and Angular-Resolved Optical Response of Molecules on Anisotropic Plasmonic Nanostructures

38 Pages·2017·8.1 MB·English

nanomaterials
Review
Polarization- and Angular-Resolved Optical Response
of Molecules on Anisotropic Plasmonic Nanostructures
MartinŠubr*andMarekProcházka* ID
FacultyofMathematicsandPhysics,InstituteofPhysics,CharlesUniversity,12116Prague2,CzechRepublic
* Correspondence:[e-mail](M.S.);[e-mail](M.P.);
Tel.:[phone](M.S.);[phone](M.P.)
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:10May2018;Accepted:7June2018;Published:9June2018
Abstract: Asometimesoverlookeddegreeoffreedominthedesignofmanyspectroscopic(mainly
Raman)experimentsinvolvethechoiceofexperimentalgeometryandpolarizationarrangement
used. Although these aspects usually play a rather minor role, their neglect may result in a
misinterpretationoftheexperimentalresults. Itiswellknownthatpolarization-and/orangular-
resolvedspectroscopicexperimentsallowonetoclassifythesymmetryofthevibrationsinvolvedor
themolecularorientationwithrespecttoasmoothsurface. However,verylowdetectionlimitsin
surface-enhancingspectroscopictechniquesareoftenaccompaniedbyacompleteorpartiallossof
thisdetailedinformation. Inthisreview,wewilltrytoelucidatetheextenttowhichthisapproach
canbegeneralizedformoleculesadsorbedonplasmonicnanostructures. Wewillprovideadetailed
summaryofthestate-of-the-artexperimentalfindingsforarangeofplasmonicplatformsusedinthe
last~15years. Possibleimplicationsonthedesignofplasmon-basedmolecularsensorsformaximum
signalenhancementwillalsobediscussed.
Keywords:SERS;Raman;plasmonics;metallicnanostructures;polarizationandangulardependences;
molecularorientation;ellipsometry
1. Introduction
Methodsofopticalspectroscopy,suchasabsorptionspectroscopy,Ramanscatteringspectroscopy
or fluorescence spectroscopy, can be performed in a wide range of experimental setups relying
on a proper choice of instrumentation, light sources, waveguides, monochromators, detectors etc.
Interplaybetweenallthefactorsmentionedabovedeterminesthefinalspectralpatternobservedin
spectroscopicexperiments.Anobviousdegreeoffreedomintheexperimentaldesignisthepolarization
and(especiallyinthecaseofRamanexperiments)angulararrangementused. Althoughformany
typesofexperimentsthisadditionalinformationisnotrequired,variationinlightpolarizationand/or
thescatteringanglegiverisetoafamilyofphenomenathatboththeoreticiansandexperimentalists
shouldbeawareofwheninterpretingtheirresults. Forexample,confusionoflaserpolarizationmay
leadtoverystrongRamanbandstoappearasratherweakintheRamanspectra,andviceversa. Thus,
understandingtheseeffectshelpstoobtaindeeperinsightintothepropertiesofthesystemstudied.
Inthisreview,themainemphasisisplacedonpolarizationanddirectional/angularcharacteristics
occurring in Raman spectroscopy and surface-enhanced Raman spectroscopy (SERS). However,
itbrieflytouchesonthepolarizationandangulardependencesofotheropticalphenomenasuchas
absorbance,reflectanceorfluorescence. Itiswellknownthatpolarization-and/orangular-resolved
Ramanexperimentsallowonetoclassifythesymmetryofthevibrationsinvolved[1–3]ormolecular
orientation with respect to a smooth surface [4–6]. We will try to extend some of these results to
moleculesadsorbedonplasmonicnanostructures. GeneralizingtheseresultstothecaseofSERSisnot
Nanomaterials2018,8,418;doi:10.3390/nano8060418 www.mdpi.com/journal/nanomaterials
Nanomaterials2018,8,418 2of38
straightforwardduetotheexistenceofonemoreelement—thenanostructure,whichimposesitsown
anisotropicpatterninthepolarizationanddirectionalpropertiesoftheSERSsignal.
Although some reviews focusing on anisotropic metallic nanostructures for SERS and their
fabricationalreadyexistinliterature[7–10],ourreviewcoversthetopicfromaratheruntraditional
point of view, focusing primarily on their polarization and directional properties. The review is
organizedasfollows. Firstly, basicoverviewofpolarizationandangulardependencesinclassical
Raman spectroscopy will be introduced. Different experimental geometries for Raman scattering
measurements will be presented and consequences for the Raman signal coming from randomly
orientedmoleculesaswellascrystalswillbederived. Wewillsummarizebasicresultsofthegroup
theory,allowingclassificationofnormalvibrationalmodesdependingonthesymmetryofthevibration
involved. Theoreticalpredictionswillbebrieflycomparedtotheexperimentalresultsmeasuredin
ourlaboratory. Itisimportanttonotethatwewillfocusonlyonthephenomenawithinthedipole
approximation,leavingasidethevastbodyofexperimentaltechniquessuchasdichroicmethodsor
Ramanopticalactivity[11]. Secondly,wewillconcentrateonthepolarizationeffectsformolecules
adsorbedonsmoothsurfaces. Wewillshowthatinterferencebetweentheincidentandthereflected
radiation must be taken into account in order to correctly describe the observed spectral pattern.
Thecrucialpartwillinvolvethederivationofthesurface-selectionrules,allowingdeterminationof
molecularorientationonsmoothsurfaces. Althoughthisapproachmayseemtogofarbeyondthe
originalscopeofthisreview,wewillshowthataverysimilarprinciplealsoappliesonroughened
metallicsurfacesandcontributestotheSERSenhancement. Finally,wewilltrytogeneralizeprevious
considerationstothecaseofsurface-enhancedRamanscattering. Wewillsummarizetheup-to-date
findingsregardingpolarization-andangular-resolvedSERSexperimentsandsurveythecontributions
tothisfieldfromthelast15years. Theinfluenceofbothdifferentcouplingefficiencyofdifferentlaser
polarizationstoplasmonsaswellassymmetryoftheRamantensoroftheanalyteonpolarizedSERS
willbecriticallydiscussed. Describedphenomenawillbesupportedbyourownexperimentalresults
aswellasbypublishedones.
