Contents

1 Introduction 9

IFI vight sttering F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W IFP sntertions led to olletive modes F F F F F F F F F F F F F F F F F F F F F F IH IFQ esonnt light sttering from dense loud of old toms F F F F F F F F F F IH IFR opis of this thesis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F II IFS gontent F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IP
2 Classical and quantum treatment of light scattering 13 PFI olrizility nd eletri (eld of resonnt dipole F F F F F F F F F F F F F F IR PFP vight sttering from n ensemle of stterers F F F F F F F F F F F F F F F F F IS PFPFI vight sttering from single tom F F F F F F F F F F F F F F F F F F F IS PFPFP vight sttering from multiple toms F F F F F F F F F F F F F F F F F F IT PFPFQ sntrodution of ensemleEverged nd )ututing eletri (elds F F F IV PFPFR feyond the menE(eld theory of optis F F F F F F F F F F F F F F F F F PH PFQ eltion etween men free pth nd system size F F F F F F F F F F F F F F F PI PFR inergy relxtion in quntum optis F F F F F F F F F F F F F F F F F F F F F F PI PFS irtul photons from lssil perspetive F F F F F F F F F F F F F F F F F F PW PFT ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QI 3 Light scattering from a dense cloud of cold atoms 33

QFI wotivtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QQ QFP tte of the rt F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QS QFQ ixperimentl setup F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QV QFQFI etomi density F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QV QFQFP hoppler e'et F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QW QFQFQ roe em nd tomi trnsition F F F F F F F F F F F F F F F F F F F F QW QFQFR wesured signl F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RH QFR petrl hnge while rossing the fol regionX qouy phse F F F F F F F F F RI QFRFI oy model for light sttering F F F F F F F F F F F F F F F F F F F F F F RP QFRFP ixplntion of symmetri spetrum F F F F F F F F F F F F F F F F F F RR QFS ttering suppression F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RS QFSFI ixperimentl results of ellegrino et al. F F F F F F F F F F F F F F F F F RS QFSFP xumeril lultions on suppression of light sttering F F F F F F F RT QFSFQ owrds oherent light sttering F F F F F F F F F F F F F F F F F F F F RV QFT petrl response of the tomi loud F F F F F F F F F F F F F F F F F F F F F F RV QFTFI snoherent spetrl response F F F F F F F F F F F F F F F F F F F F F F F RV QFTFP goherent spetrl response F F F F F F F F F F F F F F F F F F F F F F F F SH QFU uperluminl pulse propgtion through n tomi loud F F F F F F F F F F F SP gyxix P QFV ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SS 4 Polaritonic modes in a dense cloud of cold atoms RFI wotivtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SU RFP sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SU RFQ wodel F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SW RFR iigenspetrum nlysis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TI RFS vight sttering F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TR RFT i'etive dieletri onstnt F F F F F F F F F F F F F F F F F F F F F F F F F F F TS RFTFI enderson loliztion F F F F F F F F F F F F F F F F F F F F F F F F F F F TT RFU gonnetion etween polritoni modes nd mrosopi modes F F F F F F F F TU RFV vight sttering from polritoni modes F F F F F F F F F F F F F F F F F F F F F TV RFW esorption F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UI RFIH gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UP
5 Revisiting homogenization for interacting resonant scatterers

SFI wotivtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UQ SFP sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UQ SFPFI ht is homogeniztionc F F F F F F F F F F F F F F F F F F F F F F F F F UQ SFPFP romogeniztion in optis F F F F F F F F F F F F F F F F F F F F F F F F F US SFPFQ gontents F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F US SFQ ystem desription F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UT SFR xonEresonnt light sttering F F F F F F F F F F F F F F F F F F F F F F F F F F UU
SFRFI i'et of position orreltions on nonEresonnt light sttering F F F F UW SFRFP i'et of nonrditive dey on nonEresonnt light sttering F F F F F VI SFS esonnt light sttering F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VI SFSFI ttering spetr for oherent nd inoherent light F F F F F F F F F F VP SFSFP i'et of nonrditive dey nd dephsing on resonnt light sttering VQ SFT qint )ututions of the sttered power y resonnt toms in suwveE length volume F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VR SFTFI ttering spetrum for suwvelengthEsize tomi loud F F F F F F F VS SFTFP everge mode sping F F F F F F F F F F F F F F F F F F F F F F F F F F VT SFTFQ gondition for nonoservtion of singleEphoton superrdine F F F F F VU SFTFR houless riterion for enderson loliztion F F F F F F F F F F F F F F VV SFU uperrdine from orderedD dense tomi louds

F F F F F F F F F F F F F F F VW SFV gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VW
6 Theory of electroluminescence by quantum-dots based metasurface lightemitting devices

TFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WI TFP iletroluminesene from quntum dots F F F F F F F F F F F F F F F F F F F F WP TFPFI untumEdot sed eletroluminesene F F F F F F F F F F F F F F F F WP TFPFP ristory of quntumEdot sed lightEemitting devies F F F F F F F F F F WQ TFQ untumEdot sed metsurfe lightEemitting devie F F F F F F F F F F F F F WQ TFQFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WR TFQFP ixperimentl results F F F F F F F F F F F F F F F F F F F F F F F F F F F
F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHI TFSFP i'etive refrtive index lultion F F F F F F F F F F F F F F F F F F IHP TFT epplition of eletroluminesene model to experiments F F F F F F F F F F F IHR TFTFI iletroluminesene enhnement y nnorings F F F F F F F F F F F F IHR TFTFP olrized eletroluminesene F F F F F F F F F F F F F F F F F F F F F F IHS TFU gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHT

Conclusion and outlook 109

Appendices 113

A Quantization of evanescent waves in absorbing media 115 eFI untiztion of the eletromgneti (eld in dispersive nd soring medi F IIT eFP ivnesent wve opertor F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIT eFQ gommuttion reltion for evnesent wve opertors

F F F F F F F F F F F F F IIU eFR gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIV
B Numerical calculations on o-axis source in a cylindrically symmetric nanostructure 119

fFI ystem desription F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIW fFP xumeril method F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIW fFPFI iletri (eld deomposition F F F F F F F F F F F F F F F F F F F F F F F IIW fFPFP oure deomposition F F F F F F F F F F F F F F F F F F F F F F F F F F F IPI fFPFQ smproving eletri (eld lultion F F F F F F F F F F F F F F F F F F F IPP fFPFR gonvergene F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPQ
yne intertion tht ws ruil in the professionl hoies s hve mde fterwrdsD ws with ymr il qwhry t helft niversity of ehnology in the demi yer PHIHEPHIIF re orgnized journl lu on negtive refrtionF foth the topi nd the inner enthousism on physis of ymr onvined me to do oth my fF nd wF theses in the optis reserh group in helft under supervision of ul rhF sn helft s hd the opportunity to work with ilvni ereir nd eurèle edm s wellF s would like to thnk ll of them for the profound sienti( disussions we hve hdD ll of your enE thousismD pssion nd the uriosity in my wster thesis work on hyperoli metmterilsF wy next step ws doing hFhF t snstitut d9yptiqueF s m very grteful to oth tenEtques qre'et @nnophotonis nd eletromgnetism reserh groupA nd entoine froweys @quntum optis reserh groupA for hving greed on working during three yers on n interdisiplinry topi with me tht hs led to the urrent mnusriptF es from the eginning onD people told me tht s ws very luky with oth tenEtques nd entoine s eing my supervisorsD sine they re oth gret sientists nd pedgogilly strongF efter hving worked with themD s n pprove thisF he reurrent fternoon meetings tht sometimes ontinued during the eginning of the evening perfetly illustrte to wht extent we were le to lern nd disuss togetherF enother spet of oth tenEtques nd entoine tht s would like to highlight is their vililityF heir o0e doors re lwys open nd they re lwys eger to disussD even though they oth hve lot of ommitmentsF hnks for this s well3 eprt from tht s would like to thnk tenEtques for the mny opportunities he gve me to present my work outside the instituteF hnks to hving ttended severl onferenes nd workshopsD s ould lern wht other reserh is done in other groups in prne nd elsewhere nd s ould extend my professionl networkF s hve lso hd the hne to teh tutorils of ourses tht were tught y tenEtquesF re oth trusted me in tehing his tutorils in prenh nd he supported me nd the other tehing ssistnts perfetlyF hnk you very muh for hving een my ollegue nd supervisor over the lst three yers3 ou gve me plenty of useful nedotes nd lessons tht s will tke with meF omething prtiulr to entoine tht s ppreited re the pts on the shoulder s ould reeive in good nd less good momentsF s elieve he hs very unique nd soil wy of leding temF he re for your students trnslted in the ft tht youD while eing group lederD regulrly even limed two stirs to disuss with me out physisF s lso wnt to thnk entoine very muh for hving een my ollegue nd supervisor over the lst three yers3 gyxix T wy reserh eing theoretilD s oviously hd strong intertion with tenEul rugonin s wellF hnks to tenEul s ould numerilly study nonEtrivil eletromgE neti prolems in n unpreedented ury nd speedF ris door is lso lwys open nd he is lwys very hppy to disuss out new prolemF elthough we elieve tht time progresses with the sme pe for ll of usD sometimes s hd the impression tht the time of tenEul goes slowerX he relizes powerful odes in suh very short time3 ris quik responseD powerful odes nd still very modest personlityD mke him legend inside the groupF hnk you so muh for the lst three yers tenEul3 s will tke retiolo with meD wherever s will eF ghristophe uvn is my next ollegue tht s would like to thnk lotF eprt from eing very nieD kind nd modest personD he is lso very rigorousF henever he sys somethingD you n e sure tht he hs thought out it thoroughlyF s ppreited this qulity lotF yther hrteristis of ghristophe is tht he tkes re of his olE legues nd is le to rete hrmonyF hnk you very muh ghristophe for hving worked together the lst three yers nd showing me very rigorous nd kind wy of doing physisF he intertions with vn ortis were very fruitful s wellF vn hd strong nd positive in)uene on the ppers nd presenttionsF henever he nlyzes doument or presenttionD you n e sure tht everything hs e tken into onsidertionsF yn top of tht he hs good sense of humorF hnk you very muh vn for your input in the projet over the lst three yersF xextD s would like to thnk the jury memers for hving epted to red nd omment my thesisD espeilly knowing tht the heemer month is typilly (lled with mny hFhF defenses in prne3 hnk you ierre illetD émi grmintiD oin uiserD egnès wître nd eloyse hegironF huring my thesis s hve hd the opportunity to teh lworks nd tutorilsF ehing eing one of my pssionsD s m very grteful to my ollegues with whom s ould shre this pssionF sn prtiulrD s wnt to thnk wtthieu fo'etyD tenEtques qre'etD vionel tuowiezD prnçois wrquierD vn ortis nd tulien illemejneF edemi reserh groups re known for eing very dynmilD iFeFD the group memers hnge lot in ourse of timeF s hve hd the osion to tlkD disussD lugh nd shre enthousism for physis with mny ollegues during my thesisX hilippe fenEedllhD renri fenistyD wondher fesesD plorin figourdnD gédri flnhrdD wrieEghristine hheurD son hoyenD fenjmin rertD tephn tenneweinD enthony touninD wthieu vnglisD svn vtellD uentin vévesqueD prnçois wrquierD enton yvhrenkoD imilie ktD urthik sihithluD sln hlesingerD fenjmin estD véo ojszvzykD gheng hngF s would like to thnk eh of you very muh for either your input in my hFhF or @the mthemtil orA the very nie time we hve hd t the institute togetherF yne should not underestimte the importne of hving nie people surrounding you t workF ou were de(nitely very nie people nd s eome quite melnholi when thinking out the mny funny moments s hve hd with youF wo of these moments were de(nitely in the south of pin nd in olendmD where s went to with few of youF et lst s would like to thnk sln hlesinger nd fenjmin est who gurntee the ontinution of my work nd moreF fsed upon our intertions s m sure tht the projet is luky of hving oth U gyxix of you involvedD sine you form strong experimentl nd theoretil tem3 eprt from hving luk t work with my olleguesD s ws lso very luky for getting n pprtment t the gité interntionle universitire de risF yver the lst three yers s hve met n enormous mount of people over thereF ell of them forming together very interntionlD soilD intelligent nd diverse group of friendsF s would like to highlight few friends with whom s ws in the residents9 ommittee PHIRGPHIS of gollège prnoEfritnniqueX osr foussidD ilen grloniD qeorges srhimD snn svnovD ilis wkoukjiD idurdo ueiroz eresD xtly moilenko nd hiyn hoF iver sine our mndteD we sty in touh nd we orgnize tivities togetherF his nnot e onsidered ovious nd it is lso therefore tht s herish the moments with you guysF hnk you so muh for our friendship nd your support during my thesis3 huring my thesis s went home every now nd then to meet up with fmily nd friends over thereF st is during the moments you do not see them lot tht you ppreite them muh more thn eforeF iverytime s me k s ould expet some phonells of friends to meet upF wy weekends here were never oringD thnks to themF wy fmily ws lso lwys redy to mke my weekend perfet nd s sw their hppiness of hving their lost hild with themF s highly dmire my grndprents nd prents for the wy they delt with their (rst hild or grndhild hving plns to go rod for long timeF hey supported me with everything nd they hve lwys trusted in meF s ould even ount on some visits of them in risD thnk you so muh3 ot slot zou ik in het ijzonder mijn overgrootvder iet rutzezon willen ednkenF rels overleed hij enkele weken voor mijn vertrek nr rijs voor mijn promotieonderzoekD mr hij wist dt en wrom ik nr rijs zou vertrekken voor drie jrF sk voel me zeer evoorreht dt ik tot mijn PQ9ste heel veel met u he mogen sprekenF ret onvoorwrdelijke vertrouwen dt u in mij hdD zl ik nooit meer vergeten3

