Near-field to far-field characterization of speckle patterns generated by disordered nanomaterials

3.8 MB·

by  Valentina Parigi

Near-field to far-field characterization of
speckle patterns generated by disordered
nanomaterials
ValentinaParigi1,ElodiePerros1,GuillaumeBinard2,3,Ce´line
Bourdillon2,3,Agne`sMaˆıtre2,3,Re´miCarminati1,Valentina
Krachmalnicoff1,∗andYannickDeWilde1,∗
6 1ESPCIParisTech,PSLResearchUniversity,CNRS,InstitutLangevin,1rueJussieu,
1 F-75005,Paris,France
0 2SorbonneUniversite´s,UPMCUnivParis06,UMR7588,INSP,F-75005,Paris,France
2 3CNRS,UMR7588,INSP,F-75005,Paris,France
n ∗[email protected]
a ∗[email protected]
J

7

Abstract:
]
s Westudytheintensityspatialcorrelationfunctionofopticalspecklepatterns
c
above a disordered dielectric medium in the multiple scattering regime.
i
t The intensity distributions are recorded by scanning near-field optical
p
o microscopy (SNOM) with sub-wavelength spatial resolution at variable
. distances from the surface in a range which spans continuously from the
s
c near-field (distance (cid:28) λ) to the far-field regime (distance (cid:29) λ). The
i non-universal behavior at sub-wavelength distances reveals the connection
s
y between the near-field speckle pattern and the internal structure of the
h medium.
p
[ © 2016 OpticalSocietyofAmerica

1 OCIScodes:(180.4243)Near-fieldmicroscopy;030.6140Speckle;290.4210Multiplescatte-
v ring;350.4238Nanophotonicsandphotoniccrystals?
1
8
4 Referencesandlinks
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Vol.18,Issue19,pp.20105-20113(2010).
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v 5. R. Carminati, “Subwavelength spatial correlations in near-field speckle patterns,” Phys. Rev. A 81, 053804
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a 7. A.ApostolandA.Dogariu,“SpatialCorrelationsintheNearFieldofRandomMedia,”Phys.Rev.Lett.91,
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1. Introduction
Scanning near-field optical microscopy (SNOM) allows to recover the high spatial frequency
components of an electromagnetic field which are normally confined as evanescent waves at
sub-wavelength (sub-λ) distances from a surface in the so-called near-field region [1]. The
spatialresolutionofthetechniqueisdeterminedbythenano-scaletiporaperturewhichisused
tointeractwiththenearfieldatnanometerseparationfromthesample.Sinceitsfirstrealization
[2],SNOMhasbeenmainlyusedtostudysamples,whichconsistofsub-wavelengthstructures
placedatthesurfaceofhomogeneoussubstrates.
Thestudyoffieldcorrelationsinspecklepatternsproducedbythree-dimensionaldisordered
materialshasrecentlyevolvedtowardsthequestfornon-universalproperties,whichcouldbe
connectedwiththemorphologyofthemediumatsub-λ scale[3].Ithasbeenrecentlyshown
that the degree of mutual polarization for different points in speckle patterns depends on the
scatteringsample[4],whilespatialfieldcorrelationscanrevealtherelevantlengthscalesofthe
mediummorphology[5]orthedistancebetweenpointsourcesembeddedinthesample[6].In
particular the speckle pattern in the near field of a scattering medium shows sub-λ grain size
whichcanbecorrelatedwiththesub-λ structureofthemedium.