2. TheOriginofPolarizationCharacteristicsinSpectroscopy
Beforegoingdeeplyintothedefinitionsofthedepolarizationratiosorformulationsoftheselection
rules,wedevoteashortparagraphtothetheoreticaldescriptionofthepolarizationofthelightwaves.
AmonochromaticplanewavetravellingalongthezaxisinagivenCartesiancoordinatesystemcanbe
characterizedbyitselectricfieldvectorinanyinstantrestrictedtothexyplane. Thus,thevariablesin
anequationdescribingaplanewavecanbedecoupledintheform[12]:
→ → →
E(x,y,z,t) = E cos(ωt−kz)e +E cos(ωt−kz+δ)e , (1)
0x x 0y y
whereE andE denoteamplitudesoftheelectricfieldintensitiesinthexandydirection,respectively
0x 0y
(traditionally,thepolarizationdirectionisassumedtobedeterminedbytheelectricfieldvectorinstead
ofthemagneticfieldvector),kdenotesthewavevectorandωtheangularfrequency. Thewaythat
the electric field vector oscillates in space and time defines the light polarization. In the simplest
case,theseoscillationsareconfinedtoasinglespatialdirection,whichistermedlinearpolarization.
Inthiscase,theredoesnotexistanyphaseshiftbetweenthexandyconstituentsoftheelectricfield
vector. Iftheredoesexistaphaseshiftδbetweentheseconstituents(δ (cid:54)= kπ, k ∈ Z),theelectricfield
vectorexperiencesamorecomplexmotioninthexyplane,whichgivesrisetocircularpolarization
(E = E , δ = π +kπ, k ∈ Z)orellipticalpolarization(otherwise). Viceversa,anypolarizationstate
0x 0y 2
canbebuiltupfromtwomutuallyperpendicularlinearlypolarizedcomponents.
Further,itisinstructivetodelvesomewhatintoabrieftheoreticaldescriptionoftheinteraction
between (polarized) light and matter. Absorption is an optical process whereby energy of an
electromagnetic wave (a photon) is absorbed in a medium, accompanied by a transition of the
atom/molecule in a higher energetic state. Depending on the photon energy and quantum states
Nanomaterials2018,8,418 3of38
participatingintheprocess,onecandistinguishtheinfrared(IR)absorption,whichisconnectedwith
a molecular transfer to a higher vibrational level within a given electronic state, and the UV/Vis
absorption,whichinvolvesanelectronexcitationtoahigherelectronicstate. Inthecontextofthis
review, absorption will be mainly understood as the UV/Vis absorption. In order to restore the
thermodynamicequilibrium,amoleculeinanelevatedelectronicstatecanloseitsenergyandreturn
tothegroundstate,whichcanbebasicallyperformedviaoneofthetwoprocesses: (i)Theenergy
excessmaydissipateintheformofheat(non-radiativecrossing);or(ii)byaspontaneousemission
ofradiation,whichistermedluminescence(fluorescence). Ramanscatteringbelongstothegroup
oflightscatteringprocessesincludingannihilationofaprimaryphotonandcreationofasecondary
photonwiththefrequencyshiftedfromtheprimaryphotonbyfrequencyofamolecularvibration.
Thus,Ramanscatteringisatwo-photon,inelasticprocess,whichdoesnotrequiretheparticipating
photonsmatchtheresonancefrequency. Forthisreason,Ramanscatteringisaveryweakprocess,but
canbegreatlyamplifiedformoleculesadsorbedinthevicinityofnanostructuredmetallicsurfaces.
Thistechniqueiscalledsurface-enhancedRamanscattering(SERS)anditprovidesenhancementofthe
Ramanscatteringbyafactorof104orhigherduetoresonanceexcitationofplasmonslocalizedinthe
metallicnanostructures[13]. Duetoitshighsensitivity,SERShasfounduseinnumerousbio-related
applications[14,15],e.g.,forthedetectionofpharmaceuticalsanddrugs[16–18],foodadditives[19,20],
explosives[21,22],diseasebiomarkers[23–25]andmanyothers.
Although“classical”modelsoftheinteractionbetweenlightandmatterappearedalreadyinthe
1900s[12](followedbyasemi-classicaltheoryofRamanscatteringin1930sbyPlaczek[26]),interaction
oflightwithatomsormoleculescannotbefullydescribedunlessquantumtheorycomesintoplay.
Inthelowestdegreeofapproximation,transitionprobability(towhichintensityofagivenspectral
lineisproportional)betweentwoquantumstatesofanatom/moleculeisgovernedbytheFermi’s
goldenrule. Thisrulesaysthatinorderforthetransitiontobeallowed,thematrixelement
(cid:104)2|Wˆ |1(cid:105) (2)
mustbedifferentfromzero. InEquation(2),|1(cid:105)representsthewavefunctionoftheinitialstate,(cid:104)2|the
wavefunctionofthefinalstateandWˆ isaperturbationoperator,whichusuallycorrespondstothe
presenceoftheelectromagneticwave.
In the case of absorption spectroscopy, the perturbation operator can be rewritten using the
→
electricdipolemomentofthemolecule(d)andtheprobabilityofthetransitionthenderivesfromthe
transitiondipolemomentmatrixelement:
(cid:104)2|dˆ|1(cid:105). (3)
In the case of Raman spectroscopy, the situation is more complex because in order to obtain
the non-zero contribution, second-order perturbation theory has to be employed. The transition
probabilitythenderivesfromthesecond-ranktensor[27,28]:
(cid:34) (cid:35)
α12 = ∑ (cid:104)2|dˆp|s(cid:105)(cid:104)s|dˆq|1(cid:105) + (cid:104)2|dˆq|s(cid:105)(cid:104)s|dˆp|1(cid:105) , (4)
pq E −E +(cid:125)ω E −E −(cid:125)ω
s(cid:54)=1,2 1 s i 2 s i
whereE andE denotetheenergiesoftheinitial(|1(cid:105))andfinal(|2(cid:105))quantumstatesofthemolecule,
1 2
→ˆ
d is the electric dipole moment operator and the s-indices refer to all remaining quantum states
(cid:125)
of the molecule. denotes the reduced Planck’s constant and ω is the angular frequency of the
i
incidentradiation.
