Polarimetric imaging beyond the speckle grain scale

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Polarimetric imaging beyond the speckle grain scale

Lucien Pouget, Julien Fade, Cyril Hamel, Mehdi Alouini

To cite this version:

Lucien Pouget, Julien Fade, Cyril Hamel, Mehdi Alouini. Polarimetric imaging beyond the speckle grain scale. Applied optics, Optical Society of America, 2012, 51 (30), pp.7345-7356.

�10.1364/AO.51.007345�. �hal-00761108�

LuienPouget, JulienFade

∗

, Cyril Hameland Mehdi Alouini

Institutde PhysiquedeRennes,CNRS,Universitéde Rennes1,Campusde Beaulieu,35042Rennes,Frane

∗

Corresponding author:julien.fadeuniv-rennes1.fr

CompiledSeptember13,2012

Weaddress anexperimentalStokesimagingsetupallowing onetoexplore thepolarimetripropertiesof

a spekle light eld with spatial resolution wellbeyond the spekle grain sale. Wedetail how the various

experimental diulties inherent to suh measurements an be overome with a dediated measurement

protoolinvolvingaarefulspekleregistration step.Thesetupandprotoolarethenvalidatedonametalli

referenesample,andusedtomeasurethestateofpolarizationoflightineahpixelofhighlyresolvedspekle

patterns(>2000pixelsperspeklegrain)resultingfromthesatteringofaninidentoherentbeamonsamples exhibitingdierent polarimetriproperties.Evolutionof the stateofpolarization withspatialaveraging and

arossadjaentspeklegrainsiseventuallyaddressed. ^{2012 OptialSoietyofAmeria}

OCISodes: (110.5405)Polarimetriimaging;(260.5430)Polarization;(110.6150)Spekleimaging;(030.0030)Coher-

eneandstatistialoptis;(220.4830)Systemsdesign

1. Introdution

Spekleisaubiquitousphenomenoninallresearhelds

studying the interation between random media and

propagating waves, whether eletromagneti [14℄ or

aousti [5℄. Sine spekle intensity patterns often rep-

resent the simplest and most aessible observables to

gaugesuhinteration,thisphenomenonhasbeenthor-

oughlyinvestigatedfordeades[1,2,5℄butisstillwidely

studied in very ative researh elds of physis, suh

as wave loalization [5℄, ontrol of light through disor-

dered media [6,7℄, optial non-lineareets in random

media[8℄orpolarizationsingularities[9℄. Inthe optial

imaging domain, spekle has often been onsidered as

anoisedetrimental to image quality[10,11℄. Neverthe-

less,spekleontrastimagingisknowntobeaneient

remote-sensingtehniqueproviding informationonsur-

fae roughness properties [1214℄ or uid veloity [15℄

for instane. More reently, new appliations involving

spekleontrast images havebeenproposed to hara-

terizepolarizationoflight[1619℄ordiusionproperties

ofmaterials[20,21℄.

Despitethisintense researhativity,anopendebate

still remainsabouthow thepolarizationstate distribu-

tionofaspeklepatternan belearlylinkedwith ma-

terialsdepolarizationproperties.Reently,thisissuehas

oasioned a number of publiations [20,2230℄, ques-

tioning for instane the denition and measurementof

polarizationorrelationlengthsinaspatialspeklepat-

tern[2224,2830℄,orthepartialrepolarizationofun-

polarizedoherentlightbaksatteredbyadepolarizing

material [20℄.Inorder to link depolarizationproperties

ofasamplewiththesatteredlightpolarizationdistribu-

tion,experimentalstudieshavebeenarriedoutusinga

spatialmultiple-saleanalysisofthespeklepolarization

properties [23,2527℄.In theseexperiments, light depo-

larizationhasbeenstudiedfromamarosopipointof

view,byonduting astatistialanalysisofasalarpa-

rameter(DegreeOfPolarization(DOP))[31℄or(Orthog-

onalStateContrast(OSC))[32℄,overagreatnumberof

ohereneareas(speklegrains).These resultstended

toexperimentallyvalidate thatthepolarizationstateis

deterministi(lightisfullypolarized)attheloalsale

of asinglespekle grain,whereasglobal depolarization

induedby interation(reetionortransmission)with

thesampleresultsfromspatialaveragingonthedetetor

overseveralohereneareas[23,2527℄.