The connection between the field spatial correlation function in near-field speckle patterns
and the structure of the medium has been analyzed in Ref. [5] using a simple model. In this
theoretical approach, the disordered medium is described by a fluctuating dielectric function
ε(r)=1+δε(r)consideredasarandomvariable.Thedielectricfunctionsatisfies(cid:104)δε(r)(cid:105)=0
and(cid:104)δε(r)δε(r(cid:48))(cid:105)=C(|r−r(cid:48)|),withC(R)acorrelationfunctionwithawidth(cid:96) thatcharac-
ε
terizestheinternalstructureofthemedium(herethebracketsdenoteanensembleaverageover
the configurations of the random medium). In the regime (cid:96) (cid:28)λ (cid:28)(cid:96), with (cid:96) the scattering
ε
meanfreepath,thefieldcorrelationfunctionabovetheexitsurfaceofathree-dimensionalscat-
teringmediumilluminatedbyaplanewavecanbecalculatedanalytically.Moreprecisely,the
modelpredictsthebehaviorofthedegreeofspatialcoherenceγE=∑k(cid:104)Ek(r)Ek(r(cid:48))(cid:105)/(cid:104)|E(r)|2(cid:105),
taken as a measure of the overall spatial correlation of the vector field, the subscript k denot-
ing a vectorial component. Its width defines the spatial coherence length. In the experiments
presentedinthepresentstudy,wehavemeasuredthenormalizedintensitycorrelationfunction
γ =(cid:104)I(r)I(r(cid:48))(cid:105)/(cid:104)|I(r)|2(cid:105), whose width is the average size of a speckle spot. In a fully devel-
I
opedspecklepattern,thefieldexhibitsaGaussianstatistics,sothatγ =1+|γ |2 [3],andwe
I E
will identify the size of a speckle spot with the spatial coherence length. The model predicts
three different regimes for the speckle patterns measured in a plane at a distance z above the
exitsurfaceofthemedium.Forz(cid:29)λ (farfield),thebehavioroffullydevelopedspecklepat-
ternisuniversal,andthespecklespotsizedependsonlyonthewavelengthandtheillumination
geometry(forplane-waveilluminationthesizeisλ/2).Onthecontraryinthenearfieldzone
z<λ, the speckle loses its universal behavior, and the speckle spot size decreases linearly
with z. Finally, in the extreme near-field regime for which z∼(cid:96) (cid:28)λ, the speckle spot size
ε
reachesanasymptoticvalueontheorderof(cid:96) ,thatreflectstheinternalstructureofthedisor-
ε
deredmedium.Asaconsequence,inthisextremenear-fieldregime,thestructuralpropertiesof
themediumareencodedinatwo-dimensionalimageofthespecklepattern,providedthatthe
measurementisabletoresolvethespecklegrainswithsufficientspatialresolution.
Thenon-universalbehaviorofintensityspecklepatterninthenearfieldhasalreadybeenput
forward in some previous experimental studies [7, 9, 8, 10], which in particular have shown
speckle spot sizes below the diffraction limit at sub-wavelength distance from the surface. In
thispaper,weexploreforthefirsttimethecontinuoustransitionfromthefarfieldtotheextreme
nearfieldusingSNOM,leadingtoameasurementofthespecklespotsizefordistancestothe
mediumsurfacerangingfromafewnanometerstoseveraltensofmicrometers.Thebehavior
of the speckle spot size versus the distance reveals different regimes, in excellent agreement
with the predictions of the model described above. Besides the possibility of disclosing the
relevantlengthscalesofnanostructuredsamples,theinvestigationoffundamentallowerlimits
inspecklegrainsizeisanimportantissueforimagingtechniqueswhichrelyonspatialfieldor
intensitycorrelations,focusingorwavefrontcontrolindisorderedmedia[11,12,13,14,15].
Asthespatialresolutionisdrivenbythespecklegrainsize,thenon-universalbehaviorinthe
near-fieldrangecouldpushtheresolutionbeyondthediffractionlimit.