Chapter 1 Introduction sn this hFhF thesis we study theoretilly light sttering nd emission y denseD disorE deredD nd resonnt systemsF wore spei(llyD we study @IA light sttering y dense loud of old tomsD nd @PA light emission y dense (lm of quntum dots tht is oupled to plsmoni metsurfeF his hFhF projet is prt of ollortion etween the group of profF drF tenEtques qre'et nd the group of drF entoine froweysF foth groups re prt of vortoire ghrles pry t snstitut d9yptique in liseu @prneAF he group of froweys hs onduted experimentl reserh on resonnt light sttering from louds of old ruidiumEVU tomsF his work onerns the sme tomi systemsD ut the study is theoretilF sn this (rst hpter we introdue the min topis of interest of this hFhF thesisF 1.1 Light scattering vight sttering is the physil phenomenon of the de)etion of light due to the enounter of n ojetF ome exmples of everydy situtions where light sttering ours reX light sttering y moleules in the skyD re)etion of light y rough surfeD nd light sttering y milkF hese exmples illustrte tht the onsequenes of light sttering re experiened y everyone on dily sisF elthough light sttering is pereived y everyone nd there exists extensive literture of this topiD there is still lot of reserh going on in the (eld of light sttering from rndom mediF hen we onsider light sttering from disordered mediD we relize there re tully mny di'erent regimes of light stteringF ih regime is hrterized y set of prmetersF ixmples of suh prmeters reX @IA resonnt or nonEresonnt light stteringD @PA the size of the stterersD @QA the density of stterersD @RA the size of the ensemle of stterersD @SA elsti or inelsti stteringD @TA the energy level struture of the stterersD @UA position orreltions of stterersD @VA the osilltor strength of the eletri dipole trnsitionD nd @WA the intensity of the inident emF he lrge vriety of light sttering regimes mkes it n interesting topiF e re minly interested in the interply etween light sttering nd intertions etween stterersF he djetive dense in this thesis is relted to the strength of these intertionsF es will eome ler in ghpter PD loud is onsidered dense for resonnt light sttering when the verge interEstterer distne r < λ/(2π)D where λ is the resonnt wvelengthF nder this onditionD the intertions etween the stterers re signi(nt ompred to the oupling etween the stterers nd the vuum eletromgneti (eldF grei IF sxyhgsyx IH 1.2 Interactions lead to collective modes ine we re interested in the e'et of strong intertions etween resonnt stterers in disordered ensemle on light stteringD let us (rst onsider some wellEknown situtions enE ountered in solid stte physis of ordered systems in whih intertions hve n importnt in)uene on the modes of the systemF pirstD we onsider one dimensionl periodi hin of N tomsD eh tom eing le to move round its equilirium positionF hen we ssume there is no intertion etween the tomsD the toms n e onsidered independentF his simple model orresponds to the iinstein model of solid where ll toms osillte t the sme frequenyF xextD we inlude intertions nd we ssume the tomi displement to e smllD so tht the oupling etween djent toms n e modeled y hrmoni springF he introdution of n intertion leds to norml modesF he mehnil norml modes tht exist in solid stte medium re lled phononsD whih re collective lttie virtionsF he N degenerte energy levels eome phonon nd struture due to the intertionF he importnt messge here is tht olletive response of the system is due to the presene of n intertion etween the individul prtilesF his intertion hnges the wy the ensemle of prtiles should e studiedX not s n ensemle of independent prtilesD ut s n ensemle of olletive modesF imilrlyD the hrude model of eletri ondutivity trets eletrons s independent prtiE les nd the eletri resistivity origintes from ollisions of eletrons on ny ojet tht they enounter during eletroni trnsportF st is under the introdution of eletronEeletron inE tertions @goulom intertionA tht olletive mehnil exittions re preditedD lled plsmons IF lsmons re norml modes of the systemF glose to the plsmon frequeny the response of eletrons to n externl exittions strongly devites from the hrude model tht trets eletrons s independent entitiesF st is gin the presene of intertions etween individul prtiles tht gives rise to olletive modesF 1.3 Resonant light scattering from a dense cloud of cold atoms imilr to the two exmples given in the previous setionD there exists n intertion eE tween toms due to light indued dipolesF his intertion is the resonnt dipoleEdipole intertionF e n modify the intertion strength etween toms in two wysX either y vrying the verge interEtomi distne or y hnging the frequeny of the inident wve with respet to the resonnt frequeny of eh tomF e disordered loud of old toms is therefore n interesting pltform to study the interply etween light sttering nd the intertion etween the stterersF roweverD reting strongly interting loud of toms is omplitedF sn order to hve strongly interting loud of tomsD one needs to respet the following two onditionsX @IA ll toms need to hve lrge dipole momentD so they ll need to e resonntly exited t the sme timeF hereforeD they ll need to hve the sme resonne frequenyF he presene of the hoppler e'et in for exmple hot tomi vpor leds to spred of the resonne frequeniesD e'etively reduing the resonnt dipoleEdipole intertionsF reneD we need old tomi loudF @PA he old tomi loud needs to e denseD iFeFD mny toms need to e present inside volume of λ 3 F II IFRF ysg yp rs ris ixperimentllyD it turns out to e di0ult to stisfy oth onditionsF st is round the time this hFhF projet strtedD ytoer PHIQD tht the group of froweys hd demonE strted tht it is possile to lod wvelengthEsize dipole trp with few hundred old tomsD so tht the tomi loud n e onsidered dense ording to the de(nition of dense in this work P!RF fy oth vrying the numer of toms inside the trp nd the lser frequenyD we re in the position to study the trnsition of independent light sttering towrds sttering medited y olletive modesF 1.4 Topics of this thesis sn order to hve strong intertions etween dipolesD we need n tomi system where ll dipoles hve lrge dipole momentF fy studying light sttering from dense louds of old tomsD we try to stisfy this onditionF roweverD strong intertions re themselves elieved to prevent ny oherene e'ets to our for subwavelength-size tomi systems SD TF he prevention of oherene e'ets n e understood y the ft tht very strong resonnt dipoleEdipole intertions etween toms n hnge their resonne frequenyF es the toms re rndomly pled in trpD they ll experiene di'erent intertion with the other tomsD therey leding to strong inhomogeneous rodeningF yn the other hndD we might expet tht intertions generte olletive modes similr to phonons or plsmonsF hile these modesD lled polritonsD re introdued in ordered systemsD we my expet to oserve delolized modes for rndom systemsF uh delolized modes would produe sptil oherene in disordered systemF reneD we sk ouselves the questionX does collective light sttering our in dense and wavelength-size loud of old tomsc his question n e formulted di'erentlyX should dense nd old tomi loud e modeled s n ensemle of independent stterers or in terms of olletive modesD even though interE tomi intertions n e so strong tht the resonne frequeny of eh tom hngesc he ove question is relted to the question if dense loud of old toms n optilly e desried y refrtive indexF sf suh loud n e desried y refrtive index of refrtionD light di'rtionD whih is intrinsilly oherene e'etD n tke ple for wvelengthEsize loudF he mount of di'rted light depends nonElinerly on the index of refrtionF imilrlyD if the dipole moments of the toms re ligned nd the numer of toms inresesD it is expeted tht the ensemle emits more thn n ensemle of indeE pendent emitters would doD due to onstrutive interferenes etween the emitted eletri (eldsF e will study how the refrtive index emerges s the tomi density inreses nd lso how olletive eigenmodes of the tomi system emerge @or notAF fy doing soD we will e le to study the onnetion etween mrosopi model @refrtive indexA nd mirosopi model @tomsA of the tomi loudF hen one studies system for whih the density grdully inresesD question tht rises is how the system moves from few toms present in the loud tht stter light in ll diretionsD towrds homogeneous prtile desried y n e'etive refrtive index tht di'rts lightF uh prolem is lled homogeniztion prolemF e question tht will e ddressed in this work n e formulted s followsX is it possile to desrie the optil response of dense tomi loud for resonnt light sttering uniquely y n e'etive refrtive indexc enother question isX is it lwys possile to reh the homogeniztion regime y inresing the densityc heres this seems to e the se when moving from dilute gs to liquid phseD we will see tht suwvelength volumes of strongly interting old toms nnot e homogenizedF grei IF sxyhgsyx IP ine we re interested in the e'ets of strong resonnt dipoleEdipole intertions on the optil ehvior of dense ensemle of stterersD we will lso study dense ensemle of quntum dotsF wotivted y reent experiments performed in the group of drF eloyse hegiron t the snstitut d9Életronique pondmentleD yrsy @prneAD we will study light emission from dense ensemle of quntum dots emedded in system tht ws studied in tht groupF sn this groupD they study experimentlly the pplition of quntum dots in quntumEdot sed lightEemitting deviesF en importnt question for the modeling of these devies is if these quntum dots ould e modeled s individul quntum dots or if they should e modeled s n ensemle for whih olletive modes n e de(nedF he nswer to this question hs ig in)uene on the wy suh quntumEdot sed lightEemitting devie needs to e modeledF 1.5 Content his thesis is orgnized s followsX Chapter 2 desries the theory tht is needed to understnd light sttering s it is treted in wht followsF et the end of the hpter the relxtion of energy in n tomi system is treted quntum mehnillyF his tretment will llow to estlish onnetion etween @lssilA evnesent wves nd @quntumA virtul photonsF sn Chapter 3D we will use the notions s introdued in ghpter P to study theoretilly the experimentl results otined y the group of froweys RDU on light sttering from mirosopi loud of old ruidiumEVU toms for whih strong resonnt dipoleEdipole intertions re expetedF prom tht hpter it will follow tht light sttering from mirosopi loud ontining few hundred old toms ours prtilly s if the loud were homogeneous medium desried y refrtive indexF sn Chapter 4 we introdue the onept of polritoni modes in the mirosopi desription of n tomi loud nd identify these polritoni modes with pryEerotElike modes of n e'etive prtileF st follows tht light sttering is dominted y only few modes out of ll the olletive modes tht existD nmely the polritoni modesF his hpter omplements efsF VD W y studying oth the eigenvlues nd the eigenvetors t the sme time nd proving the ides tht were presented in the forementioned worksD ut this time lso for dense tomi systemsF sn Chapter 5 we will tret nother spet of light stteringD nmely uctuations in light stteringF e sk ourselves the question if resonnt tomi system n e homogenizedD iFeFD is it possile to desrie the optil response of n tomi loud uniquely y n e'etive refrtive indexc his question is importnt for the understnding of reent experimentl works on resonnt light stteringF sn the lst hpterD Chapter 6D we reple the toms y inhomogeneously rodened olloidl quntum dotsF e disuss the optil desription of dense ensemle of quntum dotsX n the quntum dots e seen s independent entities or should they e onsidered s eing dependentF he nswer to this question is needed to develop model for the study of eletroluminesene from dense (lm of olloidl quntum dots pled in ner viinity of plsmoni nnontenns IHD IIF e propose model to desrie the oupling etween quntum dots nd twoEdimensionl rry of plsmoni nnontennsF he presene of this plsmoni metsurfe leds to modi(tion of the eletroluminesene signlF he proposed model of eletroluminesene is n dpted version of the onventionl theory of eletroluminesene from lightEemitting diodeF Chapter 2 Classical and quantum treatment of light scattering sn this thesis we study light sttering from n ensemle of stterersD eFgFD toms or qunE tum dotsF sn ghpter I we noted tht there exist mny regimes of light stteringF e re interested in light sttering from strongly interting systemsD where the intertion is the resonnt dipoleEdipole intertion in this workF trong intertions my our when stE terers re in the nerE(eld region of eh otherD iFeFD when the interEprtile distne is muh smller thn the wvelength of the inident wveF hue to strong resonnt dipoleEdipole inE tertions in the tomi systems we studyD light typilly gets sttered mny times efore it leves the tomi loudF e therefore need to onsider light sttering tht goes eyond the single sttering regimeD whih is lled multiple sttering regimeF sn this hpter we introdue the neessry onepts to understnd multiple sttering theoryF his hpter supports the min results tht re presented in this thesisF sn ghpter T we will disuss the intertion etween light nd n ensemle of quntum dotsF es opposed to old toms tht ll hve the sme resonne frequenyD quntum dots re inherently inhomogeneously rodenedD iFeFD eh quntum dot hs di'erent resonne frequenyD whih is due to fritionF snhomogeneous rodening leds to weker resonnt dipoleEdipole intertionsF he disussion out multiple sttering will llow us to de(ne prmeter tht n quntittively distinguish oth regimes of strong nd wek resonnt dipoleEdipole intertionsD oth for toms nd quntum dotsF vight sttering is n exmple of topi tht is of interest for oth the lssil nd quntum optis ommunitiesF sn this hpterD we will lso disuss light sttering within the quntum optis frmeworkF st is expeted tht oth theories give di'erent results for light stteringD sine n eletri dipole trnsition in quntum optis n e sturtedD whih is not possile for lssil dipoleF elthough dipole in quntum optis is therefore fundmentlly di'erent thn one in lssil optisD we will show tht there is regime in whih oth lssil nd quntum optis predit the sme results for light stteringX the wekEexittion regimeF xo sturtion e'ets re present in the wekEexittion regimeF sn strongly interting systemsD dipoles re within eh other9s ner (eldF st is known from lssil optis tht evnesent wves exist in the nerE(eld region of ny dipole nd tht they hve mjor in)uene on the lssil resonnt dipoleEdipole intertion when two stterers re in eh other9s ner (eldF prom quntum optis point of viewD systems tht re in eh other9s viinity re interting vi oth rel nd virtul photonsF sn the lst setion of this hpterD we will disuss how @lssilA evnesent wves re relted to @quntumA virtul photonsF grei PF gvesgev exh exw iewix yp vsqr geisxq IR 2.1 Polarizability and electric eld of a resonant dipole sn this setion we rell the eletri (eld reted y n eletri dipole nd the optil response of resonnt dipole upon illumintionF por thisD we strt with more generl sitE ution whih will e useful in ghpter T on quntum dots emedded in n inhomogeneous eletromgneti environmentF e ssume to hve ontinuous eletromgneti environE ment desried y reltive permittivity or dieletri onstnt ε r D nd the presene of some eletromgneti soures @eletri dipolesD qudrupoles etFAF he soures re represented y the polriztion density distriution PD where the tilde indites tht the quntity hs een pourier trnsformed with respet to the time vrile IPX

(∇ 2 1 -∇∇) Ẽ(r, ω) + ω 2 c 2 ε r (r, ω) Ẽ(r, ω) = -µ 0 ω 2 P(r, ω), @PFIA
where ω is the ngulr frequenyD c = 1/ √ ε 0 µ 0 the speed of light in vuumD ε 0 the permittivityD µ 0 the permeilityD nd P(r, ω) is relted to the urrent density J y P(r, ω) = (-iω) -1 J(r, ω)F prom now onD we will omit the tilde ove the pourierE trnsformed quntities nd lso the frequeny vrile within rketsY oth re impliit in the nottion E(r)F e notie tht we hve n inhomogeneous di'erentil equtionD where the polriztion density is the soure of n eletromgneti (eldF he prtiulr solution of the di'erentil equtionD iqF PFID is

E(r) = µ 0 ω 2 ˚Ḡ (r, r )P(r )d 3 r , @PFPA
where Ḡ(r, r ) is the solution of iqF PFI with -µ 0 ω 2 P(r) eing repled y the hir distriE ution -δ(r-r ) 1F he doule r indites tht the quntity is mtrixF Ḡ(r, r ) is lled the qreen9s tensor or dydi qreen9s funtionF he qreen9s tensor desries the eletri (eld produed y pointlike soureF sn this thesis we minly disuss light sttering y n ensemle of stterers in homogeneous mediumF por homogeneous medium with refrtive index nD the qreen9s tensor for r = r reds

Ḡ(r, r ) = 1 + 1 k 2 ∇∇ G(r, r ), @PFQA
where

G(r, r ) = e ik|r-r | 4π|r -r | @PFRA
is the slr qreen9s funtionD with k = nω/cF nless stted di'erentlyD the stterers re pled in vuumD so n = 1F hroughout this work we onsider light sttering y n ensemle of stterersF e ssume the stterers to e very smll ompred to the optil wvelength nd therefore we use the eletri dipole pproximtionF he polriztion density in iqF PFI n then e repled y the eletri dipole moment pD ording to P(r) = i p i δ(r -r i )D where the susript i indites to whih prtile the dipole moment elongsF sn liner optisD the dipole moment nd the eletri (eld re linerly relted y the polrizility mtrix ᾱ(ω)X

p i = ε 0 ᾱ(ω)E(r i ), @PFSA
where the eletri (eld is the totl eletri (eld t the position of the dipoleD prt from the self (eldD iFeFD the eletri (eld produed y the dipole itself nd evluted t its own positionF IS PFPF vsqr geisxq pyw ex ixiwfvi yp geii sn this workD the stterers re tken to e lssil stterers nd therefore isotropi @in tomi physis lssil resonnt stterer orresponds to J = 0 → J = 1 trnsitionD where J is the totl ngulr momentum nd is the sum of the spin ngulr momentum S nd the oritl ngulr momentum LAF por isotropi stterersD the polrizility is slr quntity ᾱ(ω) = α(ω) 1F he lssil polrizility is

α(ω) = 6πc 3 Γ 0 ω 2 0 (ω 2 0 -ω 2 -iωΓ 0 ) , @PFTA
where ω 0 is the ngulr frequeny orresponding to the energy di'erene etween two energy levels nd Γ 0 is the spontneous emission rteF glssillyD Γ 0 is relted to the power emission of n osillting dipole IPF hen nerEresonne light sttering is studiedD the pproximtion ω + ω 0 ≈ 2ω 0 n e mdeD whih gives ommonly used form of the polrizility

α(ω) = 3πc 3 Γ 0 ω 3 0 ω 0 -ω -i Γ 0 2 . @PFUA
he imginry prt of the polrizility is vorentzin funtionY funtion whih will often return in the next hptersF por ompletenessD let us note tht this polrizility is lso vlid for twoElevel system IQF roweverD the optil response of twoElevel system depends on the polriztion of the eletri (eldF he digonl elements of the polrizility mtrix re therefore not the smeF 2.2 Light scattering from an ensemble of scatterers e hve now the tools to lssilly desrie light sttering from n ensemle of eletri dipolesF 2.2.1 Light scattering from a single atom pirstD let us disuss light sttering from single tom pled t position r 0 = (0, 0, 0)

in vuumF e plne wve propgting long the z xis with ngulr frequeny ω = ck

E(r) = E inc exp(ikz) is inident upon tht tomF prom iqF PFS it is seen tht the indued dipole moment is given y p = ε 0 α(ω)E inc F eording to iqF PFPD the osillting dipole retes n eletri (eld E(r) = ω 2 c 2 α(ω) Ḡ(r, 0; ω)E inc , @PFVA
where r is the oservtion pointF he totl sttered power is given y

P sc (ω) = ω 4 c 4 |α(ω)| 2 6π ε 0 c 2 |E inc | 2 . @PFWA
hen ω ω 0 D we otin the wellEknown ω 4 dependene of the sttered power for yleigh sttering IRD ISF roweverD for nerEresonne light sttering @ω ≈ ω 0 AD the presene of resonne in the polrizility α(ω) drmtilly lters light stteringF prom the totl sttered powerD we n dedue the e'etive sttering ross setionD whih is de(ned s P sc (ω) = σ sc (ω)|S|, @PFIHA grei PF gvesgev exh exw iewix yp vsqr geisxq IT where S is the oynting vetor of n inident plne wveF por plne wve |S| = ε 0 c/2 × |E inc | 2 D so tht the e'etive sttering ross setion is given y

σ sc (ω) = ω 4 c 4 |α(ω)| 2 6π .
@PFIIA e note tht on resonne σ sc = 3λ 2 0 /(2π)D with λ 0 eing the resonne wvelengthF he e'etive sttering ross setion is t resonne muh lrger thn the geometril ross setion of single tomF 2.2.2 Light scattering from multiple atoms he presene of more thn one stterer mkes light sttering in generl more omplitedF his n e seen y repling

