Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes

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by  Jon D. Swaim

Atomic vapor as a source of tunable, non-Gaussian
self-reconstructing optical modes
Jon D. Swaim1, Kaitlyn N. David1, Erin M. Knutson1, Christian Rios1, Onur Danaci1, and
Ryan T. Glasser1,*
1DepartmentofPhysics,TulaneUniversity,NewOrleans,LAUSA70118
*[email protected]
7
ABSTRACT
1
0
2
Inthismanuscript,wedemonstratetheabilityofnonlinearlight-atominteractionstoproducetunablynon-Gaussian,partially

n self-healingopticalmodes. Gaussianspatial-modelighttunedneartotheatomicresonancesinhotrubidiumvaporisshownto
a resultinnon-Gaussianoutputmodestructuresthatmaybecontrolledbyvaryingeithertheinputbeampowerorthetemperature
J oftheatomicvapor. Weshowthattheoutputmodesexhibitadegreeofself-reconstructionafterencounteringanobstructionin

6 thebeampath. TheresultantmodesaresimilartotruncatedBessel-Gaussmodesthatexhibittheabilitytoself-reconstruct
earlier upon propagation than Gaussian modes. The ability to generate tunable, self-reconstructing beams has potential

] applicationstoavarietyofimagingandcommunicationscenarios.
s
c
i
t
p
o Introduction
.
s Coherentlighttunedon,ornear,atomicresonanceshasallowedforthediscoveryandinvestigationofawealthofnonlinear
c
i optical effects1–9. In addition to cross-action effects in which the presence of one beam alters the behavior of another,
s manyself-actioneffectshavebeenthoroughlyinvestigated,includingself-phasemodulation4,self-focusing5,10,11 andself-
y
h defocusing2,3,12,continuous-wave(CW)on-resonanceenhancement6,andself-inducedtransparency13,14. Whilenonlinear
p index-inducedfocusinganddefocusingoflighthavebeenstudiedinbothcoldatomsystems15–17andhotatomicvapor18–21,
[ theseeffectshaveonlybeenstudiedindividually,andtoourknowledgenoexperimenttodatehasdemonstratedatunable

1 interplaybetweennonlinearfocusinganddefocusingatafixeddetuningfromanatomicresonance,withaninputbeamthatis
v bothconverginganddivergingthroughthenonlinearmedium. Self-actioneffectsstronglymodifythespatialstructureofan
5 opticalbeam,andmayresultinoutputmodeprofilesthatexhibitinterestingopticalpropertiessuchasdiffraction-freeand
1 self-healingbehavior22–24. Theabilityofopticalmodestoreconstructthemselvesafterencounteringanobstructionintheir
7
pathhasbeeninvestigatedinboththeclassical25–28andquantumregimes29,thelatterofwhichhasresultedinademonstration
1
0 ofmorerobustpropagationandrecoveryofquantumentanglement.
. Thefirstexperimentalinvestigationintoself-healingBessel-Gauss(BG)beamswascarriedoutbyDurninandco-authors22.
1
0 In their work, significant (annular) aperturing of the incident optical mode produced a BG beam with reduced diffractive
7 spreadingcomparedtothatofaGaussianbeam. Morerecentmethodsinvolveeitherspatiallightmodulators30,31oraxicon
1 (conical)lenses25,32. Insomecases,spatiallightmodulatorscanbedamagedatrelativelylowintensities33,34,thuslimiting
:
v theintensityofthegeneratednon-Gaussianlight. Further,whilesomeprogresshasbeenmadecombiningnumerousoptical
i elements with axicon lenses to produce tunable output self-healing modes, the cone angle of axicon lenses themselves is
X
inherentlyfixed. Incontrast,hotatomicvaporsprovideanall-opticalmethodofgeneratingnon-Gaussian,BG-likemodes9.
r Withthismethod,tworecentmotivatingworksdemonstratedsometunabilityintheshapeoftheoutput20,21.Whileself-focusing
a
anddefocusingwerethenonlineareffectsconsideredinthetheoreticaldiscussionstherein,adescriptionofthespatial-mode
structure of these modes, as well as their ability to self-reconstruct after encountering an obstruction still remains to be
investigated.