Equations(3)and(4)demonstratebasicdifferencesbetweenabsorptionandRamanmeasurements.
Whilethemoleculardipolemomentisavectorquantity,thetransitionprobabilityisexpectedtoscale
withtheprojectionofthelightpolarizationinthedirectionofthetransitiondipolemomentofthe
molecule. However,Ramanscatteringisatwo-photonprocess,whichmeansthatboththepolarization
oftheincidentbeamaswellasthatofthescatteredbeamdictatethetotalRamanintensity.
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Inthecaseofabsorptionspectroscopy,polarizationeffectsinanensembleofmanyrandomly
orientedmolecules,suchasliquidsamples,areaveragedout. Thatisbecausethetotalabsorption
intensityisgivenbyspatialaveragingoverpositionsofallabsorbingmolecules. Thiscanbecomputed
explicitlyinazimuthalcoordinates:
x =sinϑcosϕ, y =sinϑsinϕ, z =cosϑ (5)
as:
→ → 2 (cid:82) cos2ϑdΩ
(cid:104)(E·d) (cid:105) = (Ed)2(cid:104)cos2ϑ(cid:105) = (Ed)2 (cid:82) dΩ (6)
where dΩ = sinϑdϑdϕ, ϑ ∈ (0,π) and ϕ ∈ (0, 2π) are standard azimuthal coordinates.
Afterstandardintegration,weobtain:
1
(cid:104)cos2ϑ(cid:105) = , (7)
3
whichimpliesthatformanyrandomlyorientedmolecules,theabsorbancedoesnotdependonthe
directionofpolarization. Sinceanypolarizationstatecanbebuiltupfromtwomutuallyperpendicular
linearlypolarizedcomponents,absorbanceofrandomlyorientedmoleculesneverdependsonthestate
ofpolarizationincludingunpolarized(natural)radiation. Thisisnotthecaseformoleculesexhibiting
somepreferentialorientation(lineardichroism)orphenomenabeyondthedipoleapproximationin
thecaseofchiralmolecules(circulardichroism). Bothoftheseeffectsare,however,beyondthescope
ofthispaper.
InthecaseoftheRamanscattering,intensityofagivenspectrallineisgivenbyacombinationof
→i →sc
themolecularRamantensorandunitvectorsofboththeincidentfield e andthescatteredfield e :
I ∼ (escα12ei)2. (8)
p pq q
→i →sc
Directions e and e are determined by the illumination-observation geometry, including
the scattering angle and polarization of the incident beam (and possibly also polarization of the
collectedbeam). Unlikeinabsorptionspectroscopy,Ramanmeasurementsallowmuchwiderchoice
oftheexperimentalgeometryincludingthepolarizationarrangementusedaswellasthescattering
angle. Equation (8) naturally explains why the molecular quantity describing the disposition to
Raman transition must intrinsically be of tensorial nature and also predicts that detected Raman
intensitiesdependonthepolarizationarrangementusedeveninthecaseofliquids,i.e.,randomly
orientedmolecules.
InbothabsorptionandRamanmeasurements,molecularsymmetryimposescertainrestrictions
onthenumberofvibrationalmodesbeingactiveornon-activeinrespectivespectroscopictechniques.
Accordingtothenumberofsymmetryelementscharacterizingagivenisolatedmolecule,molecules
can be classified into one out of 32 molecular point groups [29]. Having assigned a molecule to
a point group, the methods of group theory allow one to decompose the vibrational signature of
the molecule into so-called irreducible representations. The number of normal modes belonging
to each irreducible representation (out of the 3N−6 normal modes in total) is calculated using a
well-establishedalgorithm,makinguseofcharactertablesofmolecularpointgroups[29,30]. Forthe
purposeofspectroscopy,itisimportanttonotethateachirreduciblerepresentationmaytransform
as a particular linear or quadratic function of coordinates. Since the constituents of the electric
dipole moment have the same transformation properties as x, y and z and elements of a Raman
tensoras x2, y2, z2, xy, xzandyz,itisrelativelyeasytopredictwhichvibrationswillbeactivein
Raman/IRabsorptionspectra: InorderforagivenvibrationalmodetobeallowedinIRabsorption
spectra, its irreducible representation must span at least one of x, y and z species (in the case of
randomly-oriented molecules). For the transition to be allowed in Raman spectra, its irreducible
Nanomaterials2018,8,418 5of38
representationmustspanatleastoneofthequadratic(x2, y2, z2, xy, xzandyz)species(againinthe
caseofrandomly-orientedmolecules).
3. PolNaarniozmaatteiroianls 2a0n18d, 8A, xn FgOuR lPaErERD ReEpVeIEnWd e ncesinRamanSpectroscopy 5 of 37
3. Polarization and Angular Dependences in Raman Spectroscopy
3.1. DerivationofRamanIntensitiesasaFunctionofthePolarizationandAngularArrangement
L3e.1t. uDserciovnatsiiodne orf aRagmeanne rIanlteenxspitieersi mas ean Ftuanlcgteioonm ofe tthrey PfoolrarRizaamtioann asncda Attnegriunlagr mAreraasnugermemenetn ts,depictedin
Figure1. LetussupposethataCartesiancoordinatesystemischosensothatthesampleisplacedinits
Let us consider a general experimental geometry for Raman scattering measurements, depicted
originin,t Fhieguerxec i1t.a Ltieotn usb esuamppotrsaev tehlast aal oCnagrtetshieanx cdoiorredcitnioatne wsyistthempo ilsa crhizoasteino nsoa tlhoantg thteh esazmdpilree cist ipolnacaendd the
detecitno ritsis oorirgiienn, ttehde etxoccitoaltlioenct bleigamht tsracvaettlse raelodnign ththee 𝑥 xdyirpecltainone watitahn paonlagrilzeaϑtiown iatlhonregs tphee c𝑧t dtoiretchteioxn axis.
