?ψ1 (

SEQ(\*ARABIC

4

)

wheref1oistheabsolutetransitionprobability,andf1maxistheapparentmaximumforthepolarizationprofileinEq.

REF_Ref137916884\h

(4)

.Thephaseangle,ψ1,ofthepolarizationprofileisidenticaltotheanimuthalangleofthemolecularsymmetryaxis,ψm.

Similarly,μ2=Rψm,θm,?mμ2o,andthetransitionprobabilityofμ2becomes

f2η=f2ocosθmcos??mcosη?ψm+sin?msinη?ψm2 (

SEQ(\*ARABIC

5

)

whichcanalsobeexpressedinacosinesquaredfunction

f2η=f2maxcos2η?ψ2 (

SEQ(\*ARABIC

6

)

Theapparentmaximumis

f2max=f2o1?sin2θmcos2?m? (

SEQ(\*ARABIC

7

)

andthephaseisψ2=ψm+tan?1tan?msecθm.Then,thephasedifferencebetweenf1ηandf2ηbecomes

Δ≡ψ2?ψ1=tan?1tan?msecθm (

SEQ(\*ARABIC

8

)

AsshowninEqs.

REFeq_f1\h

(3)

and

REF_Ref174562704\h

(5)

,bothf1ηandf2ηareinacos2form,andtheiramplitudesandphasesareexpressedwiththecommonorientationanglesofasingledirector(ψm,θm,?m).

2.2.PolarizationAngle-DependentTransmittance

Lightabsorptionbymoleculesinasamplewilllowerlighttransmission,andthedegreeoftransmissionisexpressedastransmittance(T≡IoutIin),whereIinisincidentlightintensity,andIoutistransmittedlightintensity.Therehavebeenmanytheoreticalpapersthatdescribethepolarizationangle-dependenceofIRabsorptionandtransmissionbyorientedtransitiondipolemoments

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[39-41]

.Forexample,Gregoriouetal.derivedthepolarizationdependenceoftransmissionusing3Dellipsoidmodelsofatransitiondipole

ADDINEN.CITE<EndNote><Cite><Author>Gregoriou</Author><Year>2004</Year><RecNum>1719</RecNum><DisplayText>[40]</DisplayText><record><rec-number>1719</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723587645"guid="f6891a96-294f-4107-9fc8-9beec22bcfb3">1719</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Gregoriou,V.G.</author><author>Tzavalas,S.</author><author>Bollas,S.T.</author></authors></contributors><titles><title>AngularDependenceinInfraredLinearDichroism:AReevaluationoftheTheory</title><secondary-title>AppliedSpectroscopy</secondary-title></titles><periodical><full-title>AppliedSpectroscopy</full-title><abbr-1>Appl.Spectrosc.</abbr-1><abbr-2>ApplSpectrosc</abbr-2></periodical><pages>665-661</pages><volume>58</volume><number>6</number><section>655</section><dates><year>2004</year></dates><urls></urls></record></Cite></EndNote>

[40]

.Later,toextendthelineardichroismtheory,Mayerh?ferusedaMaxwellformalismwithacomplexbirefringentindexofrefractiontocalculatethetransmissionspectrumofanIRpeakforvarioustiltinganglesandabsorbances

ADDINEN.CITE<EndNote><Cite><Author>Mayerhofer</Author><Year>2018</Year><RecNum>57</RecNum><DisplayText>[41]</DisplayText><record><rec-number>57</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723584356"guid="5e960b85-2a0a-46c8-87ea-c1d0e666c618">57</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Mayerhofer,TG</author></authors></contributors><auth-address>LeibnizInstPhotonTechnol,SpectImaging,AlbertEinsteinStr9,Jena,Germany|FriedrichSchillerUniv,InstPhysChem,Helmholtzweg4,D-07743Jena,Germany|FriedrichSchillerUniv,AbbeCtrPhoton,Helmholtzweg4,D-07743Jena,Germany</auth-address><titles><title>EmployingTheoriesFarbeyondTheirLimits-LinearDichroismTheory</title><secondary-title>ChemPhysChem</secondary-title></titles><pages>2123-2130</pages><volume>19</volume><number>17</number><keywords><keyword>Lineardichroismtheory</keyword><keyword>absorption</keyword><keyword>infraredspectroscopy</keyword><keyword>UV</keyword><keyword>Visspectroscopy</keyword><keyword>polarizedspectroscopy</keyword><keyword>PHASEPOLYCRYSTALLINEMATERIALS</keyword><keyword>MODELINGIR-SPECTRA</keyword><keyword>LARGECRYSTALLITESLIMIT</keyword><keyword>OPTICAL-SYSTEMS</keyword><keyword>DISPERSIONANALYSIS</keyword><keyword>RANDOMORIENTATION</keyword><keyword>NONNORMALREFLECTION</keyword><keyword>MATRIXFORMALISM</keyword><keyword>CALCULUS</keyword><keyword>LIGHT</keyword></keywords><dates><year>2018</year><pub-dates><date>SEP52018</date></pub-dates></dates><isbn>1439-4235</isbn><accession-num>WOS:000443680500001</accession-num><work-type>Review</work-type><urls></urls><electronic-resource-num>10.1002/cphc.201800214</electronic-resource-num><language>English</language></record></Cite></EndNote>