P(r) = N i p j δ(r -r i ) = N i ε 0 α(ω)E(r j )δ(r -r i ) @PFIPA in iqF PFIX (∇ 2 1 -∇∇)E(r) + ω 2 c 2 E(r) = - ω 2 c 2 α(ω) N i E(r)δ(r -r i ).
@PFIQA he generl solution of this di'erentil eqution is

E(r) = E inc (r) + ω 2 c 2 α(ω) N i Ḡ(r, r i )E(r i ), @PFIRA
where the inident (eld E inc is homogeneous solution of iqF PFIQF es opposed to the sttered eletri (eld of single tom @see iqF PFVAD it is seen tht the generl solution given y iqF PFIR is in impliit formF here re t lest three di'erent wys of solving this set of equtionsF he (rst one is y solving iqF PFIR numerillyF his is wht will e onsidered in the urrent setionF wo other wys re sed on series expnsion of the eletri (eldF yne expnsion is lled the forn series expnsionF ith this expnsion the eletri (eld is sustituted reursively in iqF PFIRF hen the reursion is stopped fter pplying it oneD one hs pplied wht is lled the (rstEorder forn pproximtionF wore generllyD when the reursion is stopped fter n itertionsD one hs pplied the nthEorder forn pproximtionF sn mny sesD like for light sttering from the skyD the forn series expnsion is very onvenientF roweverD when nerEresonne light sttering is onernedD the forn series expnsion is less usefulD sine it is divergent series ner resonne ITF he other series expnsion we mention in this thesis is soElled digrmmti expnsion IUF he digrmmti expnsion leds to perturtive solution of the sttered eletri (eld in terms of the numer density of stterersF st is sed on deomposing the eletri (eld in n ensemleEverged eletri (eld nd )ututing eletri (eldF he digrmmti expnsion llows to lulte the e'etive dieletri onstnt of the rndom mediumF st turns out tht the digrmmti expnsion nd the deomposition of the eletri (eld led to more intuitive interprettion of light sttering thn oth the forn expnsion nd the ext solution for systems tht re in the multiple sttering regimeF he dvntge of series expnsion s ompred to numerilly (nding the solution is tht it helps to rete IU PFPF vsqr geisxq pyw ex ixiwfvi yp geii some physil intuitionF etD one should e reful to know to wht extent the expnsion is vlidD sine otherwise wrong onlusions might e drwnF e strt with looking for the ext solution of iqF PFIQF snsted of inverting iqF PFIR immeditelyD we will pss y set of oupledEdipole equtionsD sine they will e useful lter in this hpterF he eletri (eld in iqF PFS is given y

E(r i ) = N j=1,j =i µ 0 ω 2 Ḡ(r i , r j )p j + E inc (r i ), @PFISA
where the dipole moments re the unknownsF ustitution of iqF PFIS into iqF PFS gives set of oupledEdipole equtions in the presene of n inident (eld

p i = ω 2 c 2 N j=1,j =i α(ω) Ḡ(r i , r j )p j + ε 0 α(ω)E inc (r i ).
@PFITA por lssil dipolesD there re 3N unknowns @N dipolesD eh one hving three dipole moment omponentsA nd 3N independent equtions @eh dipole moment omponent from eh dipole hs its own equtionAF his set of equtions n e written in mtrix formX

Ā(ω)P = ε 0 3πc 3 Γ 0 ω 3 0 E inc , @PFIUA
where we hve de(ned

P =      p 1 p 2 F F F p N     
, whih should not e onfused with the polriztion density vetorD

E inc =      E inc (r 1 ) E inc (r 2 ) F F F E inc (r N )      , nd the mtrix Ā(ω) =       (ω 0 -ω -i Γ 0 2 ) 1 -3πω 2 cΓ 0 ω 3 0 Ḡ(r 1 , r 2 ; ω) . . . -3πω 2 cΓ 0 ω 3 0 Ḡ(r 2 , r 1 ; ω) (ω 0 -ω -i Γ 0 2 ) 1 F F F F F F      
, of whih ll sumtries re of size 3 × 3F his form of Ā ssumes the tomi polrizility to e given y iqF PFUF prom this mtrix equtionD the vetor PD whih ontins ll dipole momentsD n e otined y inverting the mtrix ĀF fy solving the set of oupledEdipole equtionsD we (nd the dipole momentsF yne the dipole moments p i re knownD the totl eletri (eld n e lultedX

E(r) = µ 0 ω 2 N i=1 Ḡ(r, r i )p i + E inc (r)D for ll r di'erent from r i .
@PFIVA grei PF gvesgev exh exw iewix yp vsqr geisxq IV e stress tht the herey otined eletri (eld is ext nd does not introdue new pproximtions to the one lredy given in etion PFIX lssil dipole ssumptionD eletri dipole pproximtionD the dipoles hve single resonne frequeny ω 0 F his is s opposed to ny trunted series expnsionF hen series expnsion is truntedD not ll resonnt dipoleEdipole intertions re properly tken into ountF e will rie)y disuss this in etion PFPFRF sn this setion we desried n pproximte wy of lulting the sttered eletri (eld y mens of forn series expnsion nd n ext wy of solving light sttering from n ensemle of N stterersF sn the next setion we will introdue the ensemleEverged nd )ututing eletri (eld tht re neessry within the digrmmti expnsionF he introdution of these quntities will e prtiulrly useful in ghpter QF

Introduction of ensemble-averaged and uctuating electric elds

enother wy of studying light sttering from rndom medium follows from relizing tht rndom medi n e deomposed in ontinuous prt nd )ututing prtF his ide hs profound onsequenes on the wy we understnd light stteringF feuse of the rndom positions of the toms in loudD there exist density )ututions inside the gs whih give rise to )ututions of the polriztion density vetor P(r)F por single reliztion of the tomi positions r i D the polriztion density n e deomposed into n ensemleEverged polriztion density P(r) nd )ututing polriztion density δP(r)X P(r) = P(r) + δP(r)D where δP(r) = 0F he ensemle verge • • • D s used throughout this workD is n verging of quntities tht re lulted for severl reliztions of the tomi loudF por eh reliztionD the tomi positions re hnged ording to given proility distriutionF en exmple of proility distriution for eh oordinte of the tom is the qussin distriution in the se of toms eing trpped in n optil trpF imilr s for the polriztion density vetorD the eletri (eld n e deomposed into n ensemleEverged (eld E nd )ututing (eld δEX E(r) = E(r) + δE(r)F st n e shown tht the ensemleEverged eletri (eld stis(es relmholtz9 eqution with n e'etive dieletri onstnt IT!PHX

∇ 2 E(r) + ε e (r) ω 2 c 2 E(r) = - ω 2 c 2 [ε e (r) -1]E inc (r).
@PFIWA prom this eqution it follows tht the sttered ensemleEverged eletri (eld from (niteEsize gs of rndomly positioned toms tully gets di'rted from tht (niteEsize gs whih is optilly desried y n e'etive dieletri onstnt ε e F prom iqF PFIW it is seen tht the di'rted (eld keeps the sme polriztion s the inident (eldF his is in ontrst to the sttered )ututing (eld whih n lose its polriztion fter mny sttering events due to the rndomly orientted dipole momentsF he )ututing (eld is sid to hve forgotten its originF por ompletenessD we mention tht inoherent light sttering is governed y wht is lled the rditive trnsfer equtionF he rditive trnsfer eqution is similr to the foltzmnn trnsport eqution nd mounts silly on detiled lne of sttering nd sorption in smll di'erentil volume elementF snoherent light sttering ours in ll diretions s opposed to oherent light sttering tht follows di'rtion ptternF IW PFPF vsqr geisxq pyw ex ixiwfvi yp geii e quntity tht is mesured in the l is the powerF he mesured power is the timeE verged oynting vetor integrted in the fr (eld over (nite solid ngle de(ned y the optil xis of the detetion system nd its numeril perture xeX

P = ˆNA ε 0 c 2 |E(r)| 2 r 2 dΩ = ε 0 c 2 ˆNA | E(r) | 2 + |δE(r)| 2 r 2 dΩ ≡ ˆNA [I coh (θ, ϕ) + I incoh (θ, ϕ)]r 2 dΩ. @PFPHA
sn this equtionD we hve de(ned the oherent intensity s the energy )ow from the ensemleEverged eletri (eldD nd the inoherent intensity s the energy )ow from the )ututing (eldF his seprtion of intensities is very importnt throughout this thesisF he oherent intensity orresponds to light tht gets di'rted from n e'etive homogeneous mediumD wheres the inoherent intensity omes from light sttering y )ututions in the smpleF e will study oth oherent nd inoherent light sttering in ghpter QF vet us rie)y disuss some di'erent terminologies for the eletri (eldD where the ntonyms re given t eh lineX verge ! )ututingD oherent ! inoherentD ollimted ! di'useF he three words on the left re synonymsD like the three words on the rightF he words verge nd )ututing n e understood from the mthemtil nottionX E @vergeA nd δE @)ututingAF he terms oherent nd inoherent re orrowed from the mirowve ommunityD where these terms re often usedF he lst termsX ollimted nd di'useD re known in the (eld of optisF sn optis it is known tht the intensity of ollimted em inident upon lossless nd sttering medium deys exponentilly during propgtionD ording to feerEvmert lwF he power tht is lost from the ollimted em is sttered wy nd forms di'use lightF he wy we use the words oherent nd inoherent might led to onfusion etween people from the lssil nd quntum optis ommunitiesF sn ghpter S of the textook Atom-Photon Interactions: Basic Processes and ApplicationsD oherent nd inoherent light sttering re explined from quntum optis perspetive PIF sn tht hpterD the sttered oherent power of n tom is proportionl to the squre of the expettion vlue of the dipole moment opertorD wheres the inoherent power is proportionl to the squre of the )ututions of the dipole moment opertorF he lultion of the expettion vlue is done in quntum mehnil wy where the dipole moment opertor is tred over ll possile sttes of the dipoleF por usD the verge is n ensemle verge @aestimtor of the expettion vlueA over the stterer positions nd therefore fundmentlly di'erentF his is why the terms coherent nd incoherent intensity re not the sme in lssil nd quntum optisF roweverD we will use the terms oherent nd inoherent light sttering often in this thesis nd they re relted to the lssil mening of itF et lstD let us repet tht oherent nd inoherent light sttering hppens di'erentlyF goherent light sttering follows di'rtion ptternD wheres inoherent light sttering ours in ll diretionsF hereforeD oth forms of light sttering n e mesured seprtelyF here re t lest two wys for mesuring the oherent nd inoherent powers individullyF e (rst wy is y mens of the position of the detetor with respet to the grei PF gvesgev exh exw iewix yp vsqr geisxq PH proe emD sine oherent nd inoherent light sttering do not our in the sme diretionsF e seond wy of seprtely mesuring oth powers is y using polrizerD sine for strong sttering systemD the di'use (eld does not hve the sme polriztion s the inident (eldF st is the (rst wy of seprtely mesuring the oherent nd inoherent intensities tht will e disussed in the next hpterD sine two di'erent experiments hve een onduted in the group of froweys tht re sed on this ideF ith thisD we hve illustrted qulittively the intuitive understnding of light sttering tht omes with the deomposition of the eletri (eldF 2.2.4 Beyond the mean-eld theory of optics sn the previous setion we hve introdued deomposition of the eletri (eldF e hve seen thtD ording to iqF PFIWD the oherent eletri (eld gets di'rted from homogeE neous medium desried y n e'etive dieletri onstntF sn generlD the e'etive dieletri onstnt of loud of resonnt dipoles tht re deE sried y polrizility α depends on oth α nd the tomi density distriutionF his dependene n e very omplitedD ut it turns out tht in most ses it is notF he simplest possile reltionship is PP

ε e (ω) = 1 + ρα(ω), @PFPIA
where ρ is the numer density of stterersF prom solid stte physis it is known tht for dense systems one needs to tke into ount the lolE(eld orretion due to the sreening of dipole y the other dipoles surrounding it PPDPQF he reltionship ε e (ω) = 1 + ρα(ω) then turns into the vorentzEvorenz reltion

ε LL (ω) -1 ε LL (ω) + 2 = ρα(ω) 3 , @PFPPA
where we used the susript vv to indite tht it is the e'etive dieletri onstnt s given y the vorentzEvorenz reltionF e ll the sitution where n ensemleEverged eletri (eld gets di'rted y n e'etive ojet for whih the dieletri onstnt stis(es the vorentzEvorenz formul mean-eld theoryF feyond menE(eld theory is de(ned s system for whih the dieletri onstnt does not stisfy iqF PFPPF he presene of strong resonnt dipoleEdipole intertions would e possile reson tht light sttering from n tomi system should e desried y eyond menE(eld theoryF trong resonnt dipoleE dipole intertions led to dependent light stteringD whih is not inluded in iqF PFPP PRF iqution PFPP n e generlized y inluding light sttering proesses tht involve depenE dent light sttering proesses like reurrent stteringX multiple sttering events etween the sme tomsF wore generllyD the following reltionship holds IUD IVD PHD PSX

ε e (ω) -1 ε e (ω) + 2 = Σ(ω, k) 3k 2 , @PFPQA
where Σ is the mss opertor nd is in generl omplited funtionF por the studies we present in this thesisD we do not need the full mhinery of multiple light stteringF e re interested in n estimte of the importne of reurrent sttering tht leds to eyond menE(eld theory ehvior of the tomi loud ITD PTD PUF en estimte for the importne PI PFQF ivesyx fiiix wiex pii er exh iw si of reurrent sttering n e derived from iqF RFIHT of efF ITF eurrent sttering eomes importnt for resonnt sttering y old loud of resonnt stterers when 9πρ 2k 3 > 1. @PFPRA imilrlyD it n e shown tht for systems whih present strong inhomogeneous rodening @∆ω Γ 0 AD the ondition for hving reurrent sttering eomes

9πρ 8k 3 Γ 0 ∆ω 2 > 1. @PFPSA
sn this thesis we study minly light sttering y old tomi loud whih is in the regime where menE(eld theory filsD euse of the presene of reurrent light stteringF 2.3 Relation between mean free path and system size o fr we hve een disussing mrosopi ulk properties of n ensemle of stterersF xextD we disuss the in)uene of the size of the system on light stteringF ystems with typil size L n either e in the single or multiple sttering regimeF sn the former seD the eletri (eld gets on verge only sttered one inside the loudF sn the ltter seD there re more thn one sttering events inside the systemF e physil quntity tht is needed to distinguish oth regimes is the men free pth l mfp D whih is the verge distne etween two suessive sttering eventsF e system is in the single sttering regime when l mfp L nd in the multiple sttering regime when l mfp < LF his men free pth orresponds in n e'etive medium to the dey length of ollimted lightD whih is de(ned in the feerEvmert lw yX I(z) = I(0) exp(-z/l mfp )F he men free pth n e relted to the index of refrtion y noting tht the intensity of em propgting in n extended medium with index of refrtion n e deys sX I(z) = I(0) exp(-2kn e z)F fy ompring oth expressions for the light intensityD we (nd the reltionship

l mfp (ω) = 1/[2kn e (ω)].
@PFPTA st is only in system where oth ε e = 1+ρα nd ρα 1D tht one n (nd the expressionX l mfp (ω) = 1/[ρσ sc (ω)]F his identity is expeted not to e vlid in the tomi systems tht we studyF roweverD the feerEvmert lw is always vlid for the ollimted em in n extended mediumD so lso in the presene of reurrent stteringF 2.4 Energy relaxation in quantum optics es indited t the eginning of this hpterD we will now tret light sttering y using quntum optisF elthough lssil system orresponds in tomi physis to trnsition etween J = 0 nd J = 1 stteD there is ig di'erene etween the quntum optil response nd the lssil response of suh trnsition under illumintionF sn quntum optisD suh trnsition n e sturtedD iFeFD there re no eletrons left in the ground stte fter the tom hs sored photonF his mkes the system response nonlinerD sine seond photon nnot e sored fter the (rst one hs een soredF his form of nonliner response does not exist in lssil optisF sn this setion we show tht under grei PF gvesgev exh exw iewix yp vsqr geisxq PP ertin onditions oth lssil nd quntum optis hve the sme set of oupledEdipole equtionsD whih form the sis of light sttering s we hve seen in etion PFPF sn the formlism treted in this setionD we ssume the system to e initilly exitedY so there is no driving (eld throughout the derivtionF e study how suh system reE lxes under emission of photonF o study the tomi relxtionD one often relies on the eisskopfEigner theory to reover the exponentil dey in time of single tomi exiE ttion PVF en extension of the theory to system of N interting dipoles n e found in the work of vidzinsky et al. PWF sn tht work oth the dipole moments nd the eletri (eld were tken to e slr quntitiesF ine for dense tomi ensemlesD the nerE(eld intertions re importnt nd need to e inludedD one needs to tke into ount the vetoril nture of the eletri (eldF his hs een done in efF QHF sn this setion we will desrie this vetoril eisskopfEigner theoryD sine it shows the onnetion of quntum optis nd lssil optis desription of light emission from n ensemle of emitters PWF st serves therefore s ridge etween lssil nd quntum optisF Hamiltonian sn order to lulte the time evolution of given quntum stteD the rmiltonin is neededF he rmiltonin Ĥ = ĤA + ĤF + ĤI ontins three opertors tht re relted to the tom energy @ ĤA AD the eletromgneti (eld energy @ ĤF AD nd the intertion potentil @ ĤI A etween the dipoles nd the eletri (eld of the vuumF nder the eletri dipole pproximtion nd in the intertion pitureD the totl rmiltonin is given yX