Theaanngdl ethϑe idsetteercmtore dis tohreienscteadtt teor icnoglleacnt glilgehat nscdatitterceadn inta tkhee a𝑥n𝑦y plvaanleu aet farno mang0l◦e t𝜗o w18it0h◦ r.eTspheectx tyo pthlae neis
terme𝑥d atxhise. sTchaet taenrginleg 𝜗p ilsa nteer.mIendt htheef soclalottweriinngg ,awngelew ainlldr eitf ecranto tathkee azncyo vmapluoen fernomts o0°f ttoh e18e0le°.c tTrhice fi𝑥𝑦e ldas
tothepnlaonrem isa tleormrveder tthicea slc(awtteitrhinrge pslpaencet. Itno tthhee fsocllaotwteirnign,g wpel wanilel ,redfeenr otot ethdea 𝑧s cvomorp⊥on)eanntsd otfh theec eolmecptroicn ents
field as to the normal or vertical (with respect to the scattering plane, denoted as 𝑣 or ⏊) and the
lyinginthescatteringplaneastothehorizontal(denotedashor||). Ingeneral,thescatteredlight
components lying in the scattering plane as to the horizontal (denoted as ℎ or ||). In general, the
containsbothverticalandhorizontalcomponents—thatiswhyananalyserisofteninsertedbetween
scattered light contains both vertical and horizontal components—that is why an analyser is often
the monochromator and the sample to distinguish these two polarizations. In order to overcome
inserted between the monochromator and the sample to distinguish these two polarizations. In order
differentgratingresponsesfordifferentlightpolarizations,ascramblermustbeinsertedbetweenthe
to overcome different grating responses for different light polarizations, a scrambler must be inserted
analybseetrwaenedn tthhee amnaolnyosecrh arnodm tahteo mr(onnootchshroomwantoirn (nFoigt ushroew1nf oinr Fsiimgupreli c1i ftoyr) .simplicity).

Figure 1. Scheme of a general Raman scattering geometry. The excitation beam travels along the 𝑥
Figure1. SchemeofageneralRamanscatteringgeometry. Theexcitationbeamtravelsalongthex
direction with polarization in the 𝑧 direction and the detector collects light scattered in the 𝑥𝑦 plane
directionwithpolarizationinthezdirectionandthedetectorcollectslightscatteredinthexyplane
at an angle 𝜗 with respect to the 𝑥 axis. An analyser may be inserted between the sample and the
atanangleϑwithrespecttothe xaxis. Ananalysermaybeinsertedbetweenthesampleandthe
monochromator to allow only one specific Raman polarization enter the detector.
monochromatortoallowonlyonespecificRamanpolarizationenterthedetector.
With the fixed angle 𝜗, 4 different polarization combinations can be thought of to retrieve Raman
Wspiethctrtah:e Tfihxeered aarneg tlweoϑ ,b4asdiicf fpeoressnitbipliotileasr iozfa tsieotntincgo mthbei npaotliaornizsatciaonn boef tthheo uingchitdeonftt oberaemtr ie(eviethRera man
perpendicular to the scattering plane as depicted in Figure 1, i.e., in the 𝑧 direction, or lying in the
spectra: There are two basic possibilities of setting the polarization of the incident beam (either
scattering plane, i.e., in the 𝑦 direction) as well as two basic possibilities of rotating the analyser to
perpendiculartothescatteringplaneasdepictedinFigure1,i.e.,inthezdirection,orlyinginthe
pick out either horizontal or vertical polarization. Thus, we adopt the symbols:
scatteringplane,i.e.,intheydirection)aswellastwobasicpossibilitiesofrotatingtheanalysertopick
outeitherhori𝐼z(o𝜗n,⊥ta𝑠l𝑐,o⊥r𝑖)v=ert𝐼𝑣ic𝑣a, lpo𝐼l(a𝜗r,iz𝑠a𝑐t,i⊥o𝑖n).=T𝐼h𝑣ℎu,s,we𝐼(a𝜗d,⊥o𝑠p𝑐t,t𝑖h)e=s𝐼yℎm𝑣,bols𝐼:(𝜗,𝑠𝑐,𝑖)=𝐼ℎℎ (9)
to accIo(uϑn,t⊥ fsocr, ⊥eaic)h =of Ithe,se pIo(laϑr,i(cid:107)zasct,io⊥ni )ar=ranIge,menIts(.ϑ T,h⊥es cfi,r(cid:107)sti )sy=mIbol, in eIa(cϑh, b(cid:107)rsacc,k(cid:107)ei)t d=enIotes the (9)
vv vh hv hh
scattering angle, the second symbol refers to polarization of the scattered beam (with respect to the
scattering plane) and the third one refers to polarization of the incident beam. Sometimes the
toaccountforeachofthesepolarizationarrangements. Thefirstsymbolineachbracketdenotesthe
scattering angle is unimportant and as a shorthand we introduce the notation 𝐼 , 𝐼 , 𝐼 and 𝐼 with
scatteringangle,thesecondsymbolreferstopolarizationofthescatteredb𝑣e𝑣am𝑣ℎ(wℎ𝑣ithresℎpℎecttothe
the first subscript standing for polarization of the incident beam and the latter standing for
scatteringplane)andthethirdonereferstopolarizationoftheincidentbeam.Sometimesthescattering
polarization of the scattered beam. Our task is to derive expressions for intensities of a given Raman
angleisunimportantandasashorthandweintroducethenotation I , I , I and I withthefirst
line, measured under respective arrangements. It is important to note thvavt fovhr 𝜗h=v0° or 𝜗hh=180° the
subscsrciaptttesrtianng dpilnange fios rnopto wlaerlli-zdaetfiionnedo. fNtehveeritnhceildesesn, wtbe ewaimll sahnowd tthhaet ltahtet egrensetraanl dcoinngsidfoerraptioonlasr aibzoavtieo nof
thescwatitlle breedcobmeea tmri.viOalu irn ttahseksei sspteocdiael rciavseese. x pressionsforintensitiesofagivenRamanline,measured
Nanomaterials2018,8,418 6of38
underrespectivearrangements. Itisimportanttonotethatforϑ =0◦ orϑ =180◦ thescatteringplane
isnotwell-defined. Nevertheless,wewillshowthatthegeneralconsiderationsabovewillbecome
trivialinthesespecialcases.