[41]

.However,thesetheoriesaretoocomplicatedtoassociatewiththegeneralizedcos2formfortheconcurrentdescriptionofnon-paralleltransitiondipolemomentsofEqs.

REFeq_f1\h

(3)

and

REF_Ref174562704\h

(5)

.Inthispaper,wesimplifytheangle-dependenttransmittancebyassumingthattheeffectofnon-absorptivebirefringenceisnegligible.Thisassumptionallowsustodiscusslightabsorptionseparatelyfortheelectricfieldsparallelandperpendiculartoaprojectionplanedefinedbythelightpropagationdirectionandthetransitiondipolemoment.

Fig.

SEQFigure\*ARABIC

2

.Transmissionoflinearlypolarizedlightthroughall-paralleltransitiondipolemoments(μ)intheoriented-gasmodel.Amolecularprojectionplanedefinedbythebeampropagation(thez-axis)andμ.Incidentlightcanbedividedintotwoorthogonalprojections:oneisparallel(Iparo),andtheotherisperpendicular(Iperpo)tothemolecularprojectionplane.ηisthepolarizationangleoftheincidentlight.disthesamplethickness.ψandθaretheazimuthalandtiltinganglesofμ,respectively.

Beforewediscussorientationallybroadenedmolecularsystems,wederivetheformulasforanidealalignmentcasewhereallmoleculesareparalleltoeachother(theoriented-gasmodel).Asshownin

REF_Ref174576249\h

Fig.2

,theincidentlightintensitycanbedividedintotwoorthogonalcomponents:oneparalleltothemolecularprojectionplane,Iparo=Iocos2η?ψ,andtheotherperpendiculartotheplane,Iperpo=Iosin2η?ψ

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[33,41,42]

.

Io=Iparo+Iperpo=Iocos2η?ψ+Iosin2η?ψ (

SEQ(\*ARABIC

9

)

Whenthelightpassesthroughasample,IpardecreasesduetolightabsorptionwhileIperpdoesnotchange.

dIpar=?(fosin2θ)cIpardz

dIperp=0 (

SEQ(\*ARABIC

10

)

wherecistheconcentrationofabsorbingmolecules.IfdIparanddIperpareintegratedoverthesamplethicknessofd,thetransmittedlightintensitieswillbecome

Ipar=Iparoe?fmaxcd=Iocos2η?ψe?fmaxcd

Iperp=Iperpo=Iosin2η?ψ=Io1?cos2η?ψ (

SEQ(\*ARABIC

11

)

wherefmax=fosin2θisthetransitionprobabilityofanorientedmoleculeintheprojectionplane.Then,thetransmittedlightintensitybecomes

Iη=Ipar+Iperp=Io1?1?e?fmaxcdcos2η?ψ (

SEQ(\*ARABIC

12

)

anditstransmittancebecomes

Tη≡IηIo=1?1?e?fmaxcdcos2η?ψ (

SEQ(\*ARABIC

13

)

2.3.AbsorptanceorAbsorbance

Insteadoftransmittance,absorptance

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[43-45]

andabsorbance

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[46-49]

havebeenwidelyusedtodescribethepolarizationopticalanisotropyofalignedmolecularsystems.Typically,researchershaveusedasimplifiedformofcos2ηtodescribethepolarizationdependenceofbothabsorptance,αη,andabsorbance,Aη.However,later,researchersreportedthatAηdeviatesfromthecos2ηformandthatαηisnotlinearlypproportionaltothesampleconcentration

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[40,41]

.Briefly,TηinEq.