Ĥ = N j=1 ω 0 π † j πj + k,λ ω k â † k,λ âk,λ + k,λ N j=1
g k,λ,j πj e -iω 0 t + π † j e iω 0 t • â † k,λ e i(ω k t-k•r j ) + âk,λ e -i(ω k t-k•r j ) , @PFPUA where πj = |g j e j | @π † j = |e j g j |A is the lowering @risingA opertor for tom jY âk,λ @â † k,λ A is the nnihiltion @retionA opertor for the eletri (eld of plne wve with wve vetor kY λ is the polriztion of the plne wveY ω 0 is the tomi frequenyY nd ω k is the photon frequenyF he oupling strength of the dipoles to the vuum (eld

g k,λ,j = ω 0 e k,λ • pj p j 2ε 0 ω k V = g k,j e k,λ • p j , @PFPVA
where V is the quntiztion volumeD ω k = ckD e k,λ the polriztion of the eletri (eld nd p the unit dipole moment vetorD iFeFD the polriztion diretion of the dipoleF ery oftenD the terms ontining the opertors πâ nd π † â † re negleted in quntum optisF he ommon justi(tion for this pproximtion is tht these terms led to integrls similr to ´t 0 exp[±i(ω 0 + ω k )t ]dt F he integrnd of this integrl osilltes rpidly under integrtion nd therefore gives zeroF his is in ontrst with the integrnds of the form exp[±i(ω 0ω k )t ]D when ω k ≈ ω 0 D whih originte from the terms π † â nd πâ † F his pproximtion where ω k ≈ ω 0 nd therefore the terms πâ nd π † â † re negleted in the rmiltoninD is lled the rotting wve pproximtion @eAF sn this setion we will see tht the e nnot e pplied for systems where dipoles re in eh other9s nerE(eld regionF ine we re preisely interested in this kind of systemsD we do not pply the e in this thesis nd therefore keep the terms πâ nd π † â † in the rmiltoninF PQ PFRF ixiq iveesyx sx exw ysg Wave function and its time evolution -Beyond the rotating wave approximation he next step in lulting the time evolution of N toms tht initilly shre single tomi exittionD is de(ning the wve funtionF fy mens of eisskopfEigner theoryD we will e le to derive the time evolution of the system nd show under whih onditions lssil nd quntum optis led to the sme set of oupledEdipole equtionsF sn their originl workD eisskopf nd igner studied the relxtion of single tom in vuum PVF he wve funtion for single tom ws hosenD within the hrödinger pitureD to eX

|Ψ(t) = β(t) exp(-iω 0 t)|e |0 + k,λ γ k,λ (t) exp(-iω k t)|g |1 k,λ D where λ inE
dites the polriztion of the plne wve with wve vetor kF hen the e is ppliedD the number of excitations inside the system nnot vry in timeF his wve funtion spns therefore the entire vetor spe of funtions ontining single exittion initillyF sn the se of N toms nd y pplying the eD we ould similrly write the wve funtion s

|Ψ(t) = N j=1 β j (t)|g 1 , • • • , e j , • • • g N |0 + k,λ γ k,λ (t)|g 1 , • • • , g N |1 k,λ , @PFPWA
where we ssumed tht the system hs een exited initilly y single exittionF roweverD we will not pply the eF e ssume the wve funtion to e of the following formX

|Ψ(t) = N j=1 β j (t)e -iω 0 t |g 1 , • • • , e j , • • • g N |0 + k,λ γ k,λ (t)e -iω k t |g 1 , • • • , g N |1 k,λ + m<n k,λ α mn,k,λ (t)e -i(2ω 0 +ω k )t |g 1 , • • • , e m , • • • , e n , • • • , g N |1 k,λ . @PFQHA
hese dditionl sttes with oe0ients α re the only di'erene with the wve funtion of iqF PFPWD where the e hs een ppliedF st is euse of these sttes tht we will rrive t di'erent solution then derivtion for whih the e is ppliedF hese sttes re therefore importntD s we will oserve during the derivtion of the time evolution of the wve funtionF huring the mthemtil derivtion of the time evolution of the wve funtionD we will pinpoint on ples where the result di'ers from derivtion tht pplies the eF sn order to (nd the unknown oe0ients of iqF PFQHD the hrödinger eqution is written for the wve funtion given y iqF PFQHX

βj (t) = -i k,λ g k,λ,j γ k,λ (t)e -i(ω k -ω 0 )t e ik•r j -i k,λ N j =1,j =j g k,j α jj ,k,λ (t)e -i(ω k +ω 0 )t e ik•r j , @PFQPA γk,λ (t) = -i N j=1 g k,λ,j β j (t)e i(ω k -ω 0 )t e -ik•r j , @PFQQA αmn,k,λ (t) = -ig k,λ,m β n (t)e i(ω k +ω 0 )t e -ik•rm -ig k,λ,n β m (t)e i(ω k +ω 0 )t e -ik•rn . @PFQRA
foth γ k (t) nd α mn,k (t) re found y integrting their di'erentil equtions nd using the ft tht there is initilly only single exittion whih is in the tomi prt of the wvefuntion so γ k,λ (0) = 0 nd α mn,k,λ (0) = 0

γ k,λ (t) = -i ˆt 0 N j=1 g k,λ,j β j (t )e i(ω k -ω 0 )t e -ik•r j dt , @PFQSA α mn,k,λ (t) = -ig k,λ,n ˆt 0 β n (t )e i(ω k +ω 0 )t e -ik•rm dt -ig k,λ,n ˆt 0 β m (t )e i(ω k +ω 0 )t e -ik•rn dt . @PFQTA
xowD γ nd α n e sustituted in the di'erentil eqution for β j (t)X βj (t) =k,λ j ˆt 0 g k,λ,j g k,λ,j β j (t )e i(ω k -ω 0 )(t -t) e ik•(r j -r j ) dt k,λ N j =1,j =j ˆt 0 g k,λ,j g k,λ,j β j (t )e i(ω k +ω 0 )(t -t) e -ik•(r j -r j ) dt . @PFQUA e ontinue the derivtion y interhnging the sum over k with the integrl over timeF e will ssume to hve lrge volume so tht the disrete sum over ll modes k n e repled y integrls @ k → V /(2π) 3 × ´d3 kAF sn our sitution of toms inside vuumD this is resonle ssumptionF he integrls over k eomesX ˆgk,λ,j g k,λ,j e ick(t -t)±ik•(r j -r j ) d 3 k.

@PFQVA his integrl is not equlD ut n e pproximted y the form δ{t -[t -(|r j -r j |)/c]}F he typil distne etween toms is hosen to e the size of the loud RF he time integrl in iqF PFQU will led to terms like β j (t -R/c)F hen we ssume tht the dey time of the exittion is lrger thn the time of )ight of photon through the tomi loud @R/cAD then we n ssume β j (t -R/c) → β j (t)F his pproximtion is lled the wrkov pproximtionF e wrkovin system is sid to e memoryless systemF ine the tomi louds tht we will study hve size whih is on the order of mirometer @so R/c ∼ 10 fsAD nd the typil dey time is on the order of IHH nsD we hve justi(ed the sustitution PS PFRF ixiq iveesyx sx exw ysg β j (t ) → β j (t)F e go k to the di'erentil eqution for β nd mke the sustitution

β j (t ) → β j (t)X βj (t) = - k,λ j ˆt 0 g k,λ,j g k,λ,j e i(ω k -ω 0 )(t -t) e ik•(r j -r j ) dt β j (t) - k,λ N j =1,j =j ˆt 0 g k,λ,j g k,λ
,j e i(ω k +ω 0 )(t -t) e -ik•(r j -r j ) dt β j (t). @PFQWA he time integrl is now nlytil nd results in

βj (t) = i k,λ j g k,λ,j g k,λ,j 1 -e -i(ω k -ω 0 )t ω k -ω 0 β j (t)e ik•(r j -r j ) +i k,λ N j =1,j =j g k,λ,j g k,λ,j 1 -e -i(ω k +ω 0 )t ω k + ω 0 β j (t)e -ik•(r j -r j ) . @PFRHA
es n intermezzo we will disuss out the form of the di'erentil equtionF his form of the di'erentil eqution n e rewritten in more generl form @prt from some ftorsA

βj (t) ∝ m j 0| g 1 , • • • , e j , • • • , g N |H I |m m|H I |g 1 , • • • , e j , • • • , g N |0 E i -E m , @PFRIA where E i is the energy of the stte |g 1 , • • • , e j , • • • , g N |0 D nd E m
is the energy of the intermedite stte |m F he sum over the intermedite sttes |m runs over ll possile sttes within the wve funtion PFQHF he (rst term of iqF PFRH origintes from intermedite sttes hving the oe0ient γD wheres the seond term omes from sttes hving the oefE (ient αF ixmples of the ssoited proesses for whih the intermedite sttes intervene hve een visulized in pigF PFI for system of two tomsF sn tht (gure we oserve tht there re two types of intermedite sttesF yne intermedite stte @|m 1 A is stte where the initilly exited tom relxes under emission of photon nd the seond intermedite stte @|m 2 A is stte where efore the exited tom relxesD the tom in the ground stte gets exited nd emits photonF sf we did not inlude the wve funtions with oe0ient αD we would not hve the seond term in iqF PFRH nd the result would e the sme s when the e would e ppliedF prom the wve funtion s given in iqF PFQHD it n e seen tht both the sttes with oe0ients γ nd α do not neessrily hve the sme energy s the system hd initillyF snluding these soElled virtul sttesD y not pplying the e so tht we re not restrited to ω k ≈ ω 0 D is importnt when the toms re sptilly lose to eh otherF e will see tht this quntum optis tretment results in the sme set of oupledEdipole equtions s we hve found y mens of lssil optisF e go k to the derivtion of β(t)F he next step will e splitting up the (rst sum in the self ontriution nd the ontriution of other tomsX efter thisD we ssume to hve lrge volume so tht the disrete sum over ll modes k n e repled y integrls nd we lso neglet the timeEdependent osilltion termsD s they osillte fst under integrtion over k nd n e negleted ompred to the onstnt term in the numertorX

βj (t) = i k,λ g k,λ,j g k,λ,j 1 -e -i(ω k -ω 0 )t ω k -ω 0 β j (t) + i k,λ N j =1,j =j g k,λ,j g k,λ,j • 1 -e -i(ω k -ω 0 )t ω k -ω 0 • e ik•(r j -r j ) + 1 -e -i(ω k +ω 0 )t ω k + ω 0 e -ik•(r j -r j ) β j (t
βj (t) = iV (2π) 3 c ˚d3 k λ g k,λ,j g k,λ,j 1 k -k 0 β j (t) + iV (2π) 3 c ˚d3 k λ N j =1,j =j g k,λ,j g k,λ,j 1 k -k 0 e ik•(r j -r j ) + 1 k + k 0 e -ik•(r j -r j ) β j (t).
@PFRQA e writeX g k,λ,j g k,λ,j = g 2 k pj •e k,λ e k,λ • pj F he summtion over ll polriztion omponents leds toX λ e k,λ e k,λ = 1 -kk . @PFRRA rving this nd repling k 0 y k 0 + i0D we n rewrite the di'erentil eqution for β j sX

βj (t) = iV (2π) 3 c ˚d3 kg 2 k pj • 1 -kk pj 1 k -k 0 -i0 β j (t) + iV (2π) 3 c ˚d3 k N j =1,j =j g 2 k pj • 1 -kk pj 1 k -k 0 -i0 e ik•(r j -r j ) + 1 k + k 0 + i0 e -ik•(r j -r j ) β j (t).
@PFRSA e will evlute eh integrl seprtelyF sn the (rst integrl we note tht the only ngulr dependene is in the projetion opertor 1 -kk F ithout loss of generlityD we n ssume the dipole moment to e pointed long the z xisF henD one n redily showX ¨ẑ • 1 -kk ẑdΩ = 4π -4π 3 = 8π 3 . @PFRTA PU PFRF ixiq iveesyx sx exw ysg por the rdil prt of the integrlD we mke use of the identityX

1 x ∓ i0 = 1 x ± iπδ(x), @PFRUA
where stnds for prinipl vlueF he prinipl vlue prt only orresponds to freE queny shift of the sme vlue for ll β j @vm shiftA nd will e ignoredF he (rst integrl then eomesX βj (t)

First integral = - Γ 0 2 β j (t), @PFRVA
with Γ 0 = ω 3 0 p 2 /(3πε 0 c 3 ) eing the individul spontneous dey rteF he seond integrl redsX βj (t)

Second integral = -iV (2π) 3 c N j =1,j =j pj • ∇ 2 1 -∇∇ pj ˚d3 k 1 k 2 g 2 k 1 k -k 0 -i0 e ik•(r j -r j ) + 1 k + k 0 + i0 e -ik•(r j -r j ) β j (t) = -iV 2π 2 c N j =1,j =j pj • ∇ 2 1 -∇∇ pj ˆ∞ 0 dkg 2 k 1 k -k 0 -i0 + 1 k + k 0 + i0 sin(k|r j -r j |) k|r j -r j | β j (t). @PFRWA
he integrl over k n e extended from -∞ to +∞F his would not e possileD if the e were ppliedF es the wve funtions with the oe0ient α would not e thereD the term 1/(k + k 0 + i0) would not e thereF purthermoreD if the integrl hd not een pplied t llD sine stritly ω k ≈ ω 0 when the e is ppliedD the onsequenes would e even more importntF hese onsequenes of removing ll virtul sttes hve een disussed in the work of vidzinsky et al. PWF ine we do not pply the eD we n extend the integrl

βj (t) Second integral = -ik 2 0 p 2 2π 2 ε 0 N j =1,j =j pj • ∇ 2 1 -∇∇ pj • ˆ∞ -∞ dk 1 k 2 sin(k|r j -r j |) (k -k 0 -i0)|r j -r j | β j (t).
@PFSHA he sine funtion n e written in terms of omplex exponentilsD so tht the residue theorem n e pplied on βj (t)

Second integral = -k 2 0 p 2 4π 2 ε 0 N j =1,j =j pj • ∇ 2 1 -∇∇ pj • ˆ∞ -∞ dk 1 k 2 e ik|r j -r j | (k -k 0 -i0)|r j -r j | - e -ik|r j -r j | (k -k 0 -i0)|r j -r j | β j (t).
@PFSIA xote tht there seem to e P polesF roweverD we relize tht the pole t k = 0 is removle singulrity when we look t the originl expression in iqF PFRSF he integrl of the seond grei PF gvesgev exh exw iewix yp vsqr geisxq PV term vnishesD sine the ontour in the lower hlf plne does not hve pole insideF he (rst integrl does not vnishF efter the evlution of the integrlD we (ndX

βj (t) Second integral = -ip 2 2πε 0 N j =1,j =j pj • ∇ 2 1 -∇∇ pj e ik 0 |r j -r j | |r j -r j | β j (t) = -i 3Γ 0 2k 3 0 N j =1,j =j pj • ∇ 2 1 -∇∇ pj e ik 0 |r j -r j | |r j -r j | β j (t).
@PFSPA e write the di'erentil eqution slightly di'erently y using the identity ∇ 2 G(r, r ) +

k 2 0 G(r, r ) = -δ(r -r
)F e mke the following sustitutionX

∇ 2 e ik 0 |r-r | 4π|r -r | = -δ(r -r ) -k 2 0 e ik 0 |r-r | 4π|r -r | , @PFSQA
where we use the ft tht we only sum over nonEidentil dipolesD whih mens tht the delt funtion vnishes in the identity nd we n rewrite the seond integrl s followsX

βj (t) Second integral = i 3Γ 0 2 N j =1,j =j pj • 1 + 1 k 2 0 ∇∇ pj e ik 0 |r j -r j | k 0 |r j -r j | β j (t) = i 3Γ 0 2 N j =1,j =j pj • Ḡ(r j , r j )p j β j (t).
@PFSRA sn the lst eqution we hve introdued the qreen9s tensorF xote tht we would not rrive t the qreen9s tensor if we hd removed the sttes orresponding to the oe0ient αD sine the integrl of iqF PFRW ould not e extendedF pinllyD we hve found the di'erentil eqution for β j X

βj (t) = - Γ 0 2 β j (t) + i 3Γ 0 2 N j =1,j =j pj • Ḡ(r j , r j )p j β j (t).
@PFSSA e then ssume the oe0ients β to e of the formX β j (t) = β j exp(-iω j t)F ustituting this form of β j (t) into iqF PFSS gives us the eqution