Equation (8) says that each of the 4 possible polarization arrangements mentioned above
picks out different elements of the Raman tensor of the molecules. In other words, specific
illumination-observation geometry determines which elements of the Raman tensor will appear
inthefinalexpressionsdictatingtheintensityofagivenRamanline. TheCartesiancomponentsofthe
electricfieldvectorofthescatteredbeamare:
Esc = E(ϑ,(cid:107)sc)sinϑ, Esc = E(ϑ,(cid:107)sc)cosϑ, Esc = E(ϑ,⊥sc), (10)
x y z
whereE(ϑ,(cid:107)sc)andE(ϑ,⊥sc)denotecomponentsoftheamplitudeofthescatteredbeamlyinginthe
scatteringplaneandinthedirectionperpendiculartothescatteringplane,respectively.Bycombination
ofEquations(8)–(10),thefollowingexplicitformulascanbeobtained(forsimplicity,wefurtheromit
theupperindices12indenominationofthemolecularRamantensorα):
I(ϑ,⊥sc,⊥i) ∼ α2 , I(ϑ,(cid:107)sc,⊥i) ∼ (α sinϑ+α cosϑ)2, (11)
zz xz yz
I(ϑ,⊥sc,(cid:107)i) ∼ α2 , I(ϑ,(cid:107)sc,(cid:107)i) ∼ (α sinϑ+α cosϑ)2. (12)
zy xy yy
In the case of an ensemble of randomly oriented molecules, respective squares or Raman
tensorelementsinEquations(11)and(12)aretobereplacedwiththeirspatially-averagedvaluesas
contributionsfromeachindividualmoleculesumupincoherently. Thiscanbeperformedinamanner
analogoustoEquation(6),however,molecularpolarizabilityresponsiblefortheRamanscatteringis
atensorratherthanavectorquantity,whichmakesthedeterminationofspatially-averagedtensor
elements a more tedious procedure. Any spatial rotation of the molecule with respect to a given
referenceframecanbeexpressedunambiguouslybyarotationalmatrixassociatedwith3Eulerangles,
whoseelementsmustbeaveragedinamanneranalogoustoEquation(6)[31]. Analternativeand
less time-consuming approach involves computation of isotropic averages of respective products
ofdirectioncosinesbetweentwoCartesianaxissystems(againonly3outof9directioncosinesare
independent)[27,32]. Foranensembleofmanyrandomlyorientedmolecules,intensityofagiven
RamanlineinagivenexperimentalgeometrymaybeexpressedasalinearcombinationofRaman
tensorinvariants. Theseareusuallyreferredtoas:
Thesquareofthemeanpolarizabilitya:
(α +α +α )2
a2 = xx yy zz , (13)
9
theanisotropyγ:
(α −α )2+(α −α )2+(α −α )2 (α +α )2+(α +α )2+(α +α )2
γ2 = xx yy yy zz zz xx +3 xy yx yz zy zx xz , (14)
2 4
andtheantisymmetricanisotropyδ:
(α −α )2+(α −α )2+(α −α )2
δ2 =3 xy yx yz zy zx xz . (15)
4
Fornon-resonanceRamanscatteringonlytheisotropicinvariantaandtheanisotropicinvariant
γarerelevantsinceinthiscasetheRamantensorissymmetricandtheantisymmetricinvariantδis
identicallyzero. Theresultoftheaveragingprocedureis[27,32]:
45a2+4γ2 3γ2+5δ2
(cid:104)α2 (cid:105) = (cid:104)α2 (cid:105) = (cid:104)α2 (cid:105) = , (cid:104)α2 (cid:105) = (cid:104)α2 (cid:105) = (cid:104)α2 (cid:105) = , (16)
xx yy zz 45 xy xz yz 45
(cid:104)α α (cid:105) = (cid:104)α α (cid:105) =0. (17)
xz yz xy yy
Nanomaterials2018,8,418 7of38
This result shows the intuitive notion that the average values of all diagonal elements of the
Ramantensorareequalandalsoalloff-diagonalelementsareequal,irrespectiveofaspecificchoiceof
thesystemofcoordinates. AlsoallRamantensorproductswhichinvolveonecommonsubscriptare
zero. PuttingthesevaluesinEquations(11)and(12)yields:
45a2+4γ2 3γ2+5δ2
I(ϑ,⊥sc,⊥i) ∼ , I(ϑ,(cid:107)sc,⊥i) ∼ , (18)
45 45
3γ2+5δ2 45a2+4γ2 3γ2+5δ2
I(ϑ,⊥sc,(cid:107)i) ∼ , (ϑ,(cid:107)sc,(cid:107)i) ∼ cos2ϑ+ sin2ϑ. (19)
45 45 45
Equations (18) and (19) reveal that actually the only polarization arrangement in which the
intensityofagivenlinewilldependonthescatteringangleϑisthearrangementinwhichboththe
incidentbeamaswellasthescatteredbeamarepolarizedinthescatteringplane.
Inordertoavoidmisunderstanding,itisimportanttorealizethatanycombinationofRaman
tensorinvariants(13)–(15)isagainaninvariant. Inliterature,manyalternativedefinitionsofRaman
tensorinvariantsmaybefound[27]. Forexample,RamantensorinvariantsusedbyPlaczek[26]are:
2 2
G0 =3a2, Ga = δ2, Gs = γ2. (20)
3 3
A quantity which can be very easily measured experimentally is the Raman depolarization
ratio,whichisdefinedastheratioofRamanintensitiesmeasuredundertwoselectedconfigurations.