REF_Ref137917193\h

(13)

canbeconvertedintoαηbythedefinition,α≡1?T.

αη=(1?e?fmaxcd)cos2η?ψ (

SEQ(\*ARABIC

14

)

Thisisintheformofcos2η.However,theamplitude,αmax=1?e?fmaxcd,isnotlinearlyproportionaltofmaxcd.Instead,TηcanbeconvertedintoAηfromthedefinition

Aη≡?logTη=?log1?1?e?fmaxcd×cos2η?ψ (

SEQ(\*ARABIC

15

)

whichisfarfromthecos2ηform.However,itsmaximumvalue,Amax=fmaxcd,becomesproportionaltofmaxcd.

REF_Ref138071579

Fig.3

showsαηandAηcalculatedfromEqs.

REF_Ref175041740

(14

)and

REFeq

(15)

forvariousfmaxcdvalues,whicharecomparedwithacos2ηfunctionwithanamplitudeoffmaxcd.First,whenfmaxcdissmall,Aηissimilartofmaxcdcos2η?ψ.However,assampleabsorptionincreases,fmaxcdislarger,Aηbecomeslowerthanthecos2ηform.Second,asαηisalwaysinthecos2ηformforallfmaxcd.However,forhigherfmaxcd,theamplitudeofαηbecomeslowerthanfmaxcd.

Fig.

SEQFigure\*ARABIC

3

.PlotsofαηandAηcalculatedfromEqs.

REF_Ref175041740

(14

)and

REFeq

(15)

forvariousfmaxcd,comparedwiththecorrespondingfmaxcdcos2η.

Ifthesymmetryaxisofamolecularaxisispre-defined,αmaxandAmaxcanbeeasilymeasuredatthepolarizationangleparalleltothemolecularaxiswithoutscanningthepolarizationangleoftheincidentlight.However,asystemwithanunknownmolecularaxiswillrequireaseriesofabsorptionmeasurementsatdiscretepolarizationangles.Then,thepolarizationprofileofeitherαηorAηneedstobefittedwiththecorrespondinganalyticalfunctionsothatthephaseandαmaxandAmaxcanbedetermined.Thebottomplotsof

REF_Ref138071579

Fig.3

suggestthatforweakabsorption(Amax<0.1),bothαηandAηcanbefittedwithacos2η?ψfunctionwithanamplitudeproportionaltofmaxcd.Theequivalenceofαηtofmaxcdcos2η?ψfortheweakabsorptionlimitcanbederivedbyusingaTaylorseries,e?x=1?x+x2/2??,inEq.

REF_Ref175041740

(14

)

ADDINEN.CITE<EndNote><Cite><Author>Lee</Author><Year>2018</Year><RecNum>53</RecNum><DisplayText>[33]</DisplayText><record><rec-number>53</rec-number><foreign-keys><keyapp="EN"db-id="pvarx2tei2fw0oesrpvpd52gpeprf2t99vee"timestamp="1723584356"guid="3c37947b-fa91-4906-a600-4c14f6e66524">53</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Lee,YoungJong</author></authors></contributors><titles><title>ConcurrentpolarizationIRanalysistodeterminethe3Danglesandtheorderparameterformolecularorientationimaging</title><secondary-title>OpticsExpress</secondary-title></titles><periodical><full-title>OpticsExpress</full-title><abbr-1>Opt.Express</abbr-1><abbr-2>OptExpress</abbr-2></periodical><pages>24577-24590</pages><volume>26</volume><keywords><keyword>Analyticaltechniques</keyword><keyword>Infraredspectroscopy</keyword><keyword>Lightintensity</keyword><keyword>Polarizationanalysis</keyword><keyword>Ramanscattering</keyword><keyword>Secondharmonicgeneration</keyword></keywords><dates><year>2018</year></dates><publisher>OpticalSocietyofAmerica</publisher><urls></urls><electronic-resource-num>10.1364/OE.26.024577</electronic-resource-num></record></Cite></EndNote>

[33]

.Similarly,theconvergenceofAηtofmaxcdcos2η?ψcanbeprovedbyconvertingEq.