ω j β j = -i Γ 0 2 β j - 3πΓ 0 k 0 N j =1,j =j pj • Ḡ(r j , r j ; ω 0 )p j β j .
@PFSTA hen trnsforming this eqution k to the hrödinger piture nd fter putting the terms proportionl to β j togetherD the eqution eomes

ω 0 -ω j -i Γ 0 2 β j - 3πΓ 0 k 0 N j =1,j =j pj • Ḡ(r j , r j ; ω 0 )p j β j = 0.
@PFSUA sn the sene of ny intertion etween the dipolesD we reover the expeted resultD where β j (t) = β j (0)e -iω 0 t e -Γ 0 2 t . @PFSVA PW PFSF sev ryyx pyw e gvesgev iigsi iqution PFSU should e ompred with iqF PFIU when we remove the inident (eld @our system hs een exited initillyAD whih is extly the sme resultF ith this we hve seen tht in the lowEexittion limitD lssil nd quntum optis desrie in the sme wy the intertions etween lssil dipolesF elthough the intermedite sttes do not hve the sme energy s the initil stteD they turned out to e importnt in order to rrive t the lssil resultF hese intermedite sttes re llowed to exist during (nite time intervl ∆tD euse of the reisenerg9s unertinty reltionX ∆E∆t ≥ /2F he virtul sttes orrow or lend some energy from the vuumF es (nl remrkD we oserve tht dipoleEdipole intertions show up t the endD lE though they hve not een inluded expliitly in the rmiltoninF he reson for this is tht dipoleEdipole intertions re tully medited y the eletromgneti modes of the vuum QIF sn the next setion we will desrie these eletromgneti modes in more detilF 2.5 Virtual photons from a classical perspective sn the previous setion we hve enountered intermedite sttes whih were lled virtul sttesF he photons tht re the elementry exittion of the ssoited eletromgneti wves re lled virtul photonsF hese intermedite sttesD whih should e disrded when pplying the eD re however the reson of hving the 1/(k 0 r) 3 term in the dipoleEdipole intertionsF st is preisely these nerE(eld intertions tht re importnt in this thesisD sine we study systems where the interEstterer distne is smll ompred to the optil wvelengthF prom lssil eletromgnetism it is known tht in this dense regime evnesent wves re importnt for the intertionF hereforeD there should e onnetion etween evnesent wves nd virtul photonsF sn this setionD we study this onnetionF he system with whih we will study this onnetion is system where energy trnsfer tkes ple from n exited twoElevel system to twoElevel system in its ground stte y using quntum optis formlismF his quntum tretment is sed on the work of endrews et al. QPF st n e shown from seondEorder timeEdependent perturtion theory tht the energy trnsfer rte from tom e @initilly exitedA to tom f @initilly in ground stteA is given y

Γ = 2π |M | 2 δ(E f -E i ), @PFSWA with M given y M = m f |H I |m m|H I |i E i -E m , @PFTHA
where |i is the initil stteD |f the (nl stteD |m ny intermedite stteD nd E stnds for the energy of prtiulr stteF es illustrted in pigF PFID there re two possile kinds of intermedite proessesX |m 1 X he initilly exited tom relxes under emission of photon with energy ω whih is not neessrily equl to the trnsition energy of the tomD |m 2 X he initilly relxed tom gets exitedD under emission of photon with energy ωF ell proesses of this kind re neessrily virtulD sine even without tking the grei PF gvesgev exh exw iewix yp vsqr geisxq QH photon in onsidertionD oth toms re exited nd there is therefore more energy present in the intermedite system thn in the initil systemF fy following the sme mthemtil steps s in the former setionD endrews et al. found tht the quntum mplitude M n e written s QP

M = p A • V (k 0 , R)p B , @PFTIA
where the oupling tensor is the qreen9s tensorX

V (k 0 , R) = -k 2 0 e ik 0 R 4πε 0 R 1 + ik 0 R -1 k 2 0 R 2 1 + 3 -3ik 0 R -(k 0 R) 2 (k 0 R) 2 RR R 2 = -µ 0 ω 2 Ḡ(R).
@PFTPA ht we then oserve is tht we reover the lssil dipoleEdipole intertion potentilD y inluding ll possile intermedite sttes in seondEorder timeEdependent perturtion theoryF prom lssil eletrodynmis we know tht evnesent wves re present in this qreen9s tensor nd tht they re ruil for nerE(eld intertionsF e will explin y mens of lssil eletrodynmis how evnesent wves nd virtul photons re reltedF prom lssil eletrodynmis we know tht the qreen9s funtion in vuum n e presented in di'erent formsX

G(r; ω) = e i ω c r 4πr @PFTQA = ˚d3 k (2π) 3 1 k 2 -ω 2 c 2 e ik•r @PFTRA = ¨dk x dk y (2π) 2 iπ γ
e iγz e i(kxx+kyy) , @PFTSA

QI

PFTF wwe with γ = ω 2 /c 2 -k 2 x -k 2 y F he form of the qreen9s funtion s given y iqF PFTR shows tht the (eld of dipole is given y weighted sum over eletromgneti sttes de(ned in reiprol spe (k x , k y , k z )F e shemti view of reiprol spe is given in pigF PFP@AF he onEshell sttesD whih re sttes for whih k = ω/cD re most importntD sine the weighting ftor 1/(k 2 -ω 2 /c 2 A is the lrgest for these sttesF ell sttes for whih k = ω/c re lled o'Eshell sttesF enother representtion of the qreen9s tensor is the eyl9s representtion in (k x , k y ; z) speD s given y iqF PFTSF his representtion is otined fter integrtion of the integrnd of iqF PFTR with respet to k z X iπ γ e iγz e i(kxx+kyy

) = ˆdk z 2π 1 k 2 -ω 2 c 2 e ik•r .
@PFTTA vet us onsider n evnesent wve tht is hrterized y k x nd k z suh thtX (k x ) 2 + (k y ) 2 > ω 2 /c 2 see the red dot in pigF PFP@AF eording to iqF PFTTD this evnesent wve is onstruted out of ll possile sttes (k x , k y , k z )D eh stte hving di'erent weighting ftorF he integrtion pth is indited y the red line in pigF PFP@AF en evnesent wve is therefore uilt up of uniquely o'Eshell sttes @k = ω/cAF sn eppendix eD we hve ssigned quntum opertor @nnihiltion opertorA to evnesE ent (eldsF here we formlly show tht the elementry exittion of n evnesent wve is superposition of virtul photonsF irtul photons orrespond to elementry exittions of o'Eshell sttesF rereyD we hve shown the reltionship etween evnesent wves nd virtul photonsF pinllyD let us note tht there hve lredy severl works een pulished on the onE netion etween virtul photons nd lssil eletrodynmis QQ!QSD ut the pproh presented in this setion is di'erent s ompred to the othersF 2.6 Summary sn this hpter we hve summrized the si elements needed to interpret the results outlined in this thesisF e hve relled some notions of multiple sttering theoryF his disussion llowed us to de(ne light sttering tht is eyond the menE(eld theory of optisF purthermoreD we hve disussed light sttering from oth lssil nd quntum optis perspetiveF his llowed us to show the equivlene etween quntum optis nd lssil optis in the lowEexittion regimeD where only single tomi exittion is present in system of N tomsF pinllyD we gve n originl wy of ompring virtul photons with evnesent wvesF prom nnophotonis we know very well tht when the interEtomi distne is smll s ompred to the optil wvelengthD the intertions re dominted y evnesent wvesF imilrlyD the ontriution of virtul photons re importnt in tht situtionF sndeedD oth evnesent wves nd virtul photons re relted to eh otherF Chapter 3

Light scattering from a dense cloud of cold atoms 3.1 Motivation e strt with disussion of two spei( exmples of resonnt light stteringF sn the (rst situtionD plne wve is inident upon S toms tht re positioned rndomlyD following homogeneous distriutionD nd re distriuted inside ox with dimensions 10 × 10 × 10 µm 3 D so tht the verge interEtomi distne is 3 × 10 2 λ 0 F he totl eletri (eld is lulted for single reliztion of the tomi loud with the proedure tht hs een outlined in etion PFPFPF he totl eletri (eld E is deomposed sX E

(r) = E inc (r) + E sc (r), @QFIA
where E inc is the inident eletri (eld nd E sc the sttered eletri (eldF he oynting vetor in the fr (eld is given y

S(r) = ε 0 c 2 |E inc (r)| 2 + |E sc (r)| 2 + 2e[E inc (r) • E * sc (r)] r, @QFPA
where the lst interferene term leds to extintionF e lulte the sttering ross setionD y following the sme proedure s outlined in etion PFPFIF he sttering ross setion is normlized y dividing the sttering ross setion y the onEresonne singleE tom sttering ross setionF pigure QFI@A shows the normlized sttering ross setion s funtion of the lser detuning δω L = ω L -ω 0 in units of Γ 0 D where ω 0 = 2πc/λ 0 D with λ 0 = 780 nmF st is seen tht S prtiles stter S times s muh s single prtile nd tht the sttering spetrum is vorentzinD like it is for resonnt light sttering y single tomF purthermoreD the full width t hlf mximum @prwA of the sttering spetrum is equl to the singleEtom spontneous dey rte Γ 0 nd the resonne is entered round the singleEtom resonne frequenyF pigure QFI@A shows the ensemble-averaged frE(eld sttering pttern of only the sttered (eldD so this pttern is proportionl to |E sc | 2 F his sttering pttern hs doughnut shpeD whih is the sme sttering pttern s for light sttering from single dipoleX light sttering tkes ple in mny diretionsD exept long the polriztion diretionF ttering y very dilute loud s the one we study here on(rms wht we know from light sttering in the tmosphereX the sky is lue in ll diretionsF sn the forwrd diretion we lso oserve pekF his pek is hrteristi for the singleEsttering regimeD sine in tht regime ll sttered (elds hve the sme phse in the forward direction nd therefore interfere onstrutivelyF xextD we grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw QR do the sme lultionsD ut for N = 100 toms in ox with dimensions 1 × 1 × 1 µm 3 D so tht the tomi density is R orders of mgnitude higher nd the verge interE tomi distne is ∼ λ 0 /4F feuse of the muh smller verge interEtomi distneD the resonnt dipoleEdipole intertions re muh stronger in this on(gurtion thn in the previous on(gurtion with S tomsF he numeril results re presented in pigF QFI@A nd pigF QFI@dAF pigure QFI@A shows the normlized sttering ross setion for single reliztion of the tomi loudF st is seen tht the spetrum for this exmple is very di'erent from the light sttering spetrum of the previous dilute on(gurtionF wultiple QS QFPF ei yp ri e peks in the sttering spetrum pper nd they re distriuted over severl singleEtom linewidths Γ 0 F pigure QFI@dA shows the ensemble-averaged totl sttering pttern @ |E sc | 2 AF elso the sttering pttern hs hnged drstillyF es opposed to eing similr to sttering pttern from single tomD whih is doughnut likeD it is this time very similr to di'rtion ptternF prom pigF QFI it is ler tht we oserve the pperne of new physis when resonnt dipoleEdipole intertions re strongF st is in this regime tht we re minly interestedF

State of the art

sn the lst setion we hve seen tht the sttering pttern for wvelengthEsizeD denseD nd resonnt tomi louds is very di'erent from wht is expeted for yleigh stteringF feing interested y resonnt light sttering from dense tomi loudsD let us disuss severl experiments tht hve een onduted on light sttering from tomi loudsF his disussion will llow us to see wht hs lredy een done in the (eld nd to wht extent the light sttering regime in whih we re interestedD is still originlF he (rst exmple goes k to p § in who relized for the (rst timeD in IWSWD tht prt from line rodening of the spetrum of interting systemsD there is lso hnge of the eigenfrequenies tht should ourD nowdys known s the oopertive vm shift @gvA QTF he hnge of eigenfrequenies re)ets itself in hnge of oth the emission nd sorption spetr nd it ws in IWUQ tht priedergD rrtmnnD nd wnssh theE oretilly studied in detil the frequeny shifts in emission nd sorption y resonnt systems of twoElevel toms QUF he (rst experiment where the oopertive vm shift ws oserved from the sorption spetrumD hs reently een reported y ueveney et al. for thin sl loded with hot tomi vpor QVF he experimentlly oserved gv hs een otined for di'erent lyer thiknesses nd oinides with menE(eld theory lultion QUF enother oservtion done y ueveney et al. for hot tomi vpor is superluminl pulse propgtion QWF hey mesured n optil pulse dvne of > 100 ps over propE gtion distne of only QWH nmF prom thisD they dedued group index n g = -1.0 × 10 5 D whih ws t tht time the lrgest negtive group index ever mesuredF ixperimentl works on light sttering from old tomi systems re essentilly perE formed on lrge @ompred to the resonnt wvelengthAD diluteD ut optilly thik loud of old toms RH!RPF eferene RH shows n indiret oservtion of oopertive wie sttering from old tomi loud y mesuring the enterEofEmss positionsF he ide ehind this indiret oservtion is tht the olletive rdition pressure fore should e lrger thn rdition pressure fore sed on independent tomsD therey eing sigE nture of olletive sttering y the loud whih ehves s prtile with n index of refrtionF xerEresonne light sttering experiments were performed on ig loud of old toms in efF RPF sn tht workD oth the temporl nd spetrl properties of the sttered light were studiedF e very smll vorentzEvorenz shift ws mesured @δω = -0.07Γ 0 AD tht is of the order of mgnitude predited y vorentzEvorenz theory for the tomi density studied in the loudF por forml proofD the shift should e lrger nd for tht denser loud of toms would hve een neededF uwong et al. shone ontinuous wve lser on n optilly thik loud of strontiumEVV grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw QT toms nd studied the time evolution of the sttered light fter the inident lser ws swithed o' ruptly RQF hey hve shown tht for nonEresonnt light sttering from loud with lrge optil thiknessD oherent super)sh of light emerges from the loudF his oherent super)sh is speil for t lest two resonsX @IA its pek intensity is higher thn the inident intensityD nd @PA the pulse durtion is shorter thn the singleE tom spontneous emission rteF hey understood tht with suh systemD it would e interesting to rete pulse trinsF st is one yer lter tht they sueeded in reting pulse trin with oth high repetition rtes nd high intensity RRF snspired y this workD erújo et al. hve lso studied the time evolution of light sttering fter the lser got swithed o'F roweverD the min di'erene with the work of uwong et al. is the ft tht erújo et al. olleted light t 35 • ngle from the optil xisF fy doing soD they did not detet oherent sttered (eldD ut inoherent sttered lightD so thtD s they limedD light emission t di'erent ngles nnot e explined y phseEmthing ondition imposed y the initil lser (eld or ontinuousEmedium desriptionF fy doing soD they hve oserved oth singleEphoton superrdine RS nd singleEphoton surdine RTF imilrlyD oof et al. hve reported the oservtion of oth singleEphoton superrdineD nd the gv RUF et lstD let us mention reent work on optil tomi loksF he stility nd ury of urrent tomi loks re ultimtely limited y hoppler e'et nd reoil frequeny shiftsF yne wy to suppress oth e'ets is y reting n optil lttie lokD where toms re loded in lttie of strongly on(ned potentils RVF roweverD s ws mentioned in efF RVD not eh eletroni stte experienes the sme on(nement nd therefore the sme perturtionF his leds to redution in ury of the tomi lttie lokF st is therefore interesting to study motionl e'ets on oherent light sttering from wekly on(nedD dense gs of tomsF his hs reently e done y fromley et al. hey hve shown ler signtures of motionl e'ets on oherent sttering nd dipolr oupling RWF st ws shown tht the linewidth is determined y the optil depth of the loud onlyD whih is in ontrst with the frequeny shift tht seems to depend signi(ntly on motionl e'ets for frequenies with n extremely smll linewidth @in efF RWX Γ 0 = 7.5 krzAF he systems used to hieve the forementioned gols hve ll in ommon tht resonnt dipoleEdipole intertions re wekD sine the dipoles re on verge not in eh other9s nerE(eld regionF hen resonnt dipoleEdipole intertions re wekD the experimentl results gree with menE(eld theoryF he (rst four lines of le QFI summrize some eferene @ekA hensity snhomogF rodF Γ

0 C RP 5 • 10 13 HFHR 1.4 RW 2 • 10 12 2 • 10 -3 10 -2 RQ 4.6 • 10 11 QFR 10 -2 QV 4 • 10 16 ∼ 10 3 10 -4
[7] 2•10 14 0.06 5

Table 3.1: The (peak) density in [at/cm 3 ], inhomogeneous broadening and the parameter C

are given for the densest reported systems of dierent research groups. References [7,38,42] study and Refs. [43,49] used Sr-88 atoms. The last row is bold, since this is the system of interest in this thesis.