Mostcommonly,thedepolarizationratioρisassumedastheratiobetweentheintensityoftheRaman
fieldpolarizedorthogonaltotheincidentfieldandtheintensityoftheRamanfieldpolarizedparallel
tothelaserfield[27,28]. Themostcommonwayofretrievingthedepolarizationratioisfixingthe
incidentpolarizationandrotatingtheanalyser. Inournotation,thisyieldsto:
I(ϑ,(cid:107)sc,⊥i) I
ρ(ϑ,⊥i) = = vh. (21)
I(ϑ,⊥sc,⊥i) Ivv
Although many textbooks resort only to the definition of the depolarization ratio given by
Equation(21)[28,30],thisformulacanbeeasilygeneralizedandmanyalternativedefinitionsofthe
depolarizationratiocanbefoundintheliterature[27],suchas:
I(ϑ,⊥sc,(cid:107)i) I I(ϑ,⊥sc,(cid:107)i) I
ρ(ϑ,(cid:107)i) = = hv, ρ(ϑ,⊥sc) = = hv, (22)
I(ϑ,(cid:107)sc,(cid:107)i) Ihh I(ϑ,⊥sc,⊥i) Ivv
I(ϑ,⊥sc,⊥i) I
ρ(ϑ,⊥sc,⊥i,(cid:107)sc,(cid:107)i) = = vv, (23)
I(ϑ,(cid:107)sc,(cid:107)i) I
hh
etc. ThedepolarizationratioprovidesuniqueinformationontheRamanpolarizabilitytensorofa
givenmolecularvibrationandisthereforeaveryusefultool. Indeed,inthecaseofrandomlyoriented
molecules,wehave:
I 3γ2+5δ2
ρ(ϑ,⊥i) = vh = . (24)
Ivv 45a2+4γ2
SimilarformulascanbederivedforthedepolarizationratiosgivenbyEquations(22)and(23).
Itisobviousthattheintroductionofthedepolarizationratiosisveryconvenientbecauseitallowsto
excludetheinsignificantmultiplicativeconstantrelatingRamanintensitiestospecificcombinations
ofsquaresofRamantensorelements. ThedepolarizationratiodefinedbyEquation(24)isusually
termedthemoleculardepolarizationratiosinceitisanintrinsicpropertyoftheRamanprobe(contrary
tothedepolarizationratiomeasuredinpresenceofsmooth/nanostructuredmetallicsurfaceswhere
themutualinteractionbetweenlight/analyte/metalsubstantiallyaffectthemeasureddepolarization
ratio). FromEquation(24)itimmediatelyfollowsthatfornon-resonanceRamanscattering(δ = 0)
the molecular depolarization ratio is always bound between 0 and 3/4. A specific value of ρ will
Nanomaterials2018,8,418 8of38
dependonthesymmetryofthevibrationinvolved. Fortotallysymmetricvibrationsγ2 =0andthe
depolarizationratiotendstozero.Conventionally,suchaRamanlineissaidtobecompletelypolarized.
Thisaspectcanbeexplainedstraightforwardlyfromaclassicalpointofview: Fortotallysymmetric
vibrations,thepolarizabilitytensorwillbecomeascalarandthedirectionoftheoscillatingdipole
willmatchthedirectionoftheincidentfieldpolarization. SincetotalRamanintensityisproportional
tothescalarproductoftheoscillatingdipolemomentandthedetectedfield,onewouldexpectthat
whensettingtheanalyserparalleltothedirectionoftheoscillatingdipole(theincidentpolarization),
theintensitywillreachitsmaximum,whileafterrotatingtheanalyserabout90◦theintensitywilldrop
toitsminimum. Ontheotherhand,fornon-totallysymmetricvibrationsa2 =0andthedepolarization
Nanomaterials 2018, 8, x FOR PEER REVIEW 8 of 37
ratiois3/4[27,28]. Conventionally,suchaRamanlineissaidtobedepolarized. When0< ρ <3/4,
90° the intensity will drop to its minimum. On the other hand, for non-totally symmetric vibrations
theRamanlineissaidtobe(partially)polarized[27].
𝑎2=0 and the depolarization ratio is 3/4 [27,28]. Conventionally, such a Raman line is said to be
Previous considerations are valid for any angle ϑ apart from ϑ=0◦ (forward scattering) and
depolarized. When 0<𝜌< 3/4, the Raman line is said to be (partially) polarized [27].
ϑ=180◦ (backscattering). In this instance the scattering plane is not well-defined and the symbols
Previous considerations are valid for any angle 𝜗 apart from 𝜗=0° (forward scattering) and 𝜗=
⊥sc,⊥1i8,0(cid:107)°sc (banacdks(cid:107)ciaatstewrinelgl)a. sInIv vt,hIisv hi,nIshtvaannced tIhheh lsocsaetttehreinirgs epnlasnee. Nise vneortt hweelells-sd,etfhineesdit uaantdio tnheis stryimvibaollisn this
case. L⊥e𝑠𝑐t,u⊥s𝑖, re𝑠s𝑐k aentdch 𝑖t ahse wexelpl earsi m𝐼 e, n𝐼ta,l 𝐼ge oamnde t𝐼ry lionseF itghueirre s2e,nased. oNpetvinegrththeleesdsi,r tehcet isointuoatfiothne isz tarixviisalp ianr allel
𝑣𝑣 𝑣ℎ ℎ𝑣 ℎℎ
tothethwisa vcaesvee. cLteotr uosf rtehsekeintcchid tehne te(xspceartitmereendt)all iggehotm.Aetfrtye rini lFluigmuirne a2t,i nadgotphteinsga mthpe ldeirwecitthionli gohf tthwe i𝑧th axaisg iven
polaripzaartaiollnel, tcoo nthfein wedavteovtehcteorx yofp tlhaen ein,ctihdeenatn (aslcyastteerremda) yligbhet.s eAtftteor ciolllulemcitnRataimnga tnhefi esaldmppolel awriizthe dligphatr allel
orortwhoitgho an gailvteon tphoelalariszeartifoinel, dc.onAfilnteerdn taot itvheel y𝑥,𝑦R palmanaen, thmee aansaulryesmer emnatsy mbea ysebt etop ceorlfloecrtm Readmwani tfhiealdf ixed
polarized parallel or orthogonal to the laser field. Alternatively, Raman measurements may be
positionoftheanalyserandvaryingpolarizationoftheincidentbeam.Respectivearrangementslead
performed with a fixed position of the analyser and varying polarization of the incident beam.