Hereweexperimentallydemonstratetunablegenerationofpartiallyself-reconstructingnon-Gaussianbeams. Toaccomplish
this, we focus strong, Gaussian spatial-mode, nearly resonant laser light into a nonlinear medium consisting of hot alkali
atoms. Acomplexinterplaybetweenvariousself-actioneffectsinthemediumdeterminestheoveralloutputmodeshape,and
accordinglyenablessometunabilityofthegeneratednon-Gaussianbeambyvaryingtheinputopticalpowerorthetemperature
oftheatomicmedium. Theabilitytotunethemodeconversionprocessinthismannershouldallowforoptimizationofthe
self-reconstruction,whichwedemonstrateforanon-Gaussianbeamencounteringanobstructionatitscenter. Thesefindings
suggestthattunable,non-Gaussianlightgeneratedviaatomicvaporsmaybeusefulforexperimentsinquantuminformation,as
wellasdemonstratingnewapproachestoopticalcommunicationandimaging.
Partial
HWP
Ti:Sapph BS PBS 500 mm 85Rb 500 mm obstruction CCD
A B C
To polarization
spectroscopy setup Z
Figure1. Schematicoftheexperimentalsetupforthegenerationoftunable,self-reconstructingopticalmodes. The
modeimages(A,B,andC)wereobtainedwithaninputpowerofP=300mWandacelltemperatureofT =125◦C.
Ti:Sapph: Titanium-sapphireCWlaser. BS:Beamsplitter. HWP:Half-waveplate. PBS:Polarizingbeamsplitter. 85Rb:
Rubidium-85vaporcell. CCD:Camera. A:Referenceimagebeforetheobstruction. B:Obstructedmodedirectlyafterthe
obstruction. C:Reconstructedmodeseveralmetersaftertheobstruction.
Results
Experimentallayout
TheexperimentallayoutisshowninFig.1. LightfromaCWtitanium-sapphirelaseristunedclosetotheD1lineofrubidium
andcoupledintoasingle-mode, polarization-maintainingfiberinordertoproduceaGaussianspatialmode. Thebeamis
focusedtoaspotsizeof220µmatthecenterofa2.5cmlongrubidium-85cell,whosetemperatureiscontrolledbyaresistive
heatingelementandmonitoredbyathermocouple. Asmallportionofthelightissenttoapolarizationspectroscopysetupfor
laserlocking(seeMethods),andtheremainingopticalpowerisvariedbythecombinationofahalf-waveplateandpolarizing
beamsplitterpriortotherubidiumcell. Atadistance190mmafterthecell,theresultingoutputmodesareimagedwithaCCD
camera(2048x1088pixelswitha5.5µmpixelpitch)toinvestigatethespatialmodeprofilesasafunctionofinputpower
andcelltemperature. Inthereconstructionexperiments,theoutputbeamisinsteadre-collimatedwitha500mmfocallength
lensplacedonefocallengthfromthecenteroftherubidiumcell. Additionally,duetothelargesizeoftheopticalmodes,a
60mmfocallengthlensisplacedapproximately30mminfrontofthecamera. Becauseatomicvaporactsasanonlinear
lensintheconversionprocess,thedefocusingstrengthoftheoutputmodevariesdependingonthechoiceofdetuning,input
powerandcelltemperature. Whilethishasthebenefitofproducingthedesiredtunabilityofthemodeshapes,forconsistency
throughoutallexperimentsthe500mmlensiskeptfixedatonefocallengthfromthefocusoftheunconvertedGaussianbeam.
Aremovablecircularobstruction3mmindiameterisplacedinthecenterofthebeam’spath,andboththeunobstructedand
reconstructedimagesarerecordedatvariousdistancesfromtheobstruction.
Tunablenon-Gaussianbeamgeneration
AtypicaloutputspatialmodeisshownatpositionAinFig.1. ThesetruncatedBG-likemodesaregeneratedwhenthelaseris
detunedtotheredsideoftherubidiumD1line,overafrequencyrangeontheorderofoneDoppler-broadenedatomiclinewidth.
Inthemedium,theKerrnonlinearityresultsinaspatially-varyingrefractiveindex1,whichinturnleadstoaspatially-varying
phaseshiftontheincidentbeam.Wesuspectthattheoutputmodeprofilesarisefromacomplexinterplaybetweenthisnonlinear
phaseshiftandself-inducedtransparency,whichgivesrisetoaspatially-varyingsoftapertureeffectthatisdependentonthe
localintensityoftheinputmode. Interferenceinthefar-fieldthenunderliestheconversionfromaninputGaussianbeamtothe
non-Gaussianoutputmode.
InFig.2(a-b),weshowtwoseriesofnormalizedintensitycross-sectionsobtainedfromsingle-shotimagesforvarious
levelsofinputpowerandcelltemperature. Smoothedtwo-dimensionalintensityprojectionsofthemeasurementsaredisplayed
below. Wefindthattheconversionintothenon-Gaussianpatternisenhancedforincreasinglevelsofbothopticalpowerand
temperature: ringswithappreciablesignal-to-noiseratiosaregenerallyobservedatopticalpowersaboveP∼200mWand
temperaturesaboveT ∼125◦ C.Theseresultsagreewithpreviouswork20,wheretheintensity-dependentphaseshiftand
temperature-dependentatomicdensityofthenonlinearmediumwereconsidered.