QU

QFPF ei yp ri e hrteristis tht re importnt for the strength of the resonnt dipoleEdipole intertions of the densest louds tht hve een studied y most of the previously ited reserh groupsF e hve listed the @pekA tomi densityD inhomogeneous rodeningD nd the prmeter tht indites if resonnt dipoleEdipole intertions re strongD s we hve de(ned in iqs PFPR nd PFPSF e ssign to this prmeter the symol CD s it is relted to the oopertivityD nd rell for onveniene the expression of this prmeterX homogeneously rodened system : C = 9πρ 2k 3 , @QFQA inhomogeneously rodened system :

C = 9πρ 8k 3 Γ 0 ∆ω 2 , @QFRA
where ∆ω is the inhomogeneous rodeningF esonnt dipoleEdipole intertions re strong when C > 1D see ghpter PF vet us study this prmeter for di'erent experimentl setups tht hve een desried in litertureF prom olumns three nd fourD we see tht the system reported in efF QV hs huge inhomogeneous rodeningD nd is not in the strong resonnt dipoleEdipole intertion regimeF he strong inhomogeneous rodening is in the loud studied in efF QV due to the hoppler e'et tht nnot e negleted for hot tomi vporsF he hoppler e'et e'etively redues the resonnt dipoleEdipole intertionsF elthough the density of the tomi loud is highD it is not su0ient to overome inhomogeneous rodeningF prom the lst olumn we n oserve tht there re no strong resonnt dipoleE dipole intertions in the louds tht re reported in efs RQD RW eitherF he reson tht resonnt dipoleEdipole intertions re not strong in these louds is tht the density of the louds is not su0iently high to hve strong resonnt dipoleEdipole intertionsD iFeFD the nerE(eld terms of the resonnt dipoleEdipole intertions re not importntF he loud studied in efF RP stis(es the riterion for strong resonnt dipoleEdipole intertionsD however it is stis(ed only slightly t the enter of the loudX the position distriution of the toms follows qussin distriutionD sine the toms re loded in dipole trpF o fr we hve een disussing tomi systems tht were used to suessfully oserve severl physil phenomen like singleEphoton superrdine nd the gvF es we hve seenD these systems hve ll in ommon tht they re not in the regime of light sttering where strong resonnt dipoleEdipole intertions existF en experimentlly lrgely unexplored system is n tomi system where resonnt dipoleEdipole intertions re strongF st is in this kind of systems tht we re interestedF vet us stress tht the strong resonnt dipoleEdipole intertions in the tomi loud tht we refer to re nerE(eld intertions in free speD for whih the intertion potentil vries s 1/r 3 D where r is the interEtomi distneF e do not onsider modi(ed eletromgneti environment for the tomi loudF fy engineering the eletromgneti environment it is for exmple possile to indue strong dipoleEdipole intertions etween toms tht re not in eh other9s ner (eldF en exmple is the use of wveguide long whih toms re pled SHF le QFI shows in old the hrteristis of the tomi loud studied y the group of froweysF st is seen tht the loud is in the strong resonnt dipoleEdipole intertions regimeD so tht reurrent sttering e'ets re expeted from the lssil dipole modelF sn this hpter we will study theoretilly light sttering from old tomi louds tht hve experimentlly een studied y the group of froweysF he louds re di'erent from other reported louds in the sense tht resonnt dipoleEdipole intertions re expeted to e strongerF e will ompre the experimentlly otined spetrl nd temporl responses of the tomi louds with numeril lultionsF ome key experimentl oservtions tht grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw QV re disussed re negligile gv UD SID suppression of light sttering in the diretion orthogonl to the inident light em SID nd superluminl pulse propgtion through the tomi loud SPF

Atomic density

es (rst stepD oth ellegrino nd tennewein loded the toms in mgnetoEoptil trp @wyAF roweverD with suh trp one n typilly reh density of only ρ = 10 11 tGm 3 D or equivlently ρ/k 3 = 10 -4 D whih is not su0ient to hve strong resonnt dipoleE dipole intertionsF sn order to reh higher densityD the toms re eventully trpped in frEo'Eresonne dipole trp @pyA SQF etomi louds in these trps n reh higher densitiesF he tomi density of the louds studied y ellegrino nd tennewein follows the sme distriution s the qussin intensity pro(le of the trpping emF prom literture it is known tht the pek density t the enter of py is given y SRD SS

ρ peak = 2N λ trap πw 4 0 U 0 πk B T QW QFQF iiswixev i
where N is the totl numer of toms inside the trpD w 0 the wist of the trp lserD U 0 the trp depthD T the temperture of the ensemle of toms nd k B the foltzmnn onstntF sn the experiment presented in pigF QFP@AD the wist of the trp em is 1.2 µm nd fter (lling in the experimentl prmeters @λ trap = 940 nmD U 0 /k B = 1 muD T = 120 µuAD the pek density n e found to e 2 × 10 14 t/m 3 for N = 180 toms in igr shped wvelengthEsize tomi loudF e hve seen in le QFI tht this tomi density @ρ/k 3 = 0.38A is enough for the tomi loud to e in the strong resonnt dipoleEdipole intertion regimeF prom the tomi density we n estimte the onEresonne sttering men free pthF his prmeter tells us if the wvelengthEsize loud is in the single sttering or multiple sttering regimeF ith pek tomi density of ρ/k 3 = 0.38 we otin for the men free pthX

l mfp = 1 ρσ ext (ω 0 ) = k 2 6πρ = 17 nm. @QFTA
qiven the size of the tomi loud tht is on the order of the optil wvelengthD this tomi loud is lerly in the multiple sttering regimeF

Doppler eect

o hve strong resonnt dipoleEdipole intertionsD the toms need to hve ll the sme resonne frequenyF he presene of signi(nt hoppler rodening @∆ω > Γ 0 A would spred out the resonne frequeniesD therey reduing resonnt dipoleEdipole intertionsF o void strong hoppler rodeningD the tomi louds studied y oth ellegrino nd tennewein re t very low tempertures of out IHH µuF he orresponding hoppler rodening is ∼IHH krzD whih is smll ompred to the T wrz nturl linewidth of the investigted trnsitionF reneD inhomogeneous rodening due to hoppler e'et is negliE gileF st is oth the sene of inhomogeneous rodening nd the high tomi density tht mke the tomi systems s presented in pigF QFP interesting to study the in)uene of resonnt dipoleEdipole intertions on light sttering from n ensemle of tomsF

Probe beam and atomic transition

efter hving disussed the tomi systemsD let us give some informtion on the proe emF sn oth setupsD the qussin lser em hs wvelength λ L = 780 nmD orreE sponding to the trnsition wvelength of the h 2 trnsition etween the @5S 1/2 , F = 2A nd @5P 3/2 , F = 3A levels in ruidiumEVU 1 D nd spetrl width ∼ 0.3Γ 0 F prom this trnsition it is understood tht the EVU toms studied here re not lssil dipolesD sine lssil dipole orresponds to trnsition etween J = 0 nd J = 1 stte whih is fourElevel sysE tem with equl gleshEqordn oe0ientsF he toms we study here re IPElevel systems 2 D with nonEequl gleshEqordn oe0ientsF etD unless stted di'erentlyD ll lultions grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw RH tht hve een done in this thesis do ssume the toms to e lssil dipolesD sine it llows to understnd some interesting generl notions for resonnt light sttering from dense smplesF gonerning the wist of the proe emsD for the setup of ellegrino we n onsider the inident (eld to e plne wve s its wist w 0 is I mm nd therefore very lrge ompred to the size of the loudF he qussin em in the setup of tennewein is strongly foused to wist w 0 = 1.2 µmF sn oth setupsD the lser intensity is smll ompred to the sturtion intensity of the h 2 trnsition of single EVU tom @setup of ellegrinoX I/I s ≈ 0.1D where I s = 1.67 mGm 2 for irulrly polrized proeD nd for the setup of tenneweinX I/I s ≈ 0.02D where I s = 3.05 mGm 2 for linerly polrized proeAF he frequeny of the lser is tuned y the use of n oustoEopti modultor @eywA over frequeny rnge of UP wrz @= 12Γ 0 X -6Γ 0 < δω L < 6Γ 0 AD in order to oserve the spetrl response of the tomi loudF 3.3.4 Measured signal et lstD we disuss the signl tht is mesuredF por ellegrino9s experimentD n sEggh with polrizer in front ws usedD so they mesured the power orresponding to the eletri (eld omponents long the polrizer its diretionF sn the se of tennewein9s experimentD they used singleEmode (er to ollet the signlF reneD the signl otined y tennewein is not the totl energy )ow integrted over given solid ngleD ut it is the projetion of the totl (eld on the (er modeF he eletri (eld inside the singleEmode (er is given y E ber (r) = ˆEtot (r ) • g * (r )dS g(r), @QFUA where g is the normlized eletri (eld distriution of the (er modeD E tot = E inc + E sc D nd dS is di'erentil re element perpendiulr to the optil xisF he integrl in front of the eletri (eld distriution inside the (er determines how well the (er mode is exitedY it is n overlp integrlF sn the sene of the tomsD we hve E tot = E inc D the eletri (eld inside the (er is then E ber,0 (r) = ˆEinc (r ) • g * (r )dS g(r), @QFVA ht is tully mesured re powersF st n e shown tht

P P 0 = | ´Etot (r ) • g * (r )dS| 2 | ´Einc (r ) • g * (r )dS| 2 , @QFWA
where P is the mesured power in the presene of the tomsD nd P 0 the mesured power in the sene of the tomsF ithout tomsD mode mthing of the inident lser em nd the singleEmode (er hs een estlished experimentllyD so tht g ∝ E inc @the diretion of oth g nd E inc re the sme due to the polrizer in front of the (erAF ixperimentllyD there is less thn one photon deteted per inident pulseF sn order to hve good signlEtoE noise rtio @xAD detetion yle is repeted thousnd timesF he preise proedure n e found in efF UF feuse of these mny ylesD the mesured quntities n e onsidered to e ensemle vergedF o wht n e derived from the experiments is

P P 0 = | ´Etot (r ) • E * inc (r )dS| 2 | ´|E inc (r )| 2 dS| 2 , @QFIHA ✹ ✶ ✸ ✳ ✹ ✳ ❙ ❊❈❚|❆ ▲❈❍❆◆• ❊ ❲❍ ■ ▲ ❊❈|❖ ❙ ❙ ■◆•❚❍ ❊❋❖❈❆ ▲| ❊• ■❖◆ ✿•❖❯( ❍❆ ❙ ❊ ✇ ❤ ❡❡ ❤ ❡ ♦❛ •✜ ❡ • ❞❊ ♦ =❊ ✐ ♥ ❝ + ❊ ❝ +δ ❊ ❝ ✳ ❚ ❤ ❡ ♠ ❡ ❛ ✉❡ ❞ ✐ ❣ ♥ ❛ •❤ ❡❡ ❢ ♦❡❝ ♦ ♥❛ ✐ ♥ ❛ ❝ ♦ ❤ ❡❡ ♥ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ❤ ❡❡ ♥ ❝ ♦ ♥ ✐ ❜ ✉ ✐ ♦ ♥ ✳ ❇ ② ♠ ❡ ❛ ♥♦ ❢ ❤ ❡❈ ❛ ✉ ❝ ❤ ② ✲ ❙ ❝ ❤ ✇ ❛③ ✐ ♥ ❡✉ ❛ • ✐② ✱ ✐ ❝ ❛ ♥ ❜ ❡ ❤ ♦ ✇ ♥ ❤ ❛ ❤ ❡♣ ♦ ✇ ❡ P ♠ ❡ ❛ ✉❡ ❞ ✐ ♥ ❤ ✐ ✇ ❛ ② ✐ ♠ ❛ • • ❡ ❤ ❛ ♥♦ ❡✉ ❛ • ♦ ❤ ❡ ♦❛ • ♣ ♦ ✇ ❡ • ✉ ❜ ❡ ❢ ♦❡❤ ❡ ✐ ♥ ❣ • ❡ ✲♠ ♦ ❞ ❡✜ ❜ ❡ ✳❋ ♦♠❛♣ ❤ ②✐ ❝ ❛ •♣ ❡♣ ❡ ❝✐ ✈ ❡❤ ✐ ♠ ❛ ❦ ❡ ❡ ♥❡ ✱✐ ♥ ❝ ❡ ❛♣♦ • ❡ ❝✐ ♦ ♥♦ ❢❤ ❡❡ • ❡ ❝ ✐ ❝✜ ❡ • ❞ ✐❞ ♦ ♥ ❡♦ ♥ ❤ ❡✜ ❜ ❡ ♠ ♦ ❞ ❡ ✱✇ ❤ ✐ ❝ ❤ ♠ ❡ ❛ ♥ ❤ ❛ ❤ ❡ ♠ ❡ ❛✉❡ ❞ ♣ ♦ ✇ ❡ ✐❜ ♦ ✉ ♥ ❞ ❡ ❞ ❢♦♠❛ ❜ ♦ ✈ ❡❜ ② ❤ ❡ ♦❛ •♣ ♦ ✇ ❡ • ✉ ❜ ❡ ❢ ♦❡ ❤ ❡♦ ♣✐ ❝ ❛ •✜ ❜ ❡ ✳ ✸ ✳ ✹ ❙ ♣ ❡ ❝ ❛ • ❝ ❤ ❛ ♥ ❣ ❡ ✇ ❤ ✐ • ❡ ❝♦ ✐ ♥ ❣ ❤ ❡❢ ♦ ❝ ❛ • ❡ ❣ ✐ ♦ ♥ ✿ • ♦ ✉ ②♣ ❤ ❛ ❡ ❇ ❡ ❢ ♦❡✇ ❡ ✉ ❞ ② ❤ ❡ ♠ ❛ ✐ ♥❡ ① ♣ ❡✐♠ ❡ ♥ ❛ • ❡✉ • ✱ • ❡✉ ❞ ✐❝ ✉ ❢ ♦ ❝ ✉✐ ♥ ❣♦ ❢❤ ❡ • ❛❡ ❜ ❡ ❛♠ ♦ ♥ ❤ ❡❝ • ♦ ✉ ❞ ✳■ ♥ ❤ ❡ ❡♦ ♥ ❛ ♥ • ✐ ❣ ❤ ❝ ❛ ❡✐ ♥ ❣❡ ① ♣ ❡ ✐♠ ❡ ♥ ♦ ❢❏ ❡ ♥ ♥ ❡✇ ❡ ✐ ♥ ✱❜ ♦❤ ❤ ❡❛♦♠ ✐ ❝ ❝ • ♦ ✉ ❞❛ ♥ ❞ ❤ ❡ ✇ ❛ ✐ ♦ ❢ ❤ ❡ • ❛❡ ❜ ❡ ❛♠❛❡♦ ♥ ❤ ❡♦❞ ❡ ♦ ❢ ❤ ❡ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤♦ ❢ ❤ ❡ ✉ ❞ ✐ ❡ ❞ ❡ • ❡ ❝ ✐ ❝❞ ✐ ♣ ♦ • ❡ ❛ ♥✐ ✐ ♦ ♥✐ ♥ ✉ ❜ ✐ ❞ ✐ ✉♠ ✲ ✽ ✼ ✳ ■ ✐ ❤ ❡❡ ❢ ♦❡✈ ❡②❝ ❤ ❛ • • ❡ ♥ ❣ ✐ ♥ ❣ ♦❢ ♦ ❝ ✉ ❤ ❡ • ❛ ❡ ❜ ❡ ❛♠❝ ♦ ❡ ❝• ② ✱✐ ✳ ❡ ✳ ✱♦ ♥❤ ❡❝ ❡ ♥❡ ♦ ❢ ❤ ❡❛♦♠ ✐ ❝❝ • ♦ ✉ ❞ ✳ ❖ ♥ ❡ ♠ ✐ ❣ ❤ ❤ ✐ ♥ ❦ ❤ ❛ ❤ ❡ ❝ ♦ ♥❡✉ ❡ ♥ ❝ ❡♦ ❢♥ ♦ ❤ ❛ ✈ ✐ ♥ ❣ ❤ ❡❢ ♦ ❝ ✉ ❡ ① ❛ ❝• ②✐ ♥ ❤ ❡❝ ❡ ♥❡ ♦ ❢ ❤ ❡❝ • ♦ ✉ ❞❞ ♦ ❡ ♥ ♦ ❛ •❡ ❤ ❡ ♠ ❡ ❛✉❡ ❞ ♣ ❡ ❝ ✉♠ ✐ ❣ ♥ ✐ ✜ ❝ ❛ ♥ • ② ✱ ✐ ♥ ❝ ❡❤ ❡❝ • ♦ ✉ ❞✐ ♠ ❛ • •❝ ♦♠ ♣ ❛ ❡ ❞ ♦ ❤ ❡ | ❛ ② • ❡ ✐ ❣ ❤ ❞ ✐ ❛ ♥ ❝ ❡✭ z R =5. 8µ ♠ ✮❛ ♥ ❞ ❤ ❡❡ ❢ ♦❡ ❤ ❡ ✐ ♥❡ ♥✐ ②♦ ❢ ❤ ❡❜ ❡ ❛♠❡ ① ♣ ❡✐ ❡ ♥ ❝ ❡ ❞❜ ②❤ ❡❝ • ♦ ✉ ❞ ❞ ♦ ❡ ♥ ♦ ❝ ❤ ❛ ♥ ❣ ❡❛• ♦ ❛ ❤ ❡❢ ♦ ❝ ✉ ♣ ♦ ♠ ♦ ✈ ❡ • ♦ ♥ ❣ ✐ ✉ ❞ ✐ ♥ ❛ • • ② ✳ ❋ ✐ ❣ ✉❡✸ ✳ ✸ -4 0 -2 0 0 2 0 4 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 f -f 0 (MH z ) 〈 P 〉 / P 0 6 0 a t om s , z 1 6 3 a t om s , z 2 ❤ ♦ ✇ ✇ ♦ ✭ ❛ ❛ ♥ ❞❞ ♦ ✮❡ ① ♣ ❡✐♠ ❡ ♥ ❛ • • ②♦ ❜❛ ✐ ♥ ❡ ❞ ♣ ❡ ❝ ❛ ✐ ♥ ❤ ❡ ❢ ♦✇ ❛❞❞ ✐❡ ❝✐ ♦ ♥ ✳ ❊ ❛ ❝ ❤♣ ❡ ❝ ✉♠ ❝ ♦ ❡♣ ♦ ♥ ❞ ♦❛ ♥❡ ① ♣ ❡✐♠ ❡ ♥ ❤ ❛ ❤ ❛ ❜ ❡ ❡ ♥❞ ♦ ♥ ❡❛ ❛❞ ✐ ✛ ❡❡ ♥ ♠ ♦♠ ❡ ♥ ✐ ♥✐♠ ❡ ✳ ❊ ❛ ❝ ❤ ❝ • ♦ ✉ ❞❝ ♦ ♥❛ ✐ ♥ ❛ ❜ ♦ ✉ ❤ ❡ ❛♠ ❡♥ ✉♠ ❜ ❡ ♦ ❢❛♦♠ ✭ ✻ ✵❛ ♥ ❞✻ ✸❛♦♠✮ ✱❜ ✉ ❤ ❡♣ ♦✐ ✐ ♦ ♥♦ ❢❤ ❡ ❛♦♠ ✐ ❝❝ • ♦ ✉ ❞✇ ✐❤ ❡♣ ❡ ❝ ♦ ❤ ❡ ❢ ♦ ❝ ✉ ✐♥ ♦ ❤ ❡ ❛♠ ❡ ✳❏ ❡ ♥ ♥ ❡✇ ❡ ✐ ♥ ♠ ❡ ❛✉❡ ❞ ❤ ❡❞ ✐ ✛ ❡❡ ♥ ❝ ❡ ❋ ✐ ❣ ✉ ❡✸ ✳ ✸ ✿❊ ① ♣ ❡✐♠ ❡ ♥ ❛ • ❡✉ • ♦ ❜❛ ✐ ♥ ❡ ❞✇ ✐❤❛❝ ♦ • • ❡ ❝✐ ♥ ❣ • ❡ ♥❛ • ♦ ♥ ❣ ❤ ❡♦ ♣✐ ❝ ❛ •❛ ① ✐❛ ♥ ❞ ❛♥ ✉♠ ❡✐ ❝ ❛ •❛ ♣ ❡✉❡ ◆❆=0. 5 ✳ ❚ ❤ ❡♣ ♦✐ ✐ ♦ ♥♦ ❢❤ ❡ • ❛❡ ❜ ❡ ❛♠❢ ♦ ❝ ✉ ✇ ✐❤ ❡♣ ❡ ❝ ♦ ❤ ❡ ❝ • ♦ ✉ ❞✇ ❛ ❞ ✐ ✛ ❡❡ ♥ ✐ ♥❜ ♦ ❤❡ ① ♣ ❡✐♠ ❡ ♥ ✭ ✐ ♥ ❞ ✐ ❝ ❛❡ ❞❜ ②z 1 ❛ ♥ ❞z 2 ✮ ✳ ❇ ♦❤❡ ① ♣ ❡✐♠ ❡ ♥ ❛ • ❡✉ • ❛❡✜❡ ❞❜ ②❤ ❡ ❢ ✉ ♥ ❝✐ ♦ ♥ ✿y=| c 1 - c 2 1 -2 i ( x -c 3 ) / c 4 e i c 5 | 2 ✳❋ ♦❛ ♥❡ ① ♣ • ❛ ♥ ❛✐ ♦ ♥♦ ❢❤ ✐ ✜ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ✱ ❡ ❢ ❡ ♦❙ ❡ ❝✐ ♦ ♥ ✸ ✳ ✹ ✳ ✶ ✳
grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw RP of the position of the tomi louds to e z 2 -z 1 = 2.5 ± 0.3 µmD whih is smll ompred to the yleigh distneF he intensity of the em on the loud hs therefore hnged negligilyF roweverD it is seen from pigF QFQ tht lthough the louds ontin out the sme numer of tomsD nd the intensity of the proe em did not hnge signi(ntlyD oth spetr re not the smeF xotlyD the symmetry of the spetr is not the smeF vet us mention tht similr results were otined in the group of rveyF sn tht groupD it ws oserved tht the trnsmission spetrum of ig loud of toms is symmetri in frequeny nd the symmetry of the trnsmission spetrum )ips sign when the fous of the em rosses the tomi loud STF he oserved symmetri trnsmission spetrum ws pulished in the work of oof et al.D entitled wirosopi lensing y denseD old tomi smple STF he )ip of the symmetry in the frequeny response ws ttriuted to fousing nd defousing e'etsY therey ttriuting the symmetry to lensing e'et of their loudF foth fousing nd defousing exist in the studied loudD sine the tomi loud is ig @255 × 3.5 × 3.5 µm 3 AD nd the lulted refrtive index of the loud n ttin vlues smller nd lrger thn one for di'erent frequeniesF e refrtive index tht is smller thn one gives rise to defousing nd refrtive index lrger thn one leds to fousingF he tomi system studied y tennewein et al. is di'erentD sine tht loud is very smll nd therefore geometril optis is not vlidF sn this setionD we propose n lterntive explntion of the oserved dependene of the spetrum on the longitudinl position of the em fous with respet to the loudF 3.4.1 Toy model for light scattering o explin the di'erent spetr for louds tht were very similr in size nd numer of tomsD see pigF QFQD we study toy model whih only ontins essentil properties to underE stnd the physisF his toy model is single tomD whih position is vriedF e horizontlly polrized inident qussin emD propgting long the z xisD hs its fous t position (x f , y f , z f ) = (0, 0, 0)D while the tom is moved long the optil xis nd hs position (0, 0, z atom )F he eletri (eld of the emD within the prxil em pproximtionD is given y