toprobingoftheaveragesofthesquaresofthediagonal/off-diagonalRamantensorelements.Inthis
Respective arrangements lead to probing of the averages of the squares of the diagonal/off-diagonal
case,thezcomponentsoftheRamantensordonotfigureintheequations. Formally,the4respective
Raman tensor elements. In this case, the 𝑧 components of the Raman tensor do not figure in the
polarization combinations discussed above split into two groups since I(0◦,⊥sc,⊥i) = I(0◦,(cid:107)sc,(cid:107)i)
equations. Formally, the 4 respective polarization combinations discussed above split into two
and Ig(0ro◦u,(cid:107)pssc s,i⊥nci)e 𝐼=(0I°,(⊥0◦𝑠𝑐,,⊥⊥𝑖s)c,=(cid:107)i𝐼)(0in°,th𝑠𝑐i,s𝑖c)a asned, t𝐼h(e0°s,a𝑠m𝑐,e⊥𝑖h)o=ld𝐼i(n0g°,⊥al𝑠s𝑐o,𝑖f)o irnϑ th=is c1a8s0e◦, .thTeh seamdee phoolladrinizga tion
ratiofaolsroϑ fo=r 𝜗0=◦ o1r80ϑ°.= Th1e8 d0e◦pcoalnaribzeateioxnp rraetsisoe fdors 𝜗im=p0ly° oars 𝜗th=e1r8a0ti°o caonf binet eexnpsrietsiessedo sbitmaipnlye dasw thieth ractrioo ssed
polariozfa itniotennssittoieisn otebntasiinteieds wobittha icnroesdsewdi ptholparairzaaltlieolnps otola irnizteantsioitniess. oFbotraminaeldly w:ith parallel polarizations.
Formally:
I(0◦ or180◦,(cid:107)sc,(cid:107)i)
ρ = 𝐼I((00°◦ oorr 118800°,◦,𝑠(cid:107)𝑐,s⊥c,𝑖)(cid:107)i), (25)
𝜌= , (25)
𝐼(0° or 180°,𝑠𝑐,𝑖)
whichtakestheformgivenbyEquation(24)inthecaseofrandomlyorientedmolecules.
which takes the form given by Equation (24) in the case of randomly oriented molecules.

FigurFei2g.uSrec h2e. Smcheeomfae ofof raw foarrwd-asrcda-tstceartitnergingge ogmeoemtreyt.ryL. iLnienaeralrylyp poolalarrizizeedd lliigghhtt iiss ffaalllliinngg oonn tthhee pplalanne eoof fthe
the sample (𝑥𝑦) from above. Analyser can be set to retrieve either a perpendicular component of the
sample(xy)fromabove.Analysercanbesettoretrieveeitheraperpendicularcomponentofthelaser
field(lIa⊥se)ro friealdp a(𝐼r⊥a)l leolr oan pea(rIa(cid:107)l)l.elA onnael o(g𝐼o).u sAsniatluoagtoiouns soictucautrisonin otchceurcsa sine othfea cbaascek osfc aat tbearcinkgscagteteorminegt ry.
geometry.
Further, let us inspect another widely-used geometry in which 𝜗=90° (right-angle geometry).
After inserting 𝜗=90° in Equation (19), we obtain:
𝐼(90°,𝑠𝑐,⊥𝑖)= 𝐼(90°,⊥𝑠𝑐,𝑖)=𝐼(90°,𝑠𝑐,𝑖)≠𝐼(90°,⊥𝑠𝑐,⊥𝑖), (26)
𝐼 =𝐼 =𝐼 ≠𝐼 . (27)
𝑣ℎ ℎ𝑣 ℎℎ 𝑣𝑣
It means that for retrieving information on the Raman depolarization ratio (and consequently
on the symmetry of the vibration involved), 𝐼 and any of 𝐼 , 𝐼 and 𝐼 are required.
𝑣𝑣 𝑣ℎ ℎ𝑣 ℎℎ
Nanomaterials2018,8,418 9of38
Further,letusinspectanotherwidely-usedgeometryinwhichϑ =90◦ (right-anglegeometry).
Afterinsertingϑ =90◦ inEquation(19),weobtain:
I(90◦,(cid:107)sc,⊥i) = I(90◦,⊥sc,(cid:107)i) = I(90◦,(cid:107)sc,(cid:107)i) (cid:54)= I(90◦,⊥sc,⊥i), (26)
I = I = I (cid:54)= I . (27)
vh hv hh vv
ItmeansthatforretrievinginformationontheRamandepolarizationratio(andconsequentlyon
thesymmetryofthevibrationinvolved), I andanyof I , I and I arerequired.
vv vh hv hh
Nanomaterials 2018, 8, x FOR PEER REVIEW 9 of 37
3.2. SimpleIllustrationoftheDepolarizationRatioMeasurement
3.2. Simple Illustration of the Depolarization Ratio Measurement
Inthisparagraph,wedemonstratepreviousconsiderationsonasimpleexampleofliquidCCl
In this paragraph, we demonstrate previous considerations on a simple example of liquid CCl44
measuredinourlaboratoryintheright-anglegeometry(excitationwavelength532nm). Thedefault
measured in our laboratory in the right-angle geometry (excitation wavelength 532 nm). The default
polarizationoftheincidentbeaminthisexperimentalsetupisperpendiculartothescatteringplane,
polarization of the incident beam in this experimental setup is perpendicular to the scattering plane,
butmayberotatedby90◦ usingahalf-waveplateoraFresnelrhombretarder. TheCCl isahighly
but may be rotated by 90° using a half-wave plate or a Fresnel rhomb retarder. The CCl44 is a highly
symmetricmoleculebelongingtotheT pointgroup. PolarizedRamanspectraobtainedinallfour
symmetric molecule belonging to the 𝑇d point group. Polarized Raman spectra obtained in all four
𝑑
polarizationarrangementsaredisplayedinFigure3.
polarization arrangements are displayed in Figure 3.

FFiigguurree 33.. PPoollaarriizzeedd RRaammaann ssppeecctrtara oof fCCCCl4l. T.hTeh eexceixtacittiaotni ownawvealveenlgetnhg wthaws 5a3s25 n3m2 ,n lmas,elra speorwpeorw 10er0
4
mW and the accumulation time 1 min. No further corrections have been made.
100mWandtheaccumulationtime1min.Nofurthercorrectionshavebeenmade.