However, the various experimental devies used in

theserefereneswerenotspeiallydediatedtostudy-

ingthepolarizationstateofthespeklepatternatthelo-

alsale,i.e.,beyondthespeklegrainsale.Tothebest

ofourknowledge,animagingsetupapableofmeasuring

thefullStokesvetor(4 omponents)of lightsattered

byadiusivesamplebelowthespeklegrainsaleinthe

optial wavedomain has not beenlearly addressed in

theliterature. If suh study ould be arried outquite

easily in the mirowave range as suggested in [33℄, it

turnsouttobeamerehallengewhenthewavelengthis

onlya few hundredsof nanometers, as already notied

in [9℄. Indeed, Stokes measurements require polarizers

andwaveplates(atleastone)tobeinsertedandrotated

in front of the imaging detetor, thus inevitably mod-

ifying the optial wavefront of the sattered light and

henethe speklepattern itself, aswill beevidened in

thenext setion. This experimental diulty is mostly

often eluded in the literature, suggesting that onven-

tionalexperimentalshemesarenotsuitableto perform

Stokes imaging at the spekle grain level. In this pa-

per,wetaklethis problembyproposing arigorousex-

perimental setup as well as the methodology allowing

onetoaquireStokesimagesofaspeklepattern,where

eahgrainoversseveralhundredorthousandsofpixels

on the detetor surfae.The paper is organizedas fol-

lows:in Setion2,theexperimentalsetupandmeasure-

ment protool proposed to ahieveStokes imaging be-

Test samples are then desribed and haraterized in

Setion 3. Lastly, we report in Setion 4 experimental

Stokes imaging of highly resolved spekle patterns ob-

tained with this setup. The proessing and analysis of

these resultsare alsoaddressed andommentedin this

setion. Conlusions and future work diretions are -

nallygiveninSetion 5.

2. Experimental setup

Analyzing thepolarization properties of a spekle eld

beyond the spekle grain sale requires performing a

measurement of theStokes vetorof lightat anypoint

ofahighlyresolvedspeklepattern.Beforedetailingthe

experimental setupused, letusbriey realltheStokes

vetorformalismandtheprinipleofStokesimaging.

A. Stokesimaging priniple

Stokesformalism is ommonlyused to haraterizethe

lightstateof polarization (SOP)[31℄.Aordingtothis

formalism, the polarization state of light is fully de-

sribedbyafour-omponentvetor:

S ~ =









S

0

= I

x

+ I

y

S

1

= I

x

− I

y

S

2

= I

+45^◦

− I

−45◦

S

3

= I

R

− I

L









.

⁽¹⁾

The four omponents of the Stokes vetor

S

i ^an

thus be simply obtained from intensity measurements

(

I

x

, I

y

, I

+45^◦

, I

−45^◦

, I

R

, I

L^{) through six ongura-}

tionsofapolarizationanalyzer.Intheontextofpolari-

metriimagingonsideredinthepaper,thepolarization

analyzeris usedto reord sixintensity imageson ade-

tetormatrix.Fromthesesiximages,theSOP(i.e.,the

4-omponent Stokes vetor) in eah pixel of the image

isthendetermined.Aglobalandsalarharaterization

oftheSOPin eah pixelislassiallyobtainedbyom-

puting the degree of polarization (DOP) image, given

by[31℄

DOP =

p S

1²

+ S

2²

+ S

3²

S

0

= q

S

²₁

+ S

²₂

+ S

²₃

,

⁽²⁾

ortheorthogonalstateontrast(OSC)givenby

OSC = S

1

S

0

= S

1

,

⁽³⁾

where the

S

_i

= S

i

/S

0 ^{stand for the normalized Stokes}

omponents.

This method has been preferred to a Fourier anal-

ysis tehnique involving a rotating quarter-wave plate

[31℄.Indeed,thisoneimpliesreordingmultipleintensity

snapshotsthroughamovingplate,whihisinonsistent

withtheextremestabilityrequiredfortheexperimental

setup, aswill be evidened below. Another alternative

would beusingaliquidrystalvariable retarder,but it

is introdued when its onguration is hanged during

themeasurement.