REFeq

(15)

withadifferentTaylorseries,log1?x=?x?x2/2?x3/3??.However,whentheabsorptionisrelativelyhigh,αmaxisnotproportionaltofmaxcdalthoughαηcanbefittedwithacos2ηfunction.Ontheotherhand,Aηbecomesunfittablewithacos2ηfunction,andAmaxbecomesunreliable.Thus,weproposeasimpletwo-stepprocesstodetermineψandfmaxcd:(1)fittheαηwiththeformofcos2η?ψanddetermineαmaxandψ;and(2)convertαmaxintofmaxcdusingtherelation

fmaxcd=log1?αmax=Amax (

SEQ(\*ARABIC

16

)

Thus,wedeterminethephaseψandtheamplitudefmaxcdforbothprimaryandsecondarymodesfromtheirpolarization-dependentabsorptances.

3.CALCULATIONOF3DORIENTATIONANGLE

3.1.3DOrientationalAnglesforMoleculesoftheOriented-GasModel

Thetwo-stepprocessofα1ηandα2ηoftheprimaryandthesecondarymodes,respectively,willleadusto(ψ1andf1maxcd)and(ψ2andf2maxcd).Fromthefourobservedvalues,wedeterminethe3Dorientationanglesψm,θm,?m.First,ψmcanbeeasilydeterminedfromψ1,asmentionedwithEqs.

REFeq_f1\h

(3)

and

REF_Ref137916884\h

(4)

,

ψm=ψ1 (

SEQ(\*ARABIC

17

)

Second,θmand?mcanbecalculatedbycombiningfromEqs.

REFeq_f1\h

(3)

–

REFeq_delta\h

(8)

.Toremovetheextrinsicvalues,likeconcentration(c)andpathlength(d)inEq.

REF_Ref137918776

(16)

,wedefinetheratioofAmaxbetweentheprimaryandthesecondarymodes.Then,fromEqs.

REFeq_f1\h

(3)

,

REF_Ref137916884\h

(4)

,

REF_Ref137917003

(6)

,and

REFeq_f2max

(7)

A2maxA1max=f2maxcdf1maxcd=f2o1?sin2θmcos2?m?f1osin2θm=f2of1o1sin2θm?cos2?m? (

SEQ(\*ARABIC

18

)

wheref2of1ocanbemeasuredseparatelyfromtheIRspectrumofanisotropicsampleorfromanaveragespectrumoveralargeareathatcoversallmolecularorientations.

Fromthephaseangledifference,Δ≡ψ2?ψ1,inEq.

REFeq_delta\h

(8)

,

tanΔ=tan?msecθm (

SEQ(\*ARABIC

19

)

IfwedefinenewintermediateparametersfortheobservablesasB≡A2maxA1maxf2of1oandT≡tan2Δ,Eqs.

REFeq_B

(18)

and

REFeq_T

(19)

areconvertedintoaquadraticequationforsin2θm,as

BTsin2θm2?B+1T+1sin2θm+T+1=0 (

SEQ(\*ARABIC

20

)

whichleadstotwosolutions.However,oneoftwoquadraticsolutions(+)yieldssin2θm>1,whichisnon-real.Thus,theremainingsolutionforsin2θmbecomes

sin2θm=B+1T+1?B+12T+12?4BTT+12BT (

SEQ(\*ARABIC

21

)

Itmustbenotedthatthesolutionofsin2θmwillyieldtwodegeneratevalues,θmand(π?θm).Foreachθm,?mwillalsohavetwosolutions,?mand(?m+π),fromEq.

REFeq_T

(19)

.Thisdegeneracyisunavoidabletothislinearlypolarizedopticalmeasurementofaprojectedtransitiondipolemoment,showninEq.

REF_Ref137916793

(1)

.Italsomustbenotedthatthissolutionisfortheidealoriented-gasmodel.Experimentally,thismodelcanbeconsideredwhentheminimumabsorbancesofbothprimaryandsecondarymodepeaksareclosetozero,whichindicatesnegligibleorientationalbroadening.

3.2.OrientationalBroadening

Orientationalbroadeningcanberepresentedbyanorientationaldistributionfunction(ODF).Herein,weassumethattheODFofthemolecularsymmetryaxisisuniaxialtothemeanorientationdirection,whichallowstheODFtobesimplifiedasρβ,whereistheaxialangleinthemolecularcoordinate.

REF_Ref138071610

Fig.4

showsaschematicp