We use numerical fitting to extract the radial positions and intensities (relative to the central peak) of the first ring in
theoutputpatternsobtainedatvariouspowersandtemperatures. Weobservethattheatomicvaporactsasanonlinearlens,
asshowninFig.2(c-d). Withincreasingopticalpower,theatomicvaporfocusestheconvertedmodeandreducesthesize
oftheringinthefar-field. Therelativeintensityofthefirstringincreasesevenmoredramaticallywithincreasingpower,
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(a) (b)
(c) (d)
Figure2. Tunabilityofnon-Gaussianmodesobtainedfromatomicvaporbyvaryingpowerandtemperature. (a)
Imagecross-sectionsasafunctionofincidentopticalpower,keepingthecelltemperaturefixedatT =125◦C.(b)
Cross-sectionsasafunctionofcelltemperature,withaconstantinputpowerofP=275mW.(c)Radialpositions(red)and
intensitiesofthefirstringrelativetothecentralpeak(blue)versusinputopticalpower. T =125◦C.(d)Radialpositions(red)
andrelativesintensitiesofthefirstring(blue)withvaryingcelltemperature. P=250mW.Theshadedregionsindicate
uncertaintiesbasedononestandarddeviation.
reachingvaluesgreaterthanunityforinputpowersabove300mW.ThemaximumvalueofP=350mWusedthroughout
themeasurementswaschosenbasedonthemaximumallowedcouplingpowerforasingle-modefiber. Conversely,wesee
thatincreasingthecelltemperatureleadstobeamdefocusing. Thecomplexinterplayofnonlineareffectsisexemplifiedby
thefactthatincreasingthecelltemperaturealsoincreasestheatomicdensityofthenonlinearmedium,andweseethatmode
conversionisenhancedathighertemperatures,asevidencedbylargerfieldintensityinthefirstringathighertemperature.
Inourexperiments,modeconversionisoptimalatatemperatureofaboutT∼125◦C,asshowninFig.2(d). Abovethis
temperature,themediumbecomeslesstransparentandlinearabsorptionlimitstheconversion.
Giventhattheatomicvaporcreatesanonlinearlensingeffect,thefocusing(ordefocusing)strengthoftheoutputmode
dependsonthepositionofthevaporcellrelativetothefocusoftheinputbeam. Ineachexperiment,effortwastakentoposition
thecellatthecenteroftheinputbeam’sfocus. ThiswasfoundtoreliablygiveresultssuchasthoseshowninFigs.2(a-b). For
themeasurementsinFigs.2(c-d),however,wefoundthatthepositionofthecellcouldbeadjustedtooptimizetheintensity
ofthering. Inthiscase,thepositionwassettooptimizetheringintensity,whichreachedamaximumataninputpowerof
aboutP ∼325mW.Accordingly,theresultsinFigs.2(c-d)differfromthoseinFigs.2(a-b). Nonetheless,itwasfoundtobe
qualitativelytruethatincreasingthepowerandtemperaturewouldresultinnonlinearfocusinganddefocusingofthemode,
respectively.
Lastly,asexpected,wehavefoundthatthemodeshapescanbetunedbyvaryingthelaserfrequency. Theseresultsare
showninthesupplementaryinformation. Forthisreason,thelaserfrequencywaslockedtoafixeddetuningfromtheresonance
(seeMethods),topreventfluctuationsand/ordriftinthelaserfrequencyfromalteringtheshapeoftheopticalmodes.
Theseobservationsmostlikelystemfromacombinationofnumerouscompetingself-actionseffects. Thecollectionof
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(a) (b)
Figure3. Self-inducedtransparencyinatomicvapor. (a)Measuredopticaltransmissionthroughrubidium-85vaporasa
functionofinputopticalpowerwithafixedcelltemperatureofT =125◦Candwiththelaserdetuned275MHztotheredside
oftheD1line. Theinsetshowsthecorrespondingopticaltransmissionwhenvaryingcelltemperatureandkeepingtheinput
powerfixedatP=300mW.(b). OpticaltransmissionasafunctionofinputopticalpowerforT =125◦Candalaserdetuning
of-80MHz. Theshadedregionsindicateuncertaintiesbasedonstandarddeviations.
theseeffects,however,canbethoughtofasancomplexaperturewhichistunedbytheinteractionsbetweenlightandmatter.