2. Experiment
Ourexperimentalapparatus,schematizedinFig.1a,makesuseofacommercialSNOM(WITec
GmbH alpha300S). It is conceived as an atomic force microscope (AFM) combined with an
optical confocal microscope with an objective of magnification 20X and numerical aperture
NA=0.4. The AFM cantilever includes a hollow metal coated pyramidal tip with a 150nm
wide aperture carved at the apex of the pyramid. It is set under the objective, the apex being
locatedatfocaldistance.Thelightpassingthroughtheapertureandcollectedbytheobjectiveis
convoyedtoamultimodefiberwhichisconnectedwithaphotomultiplier.Thesampleisplaced
underthetiponamulti-axispiezoelectrictranslationstageandisilluminatedfrombelowbya
monochromaticfieldatλ =633nm,producedbyaHe-Nelaserwithamaximumpowerof20
mW.Thesetup,whichnormallyhasasecondobjectiveinordertofocusthebeaminalimited
region under the sample, is slightly modified by replacing the objective with a plano-convex
lenswithF=85mm.Thisproducesabeamthatilluminatesthesampleonaregionofdiameter
D=40µm.
Theapparatuscanworkintwodifferentconfigurations.Inthefirstone(contactmode),the
pyramidiskeptincontactwiththesurfaceofthesamplebyanelectronicfeedback,whilethe
transducermovesthesampleinthex-yplane.Inthisconfiguration,wesimultaneouslyacquire
the topographic (AFM) image of the surface and the spatial intensity distribution of the elec-
tromagneticfieldabovethesample,measuredasthenumberofcountsduetophotonsarriving
onaphotomultiplier(PM).Theresolutionisdrivenbythediameterofthenanoaperture.Inthe
secondconfiguration(constantheightmode),theseparationbetweenthesampleandtheapex
ofthehollowpyramid,z,isfixedandthex-yscanisperformedwithoutturningonthefeedback
loop, while acquiring only the intensity distribution. The z distance is adjusted by means of
thepiezo-transducer itself whichcanretract thesampleupto z=8µmbelow theapexof the
hollow pyramid. Larger distances can be reached by rising the microscope above the sample,
whilesimultaneouslymovingthepyramidandtheobjective.
We investigated two different samples consisting of silica spheres in a disordered arrange-
ment.Twoimagesoftheirsurfaces,recordedwithascanningelectronmicroscope(SEM)are
showninFig.1b.Thetwosamplesarecomposedofseverallayersofbeadswithdifferentav-
eragediameters:statisticsperformedonSEMimagesgiveanaveragediameterofφ =276nm
andφ =430nmforsampleAandBrespectively.Multiplescatteringoftheincidentlaserbeam
transmittedthroughthesamplegeneratesspecklepatternswithoutresidualballisticcomponent.
a) b)
MMF
A
TL
20x 1 μm
PM NA 0.4
z
B
F=85mm
1 μm
Fig.1.a)Schemeofthesetup.Thelaserbeamisfocusedonthesampledowntoadiam-
eterof∼40µm.Thesampleisplacedonathree-axisstagewhichallowsscanninginthe
x-yplaneduringtheacquisition.Thez-stageisusedincontactmodetofollowthesurface
topographywiththepyramidwhentheelectronicfeedbackisinoperation.Thelightabove
thesamplesurfacepassingthroughthe∼150nmwideholeinthepyramidiscollectedby
theconfocalmicroscope(objective20X,NA0.4).Itisinjectedbymeansofthemicroscope
tubelens(TL,focaldistance200mm)intoamultimodefiber(MMF)connectedtoapho-
tomultiplier(PM)whichmeasuresthelightintensity.Thezdistancebetweenthepyramid
andthesamplesurfaceiscontrolledbythesample-stageunitupto8µmandthenbyjointly
moving the cantilever and the objective. b) Scanning electron microscope images of the
surfacesofthetwoanalyzedsamples.
Wetypicallyperformscansof10µm×10µmconsistingof256×256points.Incaseoflarge
distancesfromthesample,aswefoundlargerspecklesgrainsizes,wetook20µm×20µmscans
with 256×256 points in order to maintain a significant number of grains within the scanned
area.