E L (ρ, z) = E 0 w 0 w(z) e -ρ 2 w(z) 2 e i[kz+k ρ 2 2R(z) -ψ(z)] , @QFIIA
where ρ = x 2 + y 2 is the distne to the optil xisD w(z) = w 0 1 + (z/z R ) 2 the distne to the optil xis over whih the intensity drops y ftor

1/e 2 D R(z) = z[1 + (z R /z) 2 ] is the rdius of urvtureD nd ψ(z) = arctan(z/z R )
is the qouy phseF yn xis @ρ = 0A it is seen from iqF QFII tht the phse does not evolve the sme s the phse of plne wveF here is n dditionl phse termD lled the qouy phseF his qouy phse is preisely the phse di'erene qussin em quires during propgtion s ompred to plne wve propgting long the z xisF he origin of the qouy phse is the ft tht round fousD the verge propgtion onstnt is smller for qussin em thn for plne wveF por plne wve propgting long the z xisD the propgtion onstnt k z = k 0 F es qussin em hs smller verge propgtion onstnt round fousD its phse evolves slower during propgtion s ompred to the phse of plne wve propgting long the optil xisF st is this di'erene in the quired phse tht is lled the qouy phseF he qouy phse is the key element of this setionF RQ QFRF igev grexqi rsvi gysxq ri pygev iqsyxX qy rei he em indues n eletri dipole moment p = ε 0 α(ω)E L (0, 0, z atom )D where α(ω) is the tomi polrizility of single tomD given y

α(ω) = 3πΓ 0 /k 3 ω 0 -ω -i Γ 0 2 .
@QFIPA he indued dipole retes n eletri (eld whih is given y the qreen9s tensorF es we mesure fr wy from the tomD iFeFD kz D 1D nd z D z atom D where the susript D stnds for detetorD we only need to onsider the frE(eld term of the qreen9s tensorF he sttered (eld long the optil xis is then given y

E sc = µ 0 ω 2 e ik|z D -zatom| 4π|z D -z atom | p ≈ µ 0 ω 2 e ik|z D -zatom| 4π|z D | p.
@QFIQA he position of oservtion is tken to e positiveX z D > 0 nd the sttered (eld is then

E sc (0, 0, z D ) = ω 2 c 2 α(ω) e ik(z D -zatom) 4πz D E L (0, 0, z atom ).
@QFIRA he inident eletri (eld is simply E L (0, 0, z D ) t the oservtion positionF hen we dd the sttered eletri (eld to the inident eletri (eldD we otin the totl eletri (eld t the position of the detetorX zatom) . @QFITA prom the totl eletri (eldD one n redily otin the normlized totl power on xis

E tot = -iE 0 z R z D e ikz D + ω 2 4πc 2 α(ω)E 0 1 1 + zatom z R 2 e -iψ(zatom) e ikz D z D @QFISA = -iE 0 z R z D e ikz D 1 + i ω 2 4πc 2 z R α(ω) 1 1 + zatom z R 2 e -iψ(
P (ω) P 0 = 1 + i ω 2 4πc 2 z R α(ω) 1 1 + zatom z R 2 e -iψ(zatom) 2 , @QFIUA
where P 0 is the inident power in the forwrd diretion when no toms re presentF hen we ssume ω ≈ ω 0 D we n (nd from this formulD nd the expression for the polrizility @see iqF QFIPAD the (t funtion y = |1 -

c 2 1-2i(x-c 3 )/c 4 e ic 5
| 2 D whih is used in pigF QFQ to (t the experimentl results 3 F ell prmeters re free prmetersF he oe0ient c 3 orresponds to possile frequeny shift of the sttererD c 4 is the linewidth of the trnsitionD nd c 5 is the qouy phseF elthough this model for the spetrum follows from light sttering from single tomD it lso orretly models light sttering from sphere with resonnt dieletri onstnt ε(ω)D tht mthemtilly hs similr resonne s single tomD nd rdius a muh smller thn the wvelength if the eletri (eld inside is uniformF he reson this model lso works for smll resonnt spheres is tht in tht se the optil response n lso e desried y polrizility α(ω) = 4πa 3 [ε(ω) -1]/[ε(ω) + 2] IRF es we hve seen in pigF QFQD the sme (t funtion works lso well for the elongted loud of tomsF 3 Because of the way the measured data points were normalized in Fig. 3.3 (the maximum value of |S(ω)| 2 is set to 1), we need to leave the rst term also as a free parameter:

y = |c 1 - c2 1-2i(x-c3)/c4 e ic5 | 2 ,
where all parameters are real.

grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw RR 3.4.2 Explanation of asymmetric spectrum vet us develop the expression given in iqF QFIUF st n e written in the form

P (ω) P 0 = 1 + A|α(ω)| 2 + B (ω 0 -ω) sin ψ(z atom ) - Γ 0 2 cos ψ(z atom ) , @QFIVA
where A nd B re oth rel nd positiveF he third termD whih is proportionl to the prmeter BD origintes from interferenes etween the inident (eld nd the sttered (eldF his is very importnt notion s it follows from this interferene term tht wht is mesured with spetrosopi mesurement re interferenes etween the inident em nd the sttered (eldF st is preisely the interferene term tht introdues dependene of the mesured power on the tomi positionF hen the tom is positioned in the fousD ψ(z atom ) = 0F ine |α(ω)| 2 is symmetri round the resonne frequenyD P (ω)/P 0 is lso symmetri round the resonne frequeny for this speil seF gontrrilyD when the tom is pled out of fousD ψ(z atom ) = 0 nd thus the spetrum eomes symmetriF st is oserved from the term (ω 0 -ω) sin ψ(z atom ) tht the symmetry of the spetrum )ips sign when the tom rosses the fousF es n illustrtionD we hve numerilly lulted the onExis sttering spetrum for qussin em with wist w 0 = 1.2 µm inident upon single tom tht is put in the fousD nd oth I µm efore nd fter the fousF he spetrum is given in pigF QFRF pigure QFR shows tht the symmetry of the spetrum indeed )ips sign when the tom rosses the fousF prom the previous disussion it follows tht wht is mesured re interferenes etween the inident nd sttered (eldsF his mens tht the spetrum does not only ontin intrinsi properties of the tom @here its polrizility nd in the work of oof