Indeed, to ensure validity of the polarimetri data

reorded at the spekle grain sale, one needs to pre-

ventany modiationin thewavefrontof thesattered

light during the measurement proedure. Understand-

ably, this last ondition is not easily ahieved sine

spekleisnothingbutaninterferenepattern,andthere-

foreanyhangeintheoptialpathoftheorderofafra-

tionof wavelength will modify the speklepattern. As

willbedetailed afterwards, unwantedwavefrontdistor-

tionsanbeausedbyairturbulenewithintheoptial

path,thermalexpansionsofthesampleormehanialvi-

brations,butthemaindiultyistokeepthewavefront

unhangedwhileswithingbetweenthesixdierenton-

gurationsofthepolarizationanalyzer.Intheremainder

ofthis setion, wewill addressthese tehnialissues in

detailandproposeanoptimizedexperimentalsetupand

measurement protool allowing one to perform Stokes

imagingbeyondthespeklegrainsale.

B. Desriptionof the experimentalsetup

Theexperimental setup proposed is depited in Figure

1.Stokesimagingofthespeklepatternisperformedin

areetion geometry (quasi-monostati onguration).

Indeed,the polarimetri imagingsystem analyzeslight

satteredbythesampleinadiretionlosetobaksat-

tering diretion. Consequently, it has to be noted that

in theremainderof this paper, theword baksattering

willnotrefertothestritsenseofsatteringintheini-

dentbeamdiretion.Thesetupomprisesseveralbuild-

ingblokswhiharedesribedin detailbelow:

Illumination: Intheexperimental ongurationho-

sen, the sattering sample is enlightened with a fre-

quenydoubled Nd:YVO4 ^{laser (Coherent Verdi)emit-}

ting a maximum output power of 2 W at

λ

^{=532 nm.}

Although the light emitted by the laser is linearly po-

larized,weuseaGlanpolarizerto ensurehighontrast

horizontallinear polarization of theillumination beam.

Thelaserbeamisthenexpandedandollimatedwiththe

assoiationofamirosopeobjetive(

×

^{10,0.25NA)and}

aonvexlens

L

0⁽

f

0^{′=200mm).Adiaphragmisusedto}

adjustthebeamdiameterandsuppressunwanteddiu-

sionsorreetions onthe edges ofthe ollimation lens

(

L

0^{).Finally,thisbeamformsa2.5mdiameterillumi-}

nationspotonthesatteringsampleunder test.

Imaging optis: A set of two onverging lenses

L

1

(

f

1^{′=80 mm)and}

L

2 ⁽

f

2^{′=40mm)is used to image the}

surfaeof the sampleon thedetetor plane. Suh opti-

al onguration makes it possible to position the po-

larizationanalyzer(PA) between thetwolenses,in the

intermediate image plane (IIP). The distane between

thetwolenses is 22.5m and thedistane between

L

2

andtheintermediateimageplaneissetto 12m.

Thoughunusualinpolarimetriimagingsetups,posi-

tioningofthepolarizationanalysisomponentsinaon-

jugate plane of the sample and of the CCD is justied

here sine it helps minimizing wavefront deformations

whenthePAongurationishanged.Indeed,thebeam

has a minimum size in the intermediate image plane,

forming anintermediate image of diameter 6.2 mm on

thepolarizationanalyzer.Providedthebeamiswellen-

teredwiththerotationaxisofthePA,suhonguration

minimizesanyspuriouseetofplanarityimperfetions

of thepolarizationanalysis omponents(see paragraph

Polarization analyzer below).

Imagingpinhole: Toensureasuientsizeoftheo-

herene areaonthe detetor, suh that a speklegrain

overs a few thousands of pixels, the aperture of the

imagingsystemisshrunkbyinsertingairularpinhole

of diameter

φ = 200 µ

^{m on the imaging lens}

L

2^{. The}

typial size of a spekle grain on the detetor is then

givenby

δ = 1 . 22 λ 2 D

φ ≃ 0 . 4 mm ,

⁽⁴⁾

where

D = 6

^{m orresponds to the distane between}

imaginglens

L

2^{andthedetetorplane.Inthisongu-}

ration,eahspeklegrainofirularshapeoversinthe

nal image approximately2300square pixelsof dimen-

sion7.4

µ

^m.