Figure2showstwotypicalresultsobtainedincontactmodeonbothsamples.Theleftcol-
umn of the figure shows the topographic images, while the right column shows the intensity
maps,whichhavebeenacquiredatthesametimeasthetopography.Theycorrespondtofully
developed speckle patterns generated by the He-Ne laser transmitted through the disordered
volumeofthesample.AscanbenoticedinFig.2theintensitypatternsarenotdirectimagesof
thebeadsdistributionatthesurface,sincetheyresultfromthemultiplescatteringprocessinto
thevolume.Lookingatthescanscaleonecanalreadynoticethatthetypicalbrightgrainsizeis
belowthemicrometerrange.Inordertogetquantitativemeasurements,wedefinethespeckle
grainsizeasthehalf-widthofthebaseoftheautocorrelationpeak.Withsuchdefinitionthehalf-
width of the autocorrelation function in the far-field, in the case of a fully developed speckle
generatedunderplanewaveillumination,isδ =λ/2,correspondingtothefirstzerovalueof
0
γ (r,r(cid:48)).Infact,theautocorrelationhastheformofγ (r,r(cid:48))=sinc(k |r−r(cid:48)|)2=sinc(k ρ)2,
I I 0 0
withk =2π/λ.DataanalysisfollowstheproceduredepictedinFig.3:wecalculatethenor-
0
malized autocorrelation function of the intensity scan and we cut it with a plane at 0.2. Note
thatthethresholdvalueof0.2hasbeenchosenratherthan0inordertogetridofnoise.Then
(cid:112)
wekeepasgrainsizetheaverageradiusδ oftheresultingsurfaceSbycalculatingδ = S/π.
a) b)
5 5
0 0
-5 -5
-5 0 5 -5 0 5
c) d)
5 5
0 0
-5 -5
-5 0 5 -5 0 5
Fig. 2. Two scans in contact mode for the two samples. a) Topography, and b) intensity
imagesmeasuredinonezoneofsampleA.c)andd)thesameforsampleB.Thevertical
andhorizontalscalesunitsareµm,whilethevaluesonthecolorbararenumberofcounts
ofthePM.
a) 5 b)
1.0 1.0
0.5 0.5
0
0.0 0.0
-1.5 1.5 -1.5 1.5
1.0 0 0 0 0
-5-5 0 5 1.5 -1.5 1.5 -1.5
c)
0.5
1.5
0.0
8 8 0
4 4
0 0
-4 -4 -1.5
-8 -8 -1.5 0 1.5
Fig. 3. Evaluation of speckle grain size. a) On upper part left: typical intensity speckles
pattern.Bottom:itsautocorrelationwithazoomofthecentralregion(upperright).b)The
autocorrelationiscutwithaplaneataheightof0.2.c)Thesurfaceresultingfromthecut:
thespecklesizeδ isobtainedastheaverageradiusofthesurface.Scalesunitsontheplane
areµm.
a)
5 5 5
0 0 0
-5 -5 -5
-5 0 5 -5 0 5 -5 0 5
z = 0 z = 0.150 μm z = 0.450 μm
b)
γ
5 5 I
z = 0
z = 0.150 μm
z = 0.450 μm
0 0 z = 0.750 μm
z = 10 μm
-5 -5 x (μm)
-5 0 5 -5 0 5
z = 0.750 μm z = 10 μm
Fig. 4. a) Series of intensity speckle patterns in one zone of the sample B at different
distancesfromthesurfacerangingfromthenear-fieldtothefar-fieldregime.Thevertical
andhorizontalscalesunitsareµm,whilethevaluesonthecolorbararenumberofcounts
ofthePM.b)Sectionsalongthexdirectionoftheautocorrelationfunctionobtainedfrom
thefivepatterns.
3. Results
Figure 4a shows a typical series of measurement, obtained by changing the distance z from
a given region of the sample B. The first two measurements, for z=0µm and z=0.150µm,
belongtothenearfieldregimez<λ/2.Thetwofollowingmeasurements,atdistanceszclose
to λ =0.633µm, show a similar speckle size, while the last picture at larger distance shows
a clearly increased grains size. Figure 4b shows the section of the autocorrelation function
for each distance and allows us to measure the speckle grain size, obtaining the following
seriesδ =(0.208,0.255,0.262,0.273,0.354)µm.Inordertoconfirmthisbehaviorandtostudy
the transition between the different regimes, we performed several series of measurements in
differentregionsofbothsamples,obtainingthecurveinFig.5,whichspansbetweenz=0and
z=30µm.Thisisthemainresultofthiswork.Itconstitutes,tothebestofourknowledge,the
first measurement of spatial correlations in speckle patterns continuously spanning from the
nearfieldtothefarfield,maintainingthesameexperimentalconfiguration.