Analysis of respective depolarization ratios suggests that the band centered around 465 cm−1 can
Analysisofrespectivedepolarizationratiossuggeststhatthebandcenteredaround465cm−1can
be classified as totally symmetric (the “disrupted” structure of the 465-cm−1 band is owed to different
beclassifiedastotallysymmetric(the“disrupted”structureofthe465-cm−1bandisowedtodifferentCl
Cl isotopes which lift the degeneracy of this peak) while the bands around 222 and 319 cm−1 as non-
isotopeswhichliftthedegeneracyofthispeak)whilethebandsaround222and319cm−1asnon-totally
styomtalmlye tsryimcmmoedtreisc. mInoddeeesd. ,Ignrdoeuepd,t hgerooruypp trheedoircyts ptrheedΓic3tNs− th6ere p3rNe−se6 nrteaptiroenseonftaCtCioln toof bCeCAl4 t⊕o bEe⊕ A12F⊕
E⊕2F (E being doubly degenerate and F being triply degenerate, making 3∙54−6=9 1vibrationa2l
(E being2 doubly degenerate and F being triply degenerate, making 3·5−6 = 9 vibrational modes
modes in total).
intotal).
Basically, those values obtained from the experiment confirm that 𝐼 , 𝐼 and 𝐼 configurations
𝑣ℎ ℎ𝑣 ℎℎ
Basically,thosevaluesobtainedfromtheexperimentconfirmthat I , I and I configurations
are equivalent. However, closer analysis (inset of Figure 3) reveals slivghht hdvifferenhches in respective
are equivalent. However, closer analysis (inset of Figure 3) reveals slight differences in respective
polarization arrangements, which is obviously attributable to transfer optics imperfections. The
polarization arrangements, which is obviously attributable to transfer optics imperfections.
depolarization ratios obtained from measurements with varying incident polarization are larger than
Thedepolarizationratiosobtainedfrommeasurementswithvaryingincidentpolarizationarelarger
from the measurements where the incident polarization was kept constant, which shows some subtle
thanfromthemeasurementswheretheincidentpolarizationwaskeptconstant,whichshowssome
imperfection of the half-wave plate, transmitting a small fraction of orthogonal polarization as well
(making the relative error particularly large for highly polarized lines where 𝜌≈0). Thus,
polarization-dependent Raman spectra of CCl4 serve well for checking the correct function of the
scrambler and other optical components used for more complex polarization-dependent
measurements.
Last but not least, the literature is not consistent about whether peak heights (above spectral
background) or rather integrated areas should actually enter Equations (21)(23). In many instances
(such as in the case of CCl4) these two ways produce almost identical results as the line width is not
expected to change with polarization. On the other hand, in the case of several overlapping bands,
determination of the spectral background and delimiting the peak area may be difficult and the two
methods may give somewhat different results. In these cases, fitting the spectrum with the sum of
the Lorentz curves would be advisable.
Nanomaterials2018,8,418 10of38
subtleimperfectionofthehalf-waveplate, transmittingasmallfractionoforthogonalpolarization
as well (making the relative error particularly large for highly polarized lines where ρ ≈ 0). Thus,
polarization-dependent Raman spectra of CCl serve well for checking the correct function of the
4
scramblerandotheropticalcomponentsusedformorecomplexpolarization-dependentmeasurements.
Last but not least, the literature is not consistent about whether peak heights (above spectral
background)orratherintegratedareasshouldactuallyenterEquations(21)–(23). Inmanyinstances
(such as in the case of CCl ) these two ways produce almost identical results as the line width is
4
not expected to change with polarization. On the other hand, in the case of several overlapping
bands,determinationofthespectralbackgroundanddelimitingthepeakareamaybedifficultandthe
twomethodsmaygivesomewhatdifferentresults. Inthesecases,fittingthespectrumwiththesumof
theLorentzcurveswouldbeadvisable.
3.3. OtherPossiblePolarizationArrangementsUsed
Previoussurveyofthemostwidely-usedgeometriesforthedepolarizationratiomeasurements
is still not exhaustive. All other possible geometries can be resolved exploiting the fact that any
polarizationstatecanbedecomposedintoasuperpositionoftwoindependent, linearlypolarized
waves with their electric field vectors perpendicular to each other. Other possible experimental
geometries (maybe even the most frequent) involve the situations when no analyser is used, i.e.,
the excitation field is either vertical or horizontal, but polarization of the detected field is not
characterized (or a far less common case when the excitation field is unpolarized and respective
scatteredfieldcomponentsaredetected). Forthesakeofcompleteness,weadoptthesymbols:
I(ϑ,nsc,⊥i)= I(ϑ,⊥sc,⊥i)+I(ϑ,(cid:107)sc,⊥i), (28)
I(ϑ,nsc,(cid:107)i)= I(ϑ,⊥sc,(cid:107)i)+I(ϑ,(cid:107)sc,(cid:107)i), (29)
andthelesscommonlyusedsymbols:
I(ϑ,⊥sc,ni)= I(ϑ,⊥sc,⊥i)+I(ϑ,⊥sc,(cid:107)i), (30)
I(ϑ,(cid:107)sc,ni)= I(ϑ,(cid:107)sc,⊥i)+I(ϑ,(cid:107)sc,(cid:107)i). (31)
Equations(28)and(29)showthatinvestigatingatotalRamanintensitywithoutananalyseris
equivalent to performing two experiments in a row: setting the analyser to collect the horizontal
polarization,thenrotatingtheanalyserby90◦tocollecttheverticalpolarizationandsummingupboth
results. AnanalogoussituationistrueforEquations(30)and(31). Moreover,respectiveleft-handsides
ofEquations(28)–(31)canbecombinedandyieldfurtheralternativedepolarizationratiodefinitions.
Surprisingly, information on the orientation-averaged squares of the Raman tensor elements (and
consequentlyinformationonthesymmetryofmolecularvibrationsinvolved)canbeextractedeven
withoutananalyser—allthatoneneedstodoistorecordtheRamanspectrawiththeincidentfield
polarizationperpendiculartothescatteringplaneandparalleltothescatteringplane,respectively.
Let us, for instance, determine the expected depolarization ratio in the right-angle geometry with
varyingincidentpolarizationandnoanalyserused. BycombiningEquations(18),(19),(28)and(29),
weobtain:
I(90◦,nsc,(cid:107)i) 6γ2+10δ2
ρ(90◦,nsc) = = . (32)
I(90◦,nsc,⊥i) 45a2+7γ2+5δ2
Thedepolarizationratiofornon-resonanceRamanscattering(δ2 =0),definedinthemannerof
Equation(32),isalwaysboundbetween0and6/7.