Imagingdetetor: Duetothetinyimagingaperture

used in the experiment, the ameramust exhibit good

noise properties under low illumination levels. Conse-

quently, we hoose to reord the images on a 12-bits

monohromati 659 x 494 pixels CCD amera (Basler

sA640-70fm).Moreover,toenhanethedynamirange,

a dual exposure with two dierent aquisition times

(

T

2⁼¹⁰

T

1^{)is used to extrat data from thelowinten-}

sity areas. Image proessing is neessary to reombine

the two snapshots,nally providing anintensityimage

with higher dynamis thanthe 4096 graysale levelsof

the12-bitamera(seepixelsgraysalevaluesofFig.9.a

forinstane).

Polarization analyzer (PA): Mostonventionalpo-

larimetri imaging experiments use a single adjustable

polarizertomeasure

I

x

, I

y

, I

+45^◦

, I

−45^{◦,andthenadd}

aquarter-waveplatetoaessthetwoimages

I

R ^and

I

L

assoiatedwithirularpolarizationoflight.Thissimple

and natural approah is not satisfatory in our experi-

ment. Indeed,insertingan optialelementbetweenthe

sampleandtheCCDdetetorwillneessarilymodifythe

optial path of the baksatteredlight and modify the

spekle pattern, thus inevitably spoiling the measure-

mentaswillbeevidenedbelow.Tomaintainaonstant

optial path during the 6 intensity measurements, the

polarizationanalyzer(PA)usedonsistsofananoparti-

les linearlm polarizerplate (2mmthikness)andan

ahromatiquarter-waveplate(400-700nm,

λ

^/10wave-

front distortion, 1mm thik) losely paked together.

These omponents are insertedin asingle rotatingop-

tial mount with the axes of the polarizer and of the

quarter-waveplateforminganangleof45

◦

.Therelative

positionningofthepolarizerandquarter-waveplate,as

wellasthealibrationofthePAaxeswereoperatedpre-

liminarytoanymeasurement,usingthevertiallypolar-

izedilluminationlaserbeam.Calibrationisanimportant

stepin thedesign of a polarimeter due to possible po-

larizationartifatsintroduedbytheoptialelementsof

thesystem[34℄.

Toswithfromalinearpolarizationanalysisto air-

ularpolarization analysis, the whole mount is turned

around so that the analyzed light enters the polarizer

rst,orthequarter-waveplaterstrespetively.Tothis

aim,thePAissetonamotorizedrotatingstageforpre-

ise and repeatable positioning. This onguration al-

lowsus to aess thewhole sixintensitymeasurements

needed to determine the full Stokesvetor [31℄.In ad-

dition, twoangulardegreesoffreedomareaddedtothe

polarizationanalyzermounttoenableneadjustmentof

the orientation of the polarization analysis plates with

respet to the optial axis(One rotation about

~e

x ^and

oneabout

~e

y^).Thelens

L

1⁽

f

1^{′=80mm)isalsoplaedon}

amirometri translationmountallowingne position-

ingofthefoalspotonthepolarizationanalysis plates.

These mehanial degreesof freedom assoiated to the

PAareskethedinFigure2.Aswillbeevidenedinthe

nextsubsetion,these additionaldegreesoffreedomare

indispensableto rene therelativepositioning between

theenterofthePAandthefoalspotintheintermedi-

ateimage planeand thus minimizewavefrontdeforma-

tions betweentwosuessiveaquisitionswith dierent

ongurationsofthePA.

Mehanial stability and housing: As will be de-

tailed below, mehanial stability and air turbulene

must be taken into aount to ensure stability of the

spekle pattern. To this end, the optial setup is om-

patly built on an optial table and breadboard. The

wholesetup(apartfromlaser)isinsertedwithinaPlex-

iglas housingto protetitfromairows.

C. Measurement protool and spekle pattern registra-

tion

A ruial point to ahieve Stokes imaging beyond the

spekle grain sale is to ensure that the wavefront is

not modiedby themovementof thepolarization ana-

lyzerduringtheompleteStokesmeasurement.Inother

words,when theongurationof thePAis swithed to

measure the dierent Stokes omponents, the spekle

pattern must nothange. As the PA is neither stritly

planenorplaedin aplanestritlyperpendiulartothe

optialaxis,thisonditionisnotapriori satised.This

is evidened in Figure 3 where a rst Stokes intensity

image

I

x^{isplotted(Fig.3.a)andanbeomparedwith}

a seond intensity image

I

+45◦ ^{obtained after rotating}