Inthefarfield,z(cid:29)λ,thespatialpropertiesoffullydevelopedspecklepatternsareuniversal
[16,3].BeingDthetransversesizeofthebeamilluminatingthesample,fordistancesz(cid:28)Dthe
beamcanbeapproximatedasaninfiniteplanewave.Therefore,theautocorrelationfunctionhas
thetypicalsincform,previouslymentioned,correspondingtoagrainsizewhichonlydepends
on the illuminating wavelength (δ =λ/2). For distances to the sample of the same order as
or larger than the beam size, the autocorrelation function has a more complex, although well
known,behavior[16]andthespecklegrainsizevariesasδ =λz/D.Weobservedbothregimes.
SinceD(cid:39)40µm,forλ/2≤z≤10µmthegrainsizeismainlyconstantandnotsofarfromthe
δ =λ/2value.Fordistancesz>10µmthegrainsizestartstodependonthebeamgeometry
andthedistancefollowingtheδ =λz/Dlaw.Fittingondataatdistancesz>10µmwefound
D=38.6µmandD=39.3µmforsampleAandBrespectively,whicharecompatiblewiththe
estimatedbeamdiameter.
We further studied the far field range in order to check the reliability of our experimental
apparatus.Atdistancesz>λ,wherethesuper-resolutiongivenbythenanoscaleprobeisnot
strictly necessary, we collected the speckle patterns both in SNOM configuration (by collect-
inglightthroughtheapertureofthecantilever)andinfar-fieldmicroscopeconfiguration.The
secondoneisobtainedbyremovingthecantileverandusingthesetupasaclassicalconfocal
microscope in which the fiber core constitutes the pinhole. In that case, in order to achieve a
spatialresolutionof≈λ/2,wereplacedthe20Xobjectivebya100X,NA=0.9objectiveand
we used a single mode fiber (5µm core) instead of the multimode fiber. In this large distance
regime,wefoundnosignificantdifferencebetweenimagesobtainedintheconfocalmicroscope
andintheSNOMconfiguration.
Thez<λ rangeisthemostinterestingone:itcontainsthefar-fieldtonear-fieldtransition,
whichshowsupatz∼λ/2,asitcanbeobservedinFig.6a,whichisazoomofthenear-field
zoneofthecurveinFig.5.Belowz=λ/2thespecklesizelosesitsuniversalnatureandstarts
decreasing by approaching the surface. Data acquired for samples A and B are close to each
otherandhavealineardependencewithz,aspredictedbytheory[5].Intheextremenearfield,
i.e.inthelimitofz→0,thetwodatasetsareclearlyseparatedinthelimitoftheexperimental
errors,andwemeasureδ =225±10nmandδ =204±10nmforsampleAandBrespectively.
Thez=0casecorrespondstotheconfigurationwherethehollowpyramidiskeptincontact
by means of the electronic feedback. This is hence the most repeatable configuration and we
tookseveralmeasurements:thefirstpointofthetwocurvesistheaveragevalueover9andover
3measurementsforthesampleAandBrespectively.Theerrorbarintheplotiscalculatedas
the maximal deviation over the 9 measurements for the sample A. All the other points in the
curves are calculated from one single acquisition of speckle pattern at the given distance. In
order to evaluate the error in the estimation of the z value we have to take into account that,
onceleftthecontactconfigurationwiththefeedbacklocking,thecantileverundergoespossible
drifts due to any air flow or thermal and mechanical disturbance. Even if vibrations and air
flow are reduced by opportune isolation, thermal drifts are unavoidable due to the proximity
ofelectronicpartswhichdissipateheatwithintheenclosureoftheSNOM.Experimentaltests
showathermaldriftoftheorderof200nmperhour,whichmeans20nmina6-minuteslong
acquisition.Forthisreasonacquisitionsaretakenoneaftertheotherandbyalwayschanging
z in the same direction (either increasing or decreasing the value). Thermal drift is anyway
difficulttocontrolandwearenotexempttopossiblefluctuations,whichisthereasonwhywe
didnotaddhorizontalerrorbarsinFig.5.