Amongtheseinteractions,wefindthatoneofthedominanteffectsisaself-inducedtransparencyofthemedium13,14,35,which
effectivelybehavesasasoft-aperturefortheinputbeam6. InFig.3(a),weshowthemeasuredtransmission(i.e.,theratioof
theoutputopticalpowertoinputpower)throughtheatomicvaporwhilethecelltemperatureiskeptfixedatT=125◦C.Atlow
(∼mW)powerlevels,opticallossesresultingfromatomicabsorptionstronglyattenuatethebeam. Asthepowerisincreased,
self-transparencyoccursandthetransmissionincreasesrapidly,reaching50%ataninputpowerofP∼50mW.Forhigher
powers,therateofincreaseintransmissionslowsdownandapproachesunity. Acorrespondingtemperaturemeasurement
isshownintheinsetofFig.3(a), foraninputpowerofP=300mW.Here, thetransmissionfallsoffpredictablyathigh
temperatures. Weconfirmedthetransparencyeffectforfrequencydetuningsof-275MHz(Fig.3(a)and-80MHz(Fig.3
(b),thepredominantfrequenciesusedthroughoutthiswork. Theeffectistypifiedbythefactthat,atagiventemperature,the
transmissioncanbeboostedsimplybyincreasingtheintensityoftheopticalbeam.
Reconstructionofnon-Gaussiansbeamsfollowinganobstruction
Theabilityofanopticalbeamtoregenerateitselffollowingadisturbanceis,initself,aparticularlyinterestingpropertygiven
thediffractivenatureoflight. Hereweshowthatournon-Gaussianpatterncanreconstructitselfafterencounteringacircular
obstructionplacedatthecenteroftheopticalmode. InFig.4(a),weshowaseriesofnormalizedimageswhichzoominonthe
obstructionareaandillustratethereconstructionofthecentralportionofthemodeasittravelsalongthepropagationdirection.
ImagesoftheunobstructedandobstructedmodesareshowninthetopandbottomofFig.4(a),respectively. Atz=0m,the
(removable)circularblockattentuatesapproximately20%ofthelight,andtheremaininglightincreasinglyregeneratesitself
forz>0m. (AseriesoffullimagesisshownintheSupplementaryinformation.) Wequantifythereconstructionbycalculating
thetwo-dimensionalcorrelationbetweentheimagesofthe(unobstructed)referenceatz=0mandthepropagatingobstructed
beam(seeSupplementaryinformation). TheresultisshowninFig.4(b),wherewehavecarriedouttwocalculations: onetaken
overtheentireareaoftheimage,andasecondusingonlytheobstructedarea. Withintheobstructionarea,moderegeneration
increaseswithz,reachingamaximumcorrelationof∼94%atz∼7.5m. Ifthecalculationistakenovertheentireimage,
maximumcorrelationisobservedatz∼8m,butasawhole,thecorrelationdecreaseswithz. Giventhatthemodesresemble
truncated(afterthefirst-andsecond-orderrings)BGmodes,reconstruction,albeitimperfect,afterafinitepropagationdistance
isexpected.
Next,wecomparethereconstructionofourBG-likemodewiththatofaGaussianmode. Studyingthepropagationof
theunobstructedmodesinFig.4(a),oneseesthatthecenterofthemodereachesafocusaroundz ∼5.5m. Therefore,
focus
we consider the reconstruction of a focused Gaussian mode which has also been attenuated by approximately 20% via a
centralobstruction. WeemployGaussianfittingtoextractthefullwidthhalfmaxima(FWHM)ofthecentralpeaksfromthe
reconstructingmodesinbothcases. InFig.4(c),weshowtheFWHMfromsingleshotimagesnormalizedtotheirwidthsat
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(a)
(b) (c)
Figure4. Self-reconstructionofanon-Gaussianmodegeneratedwithatomicvapor. (a)Imagesofunobstructed(top)
andobstructed(bottom)modesatvariouspositionsalongthepropagationcoordinate. Thesizeofeachimageis1.5mm×1.5
mm,andtheimagescorrespondtotheregionoverwhichtheobstructedareacorrelationiscalculated. (b)Calculated
two-dimensionalcorrelationbetweenthe(unobstructed)referenceatz=0mandtheobstructedmodesatvariouspositionsfor
thefullimage(blue)andtheobstructionareaonly(red). Inthelattercase,aparabolicfitrevealsanoptimalcorrelationat
z∼7.5m. Theshadedregionsindicateuncertaintiesbasedononestandarddeviation. (c)Fullwidthhalfmaximaofcentral
peaksnormalizedtotheirwidthsatz=0m,foraBG-likemodewithafocalpositionz =5.5mandaGaussianmodewith
focus
z =1.1m. Inbothcases,approximately20%ofthelightisblocked. Theinsetshowscross-sectionsofthereferenceand
focus
obstructedBG-likemodesattheoptimalreconstructiondistanceofz=7.5m.