Wecanreasonablyassumethatthetwoaveragegrainsizesatz=0havebeenestimatedata
distancefromthesurfacewhichissmallerthanthetypicallengthscale(cid:96) describingthefluctu-
ε
ationofthedielectricfunctionofthemediuminasimpletheoreticalapproach[5].Thislength
shouldbeconnectedwiththenano-scalestructureofthesample,whichinourcaseshouldbe
driven by the size of silica beads and their arrangement within the sample. The presence and
the size of air gaps between the beads depend on parameters such as the homogeneity of the
beadsizes,homogeneityofthebeadsdistributioninsidethemedium,etc.Theaveragesizeof
thebeadsandtheirdispersionaroundthecentralvaluecouldbeinferredfromtheSEMimages
ofthesurface,anditisalsopossibletohaveanideaonthearrangementofbeadsonthesurface.
However,itisnotpossibletoknowhowthedifferentlayersofbeadsarearrangedinthevolume,
so it is not possible to predict the correlation length for the two samples. Nevertheless, from
thedifferentvaluesofδ foundexperimentallyonthetwosamplesatz=0,i.e.intheextreme
near-fieldregime,onecanclearlyaffirmthatthetwosampleshaveadifferentmicro-structure,
independentlyofthedifferentaveragebeaddiameters,provingthatthismethodrevealsmaterial
propertiesthatwouldremainhiddenwithotheropticalmicroscopytechniques.
δ(μm)
z(μm)
Fig.5.Measurementofautocorrelationsizefromthenear-fieldtothefar-fieldregimein
thecaseofsampleA(redpoints)andsampleB(black)points.Thecurvesarecomposed
byseveralseriesofmeasurementsvaryingzindifferentzonesofthesamples.
a) b)
δ(μm) γ0
z(μm) ρx (μm)
Fig.6.a)Near-fieldsubsetofdatainFig.5.Thetwopointsatz=0resultfromaveraging
over 9 measurements in the case of sample A (red) and over 3 measurements in case of
sampleB(black).b)Sectionalongthexdirectionoftheautocorrelationfunctionsforz=0.
Red curve: average over the measurements of sample A, black curve: average over the
measurementsofsampleB,yellowcurves:the9measurementsforsampleAatz=0.
4. Conclusion
In conclusion we have measured the intensity spatial correlation function in speckle patterns
produced by disordered dielectric materials, made of sub-micrometer beads. We have per-
formed series of measurements spanning from the near-field (z(cid:28)λ) to the far-field (z(cid:29)λ)
regimeusingascanningnear-fieldopticalmicroscope.Theaveragedspecklespotsize,defined
as the width of the intensity correlation function, has been measured from the extreme near
fieldtothefarfieldonanunprecedenteddistancerange,revealingdifferentregimesinexcel-
lentagreementwiththeoreticalpredictions.Inthenearfieldregime,thenon-universalbehavior
ofspecklepatternshasbeenclearlydemonstrated,showingtheabilityofnear-fieldspecklesto
discriminatebetweenmaterialswithdifferentinternalmicroscopicstructures.
Acknowledgment
TheauthorsthankC.AydinandL.Greusardforhelpinsettinguptheexperiment.Thiswork
was supported by LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) within the
FrenchProgram“InvestmentsfortheFuture”underreferenceANR-10-IDEX-0001-02PSL*
andbytheFrenchNationalResearchAgency(ANR“CALIN”andANR“GOSPEL”).