z=0m,asafunctionofthenormalizeddistancez/z . Inbothcases,moderegenerationisassociatedwithanincreaseinthe
focus
FWHM,whichslowsafterthefocusandeventuallyincreasesagainduetodiffraction. Importantly,thenormalizedFWHMof
thenon-GaussianmodereachesunitypriortotheGaussian,bymorethanafactoroftwoinnormalizeddistance. Thepointof
maximumimagecorrelationforthenon-Gaussiancase(z∼7.5m)correspondstothepointz/z =1.4inFig.4(c). Atthis
focus
point,thenormalizedFWHMis∼0.8forthenon-Gaussianmode(seetheinsetofFig.4(c)),whereasitisonly∼0.7forthe
Gaussianmode.
Discussion
Inthismanuscriptwehaveexperimentallydemonstratedthatatomicvapormaybeusedtogeneratetunableself-reconstructing
opticalbeams. Whilethecompletedynamicsofnear-resonantlightinteractingwithmulti-levelatomicsystemsisreasonably
complex,theseresultssuggestthatnonlinearphaseshiftsandself-inducedtransparencyaredominanteffects,producingan
effectivesoftaperturingoftheincidentopticalmode,whichleadstotheconversionintoatruncatedBGmode. Althoughthis
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truncationisexpectedtoworsenthemode’sself-healingabilitywhencomparedtoanidealBGmode,wenonethelessobserve
imagecorrelationsofupto94%inthecentralportionoftherecoveredmodeprofileandupto61%acrosstheentireimage.
Theflexibleapproachtakenhereisadvantageouslyrobusttohighlevelsofopticalpower,andthusallowsonetocontrolthe
outputmodeprofilebyvaryingtheopticalpower,aswellasthetemperatureoftheatomicvaporandthelaserfrequency. Inthis
manner,thereconstructionofthemodeafterinteractingwithanobstructioncanbeoptimized,whichmayseeapplicationsin
futureexperimentsinvolvingtherecoveryofinformationinthespatialprofileofthemodes. Weexpectthatsuchoptimization
couldfurtherimprovethecorrelationsreportedhere.
Methods
LockingtotheatomicresonancesofrubidiumisachievedwithapolarizationspectroscopysetupasdescribedinRefs.36–38.
Imagesarerecordedusingabeamprofiler(EdmundOptics89-308)withthelaserlockedatafixeddetuning∆/2π=−80MHz
fromtheredsideoftheD1lineofrubidium-85. InFig.3,dataisalsoshownforaseconddetuningof-275MHz. Ineach
measurement,aseriesoftenimagesacquiredoveraspanofapproximately1sarerecorded,andthedataanderrorbarsin
Figs.2and 4representmeanvaluesandstandarddeviations,respectively,basedonthesetenimages. Everyimageisacquired
usinganintegrationtimeof10ms,andalloftheactualimagesshowninthismanuscriptaresingleshots.
Intheself-healingexperiments,theobstructionismadebycoatingthesurfaceofacircularobject(∼3mmindiameter)
withacrylicpaint, andthencontactingtheobjectwitha170 µmthick, 22mm×22mmprecisionmicroscopecoverslip
(ThorlabsCG15CH).Theobstructionisplacedonaflip-mountsothatboththereconstructingandtheunobstructedpropagating
modes may be imaged. In comparing the reconstruction with that of a Gaussian beam, the obstruction is made using the
sameapproach,butwiththediameterchosensothattheobjectblocks∼20%ofthelight,asinthenon-Gaussianmodecase.
The FWHM in Fig. 4 are found via Gaussian fitting, and the focal positions z are taken to be the positions where the
focus
unobstructedFWHMofthecentralpeaksareminimal.
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Acknowledgments
WeacknowledgefundingfromtheLouisianaStateBoardofRegentsandNorthropGrummanNG-NEXT.
Author Contributions
J.D.Sperformedtheexperiments,withcontributionsfromK.N.D.,E.M.K.andC.R.J.D.Sperformedthedataanalysis,with
contributionsfromO.D.,andJ.D.SandR.T.Gwrotethemanuscript. R.T.G.conceivedtheproject.
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Additional Information
SupplementaryinformationaccompaniesthispaperatINSERTURL.
Competingfinancialinterests: Theauthorsdeclarenocompetingfinancialinterests.
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