RS

QFSF geisxq isyx the index of refrtionAF he spetrum therefore re)ets oth the intrinsi properties of the tom nd the geometril on(gurtionF st is importnt to relize tht when we would hve deteted ll the sttered light round the tomD the reltive power P/P 0 would e one for ll frequeny detuningsD sine we onsider elsti light sttering for whih ll inident light is stteredF he interferenes etween the inident nd sttered (eld re responsile for redistriution of the energyF purthermoreD for (nite perturesD where the numeril perture is smller thn the em spredingD it is possile to hve frequeny rnge in whih the trnsmission is superior to oneF sn the words of oof et al. this origintes from lensing e'et STF eprt from the symmetri spetrum whih rises due to displement of the tomD the dip lso shifts in frequenyD whih (nds its origin gin in interferenesF hen the tom is positioned in the fol pointD the dip frequeny orresponds to the tomi frequenyF his is not nymore true when the tom gets displedF he dip position therefore does not neessrily re)et the resonne frequenyF xow tht we hve disussed the in)uene of the qouy phse on the symmetri spetrumD we ome k to the experimentl results tht re shown in pigF QFQF elthough the two louds were of the sme size nd ontined more or less the sme numer of tomsD the symmetry is not the smeF hen we (tted the experimentl dt with the funtionX y = |c 1 -c 2 1-2i(x-c 3 )/c 4 e ic 5 | 2 D we dedued tht the distne etween the enters of the two di'erent louds ws 2.0 ± 0.6 µmF ixperimentlly this distne ws mesured to e 2.5 ± 0.3 µmD y whih we on(rm tht the qouy phse is indeed possile explntion for the oserved di'erene in spetr etween two similr loudsF purthermoreD it hs experimentlly een veri(ed tht the (tted qouy phse @c 5 in the (t funtionA ws independent of the numer of toms in the loudF his rules out the possiility of lensing e'et in our loudD sine the quired phse during propgtion would depend on the index of refrtionD nd therefore on the tomi densityF reneD the retrieved phse would depend on the tomi densityD if tennewein et al. oserved lensing e'et UF sn this setion we hve shown tht the results of pigF QFQ re n experimentl demonE strtion of the in)uene of the qouy phse on the mesured spetrumF e simple model tht tkes into ount the qouy phse is le to determine the distne etween the fol points of di'erent experimentl reliztionsY therefore showing the ility to filitte the fousing of the lser emF elthough this notion is lredy known in for exmple the (eld of goherent ntiEtokes mn sttering @geAD the qouy phse is known for giving rise to symmetri spetr SUD it is not to the est of our knowledge in the (eld of old tomsF 3.5 Scattering suppression e now study the experimentl results otined y toseph ellegrino SI nd tephn tennewein UF e follow the hronologil order in whih the experiments were done nd therefore strt disussing the experimentl results of ellegrinoF 3.5.1 Experimental results of Pellegrino et al. sn the setup s shown in pigF QFP@AD ellegrino olleted the sttered light t 90 • with respet to the optil xisF pigure QFS@A shows the ensemleEverged sttered power s grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw RT funtion of the numer of toms inside the loudF ixperimentllyD n ensemle verge orE responds to the mesurement of given quntity over severl experimentl runsF he power mesured y n xe = 0.5 optil system is normlized to the mesured sttered power for single tom under illumintion with the sme frequeny detuningF he quntity P /P 0,∆ n therefore e understood s the verge numer of toms e'etively sttering lightF he experiments were done for the frequeny detunings ∆ω L = {-2.5, -1, 0, 1, 2.5} × Γ 0 D whih re represented y di'erent mrkers in pigF QFS@AF prom pigF QFS@A it n e seen tht the loser the lser frequeny is to the resonne frequenyD the smller the e'etive numer of toms sttering light t 90 • isF ewy from tomi resonneD the sttered power seems to tend towrds the dshed lineF he dshed line represents yleigh stteringD for whih the oserved sttered power would inrese linerly with the numer of tomsF gompred to gs of nonEinterting tomsD ellegrino hs therefore oserved suppression of light stteringF he lrgest suppression of light sttering ours on resonneD where resonnt dipoleEdipole intertions re the strongestF st is therefore the resonnt nture of the intertions tht drmtilly lters the optil response of the ensemle of tomsF his oservtion is remrkleD sine light sttering is lerly very di'erent in the EVU loud s ompred to yleigh sttering from the skyY knowing tht the sky is (ve orders of mgnitude denser thn the loud of ruidium tomsF e now proeed to numeril lultions on similr system where the toms re modeled s lssil dipolesF 3.5.2 Numerical calculations on suppression of light scattering en dvntge of numeril lultions over experiments in this prtiulr se is tht we hve ess to the full sttering pttern nd tht we n seprte oherent nd inoherent light sttering esilyD s ws explined in etion PFPFQF por n onEresonne irulrly polrized inident plne wveD we lulte the sttered power t 90 • with respet to the propgtion diretion of the plne wve nd with numeril perture xe = 0.5F pigure QFS@A shows the sme quntity s hs een mesured experimentllyX the totl sttered power reltive to the sttered power from single tomF he dshed line repreE sents gin yleigh stteringF st is seen tht s the numer of stterers ugmentsD the sttered power t 90 • inresesD ut the inrese delinesF oD numerilly we lso (nd suppression of light stteringD ut it is stronger thn ws found experimentllyF sn order to understnd why light sttering gets suppressedD we hve lulted the onEresonne ensemleEverged sttering pttern for loud ontining N = 300 tomsF he result is presented in pigF QFS@AF es n e seenD there is ig loe in the forwrd diretionD remE inesent of diraction from the tomi loudF eprt from the di'rtion ptternD there is lso light tht gets sttered qusi isotropillyD s n e seen in the (gure t the height of the rrow inditing 90 • F st is seen tht di'rtion is n importnt prt of light stteringF his mens tht the e'etive refrtive index is very di'erent from IF he density )utuE tions inside the tomi loud get illuminted y smller eletri (eld thn the inident (eldD due to whih inoherent light sttering stops growing linerly with the numer of tomsY it gets suppressedF prom di'rtion theory it is known tht the di'rtion pttern of suh loud is diretionl nd peks in the forwrd diretionF roweverD t 90 • there is no di'rtionD nd it is inoherent light sttering tht domintesF hen the density of the tomi loud inresesD we oserve tht di'rtion strts plying n importnt role for light stteringD wheres the importne of inoherent light sttering gets smllerF qoing from the regime where inoherent light sttering domintes to regime where oherent light sttering domintes forms homogeniztion prolemF sn ghpter S we will disuss the homogeniztion prolem for resonnt light sttering from dense loud of old tomsF grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw RV 3.5.3 Towards coherent light scattering e re prtiulrly interested in the oherent response of the tomi loudD sine we n ompre this form of light sttering to the mrosopi wxwell equtions where the loud is modeled y n e'etive refrtive indexF prom di'rtion theory it is known tht in the forwrd diretion onstrutive interferenes existD due to whih enhned light sttering n e oservedF tennewein et al. hve studied the oherent optil response y putting the detetor long the optil xisF por the remining prt of this hpterD we will thus lso disuss the experimentl results otined y tenneweinF sn prtiulrD we will disuss the frequeny response of oth the louds studied y ellegrino nd tennewein in etion QFTF 3.6 Spectral response of the atomic cloud sn the lst setion we were minly interested in wht hppens to light sttering when the numer of toms inresesF e hve seen tht suppression of light sttering oursF sn this setionD we fous on the study of the spetrl response of oth the tomi louds of ellegrino @detetion t 90 • from optil xisA nd tennewein @detetion in the forwrd diretionAF he former will e refered to s inoherent spetrl response nd the ltter to oherent spetrl response for resons tht hve een explined in etion QFSF 3.6.1 Incoherent spectral response he experimentlly otined powers y ellegrino hve een plotted in pigF QFT@AD where the olor of the dots nd the orresponding vorentzin (ts orresponds to given numer of toms in the loudF e onsider louds with N = {1, 5, 20, 50, 200, 325, 450} toms insideF he mesured power hs een normlized to the mesured sttered power from single tom in the loud illuminted on resonneF por ll frequeny detunings it is oserved tht s the numer of toms ugments @ottom to topAD the sttered power inreses s wellD ut sturtesY n oservtion tht we hve lredy mde in etion QFSF ome other oservtions re spetrl rodening nd very smll red frequeny shiftF e spetrl rodening is in generl expetedF o (rst orderD this n e understood y the ft tht resonnt dipoleEdipole intertions slightly move the energy levelsF his e'etively inreses the linewidth of the ensemle of tomsF ith regrd to the smll frequeny shiftX this seems to e in qulittive greement with reently pulished work where the uthors lim tht no overt vorentzEvorenz lol (eld shift of the resonneD nor oopertive vm shift exist in homogeneously rodened tomi smples SVF e hve lso lulted the normlized optil response t 90 • F pigure QFT@A shows the theoretil resultsD where the vertil xis hs log sleF he onditions of the lultions re the sme s desried in the previous setionF st is oserved tht the theoretilly otined spetr re vorentzin for louds with smll numer of tomsF es the numer of toms inresesD the spetr strt to devite from vorentzinY n oservtion tht is in disgreement with the experimentl resultsF he origin of the hnge of shpe of the spetrum for inresing tomi density is the inrese of the resonnt dipoleEdipole intertions tht e'etively redue the sttering men free pth nd therey tke the wvelengthEsize loud out of the single sttering regimeF his oservtion does not question the vlidity of the rgument used to explin suppression of light stteringD ut it does provoke the following questionX why does ellegrino not mesure spetrum tht is di'erent RW QFTF igev iyxi yp ri eywsg gvyh from vorentzinc st is most likely tht the resonnt dipoleEdipole intertions in the tul experiments re weker thn in our modelF ome possile resons for this re the ft tht EVU hs in relity multilevel strutureD nd tht dipoleEdipole fores might hinder the toms to pproh eh otherD therey e'etively reduing the intertionF grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw SH 3.6.2 Coherent spectral response sn the experiments done y tennewein et al.D light is deteted in the forwrd diretionF prom etion QFSD it is understood tht in tht experiment the oherent optil response of the loud is mesuredF pigure QFU shows shemtilly linerly polrized foussed qussin em @redA tht is inident upon the @lueA loud of EVU tomsF xote tht the tomi loud of tennewein does not hve the sme size s the tomi loud studied y ellegrinoD see pigF QFPF yn the right of pigF QFUD the di'rtion pttern @| E | 2 A of loud with N = 180 toms on tomi resonne hs een shownF he numeril perture of the optil system orresponds to full ngle of 60 • nd is indited y the old dshed linesF st is oserved tht essentilly ll the oherently sttered light in the forwrd diretion rehes the lensF pigure QFV@A shows the mesured normlized powers tht were otined y tennewein for di'erent lser detuningsF he normliztion mounts to dividing the mesured power y the mesured power in the sene of the tomsF he (lled mrkers re the experimentl dt points tht elong to the loud of tennewein with N = {10, 83, 180} toms @top to ottomAF e oserve similr hrteristis for the spetrum s we hve previously seen for the setup of ellegrino in pigF QFTX spetrl rodening see pigF QFV@dAD where the spetrl width hs een plotted s funtion of the numer of toms N in the loudD very smll frequeny shift of the dip see pigF QFV@AD nd minimum vlue of the dip of the normlized power whih sturtes see pigF QFV@AF hen we look t the spetrl shift in pigF QFV@AD we note tht it is it lrger thn the tul position of the dip in pigF QFV@AF e likely reson for this is tht the lser ws not perfetly foused on the enter of the loudD giving rise to hnge in the spetrum due to the qouy phseF The theoretical spectra are obtained by using the coupled-dipole equations and assuming the atoms to be classical dipoles. The experimental data have been tted with the function

y = |1 - c 2 1-2i(x-c 3 )/c 4 e ic 5
| 2 and the theoretical data have been interpolated with the function y = |1 - with the predition of menE(eld theory QVF imilrlyD efF RU disusses light sttering from loud of old EVU toms with pek density vrying up to ρ peak = 6 × 10 12 tGm 3 D whih orresponds to ρ/k 3 = 0.01D nd they hve lso oserved frequeny shift tht grees with menE(eld theory QUD SWF he mesured frequeny shift is sustntilX 2Γ 0 RUD lthough the system is more dilute thn the system studied y tennewein et al. feuse of wek resonnt dipoleEdipole intertionsD the tomi loud is not in the regime s desried in efF SV nd menE(eld theory preditions re therefore expeted to e orretF prom the qulittive greement of tennewein9s results with the preditions of tvninen et al.D it seems tht light sttering from the loud of tennewein indeed needs to e desried y eyond menE(eld theoryF e will therefore hek if we n reprodue similr results with n ext lultion of the sttered power y mens of the set of oupledEdipole equtionsF he inident (eld is tken to e qussin emD whih hs n ngulr spred of sin(θ) = λ L /(πw 0 ) = 0.21F ine it is strongly foused em @wist of IFP µmAD we do not use the prxil em pproximtionF he eletri (eld is written s plne wve deomposition THX

c 2 1-2i(x-c 3 )/c 4 e ic 5 - c 6 1-2i(x-c 7 )/c 8 e ic 9 | 2
E L (r , z) = w 2 0 k 2 4π ¨e-w 2 0 k 2 s 2 4 u s/p (ν)e iks•r +ikcz d 2 s, @QFIWA
where

k = ω L c D ν = s, √ 1 -s 2 D s = k k D nd the polriztion u s = z×ν |z×ν| or u p = ν × u s @s
stnding for iEpolrized em nd p for wEpolrized emAF sn our on(gurtion there is not ig di'erene etween i nd w polriztionD euse of the rottionl symmetry of the prolem fter the ensemle vergingF he integrl is rewritten in polr oordintes @|s| nd θAD where the ngulr integrl is done nlytillyF fy doing soD there is only single integrl to e evluted numerillyF e hve lulted the power for severl detunings nd like for the experiment we lulted light sttering for N = {10, 83, 180} tomsD while tking into ount the presene of the singleEmode (erF he results of the lultions re shown in pigF QFV@A y the open mrkersD where the dshed lines re used to guide the eyeF st is oserved tht these numeril results do not gree with the experimentl resultsF xotlyD seond resonne dip ppers for oth the N = 83 nd N = 180 loudsF his seond resonne is not singleEtom resonneD ut resonne of the shpe of the loudF ixperimentllyD the presene of seond resonne hs never een oserved y tenneweinF e question whih remins open is why this seond resonne hs not een oservedF por more thorough disussionD refer to tennewein et al. in efF UF

Superluminal pulse propagation through an atomic cloud

sn this setion we disuss the temporl properties of pulse propgting through the tomi loudF st is known tht pulse propgtion in resonnt medium n led to fst or slow lightD due to the dispersive response of suh resonnt medium TI!TRF pst @slowA light orresponds to sitution where the pek of the outoming pulse ould e deteted erlier @lterA thn the pek of pulse whih would trvel through vuumF pst light orresponds therefore to sitution where the peak of the pulse trvels fster thn the speed of light in vuumF en interesting pplition for slow light would e n optil SQ QFUF ivwsxev vi yeqesyx ryqr ex eywsg gvyh dely lineF pst light lso (nds interest in the (eld of quntum informtion proessingD sine it is ompnied with n imprinted phse shift on the lser (eldD whih is strongly frequeny dependentF elthough there hve een severl exmples of superluminl pulse propgtion in n tomi loud TSD TTD we highlight the pulse dvnement tht hs een reported in hot vpor of EVU y the group of edmsF ueveney et al. mesured n dvne of n optil pulse of more thn 100 ps over propgtion distne of QWH nm QWF tennewein hs done n experiment where the temporl properties of pulse propgtion through the old tomi loud were studiedF re mesured the group dely for N = 170 toms in the loud nd for di'erent lser detuningsF he group dely is de(ned y the dely of the peak of the pulseF pigure QFW shows the mesured group dely s funtion of the frequeny detuningF he solid urve follows from (tD s explined in tennewein et al. SPF sn the sme workD one n (nd disussion out the symmetry visile in this urveF por our disussionD this spet is irrelevntF tennewein hs mesured negtive group delysD whih mens group dvneF re mesured group dvne s lrge s IH nsD whih is two orders of mgnitude lrger thn oserved y ueveney et al.D lthough the loud is only one order of mgnitude longerF The data points have been tted. For an explanation of this t, refer to Ref. [52]. Group delays as large as -10 ns have been measured.

xextD we lulte the group dely inside the tomi loud of tennewein y mens of lssil eletrodynmis lultionF he lultion is stedyEstte lultionD where monohromti qussin em is inident on the loudF st is essentilly the sme lultion whih led to pigF QFUF he phse φ of the eletri (eld in the forwrd diretion @fr (eldD z z R , LD where L is the typil size of the loudA is lulted s funtion of the normlized lser detuning ∆/Γ = (ω L -ω 0 )/ΓF prom thisD the group dely τ n e lulted with the formul TI τ = dφ dω . @QFPHA ine the phse φ vries strongly over frequeny rnge on the order of Γ 0 D this equlity is understood to e vlid only when the ndwidth of the optil pulse is very smll ompred to the singleEtom spontneous dey rteF ememer tht the ndwidth of the optil pulse is 0.3 × Γ in the experimentsF his might redue the oserved time dely during experimentsF grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw SR en exmple of the phse φ for N = 12 toms inside the loud of tennewein s funtion of the detuning is given in pigF QFIH@A @lk irlesAD where the lue solid line is phenomenologil (tF st is oserved tht the phse of the totl (eld vries lose to tomi resonneF por denser loud of tomsD eFgF loud with N = 170D the phse hs lso een plotted in pigF QFV@A @red dotsAF he smll step visile for N = 170 is reminesent of the presene of seond resonne in the systemF prom φ(ω)D we n derive the group delyF sn order to lulte the derivtiveD we mde use of the phenomenologil (t for N = 170 see pigF QFIH@AF e oserve in pigF QFIH@A tht numerilly the sme order of mgnitude for the group dely is found s ws found experimentllyD yet ig di'erene is seenX there re two dipsD s opposed to only one for the experimentl results given in pigF QFWF his is onsistent with the spetrl dt of pigF QFV@AD where seond resonne ws lso sent in the experimentF his shows tht we systemtilly do not oserve experimentlly the presene of seond resonneD resonne tht is predited y lssil theoryF xextD we de(ne the mximum dvne y the vlue of τ t the dipF sn the se of two dipsD we tke the left dipD s indited y the dshed line in pigF QFIH@AF he lultion hs een done for severl numers of toms inside the loudF pigure QFIH@A shows tht the lrgest dvne rnges from few nnoseonds to the order of ten nnoseondsD depending on the

-1.8 -1.6 -1.4 -1.2 -1 φ[rad] -1.5 -1.45 -1.4 -1.35 -1.3 φ[rad]

SS

QFVF wwe numer of toms inside the loudF st is oserved tht the mximum time dvne redues por N > 50F he reson for this is the spetrl rodening of the systemD whih mkes the slope of the φ(ω)Eurve smootherF sn the experimentD redution of the time dvne when the numer of toms inreses hs never een oservedD sine one nnot lod n ritrrily lrge numer of toms in the loudF 3.8 Summary sn this hpter we hve desried nd nlyzed the experiments tht hve een onduted y ellegrino nd tenneweinF he tomi louds re oth dense nd oldD mking it thereE fore interesting to study the in)uene of resonnt dipoleEdipole intertion on light stE teringF he wy the lser em ws foused on the loudD ws sed on the experimentl ft tht the symmetry of the spetrum )ips sign when the fous rosses the enter of the loudF sn this hpter we hve proposed physil explntion of this phenomenon tht is sed on the qouy phseF prom theoretil onsidertions it ws understood tht ellegrino studied inoherent light stteringF tennewein hs hnged the experiment in order to lso mesure oherent light sttering in the forwrd diretionF he experimentlly otined spetr from elleE grino nd tennewein show very smll shift of the trnsmission dipF fy solving the set of oupledEdipole equtionsD we lso found very smll frequeny shiftF rowever for oth the spetrum nd the group dely we oserve in the numeril simultions the presene of seond resonneD s opposed to wht hs een oserved experimentllyF o fr it remins n open question why the experimentl results do not show seond resonneF prom literture studyD it follows tht wekly interting tomi systems re properly desried with menE(eld theoryF es opposed to these systemsD the systems studied y ellegrino nd tenneweinD where intertions re strongerD nnot e understood from menE(eld theoryF xo theoretil modelD tht we re fmilir withD exists so fr tht is le to reprodue the experimentl resultsF Chapter 4

Polaritonic modes in a dense cloud of cold atoms 4.1 Motivation sn the previous hpter we studied light sttering from loud of old tomsF elthough the tomi loud we study is muh more dilute thn the skyD we hve seen tht resonnt light sttering from old wvelengthEsize tomi loud is very di'erent from yleigh stteringF e hve disussed in the lst two hpters tht the reson for this lies in the presene of strong resonnt dipoleEdipole intertions in the system under studyF e hve seen tht s the tomi density inresesD light sttering hnges from purely di'usive to prtilly di'usive nd prtilly di'rtive regimeF sn the ltter regimeD light sttering from the loud ours s if the loud were n ojet tht is desried y n e'etive refrtive indexF hen (niteEsize ojet is desried y n index of refrtion tht is di'erent from its optil environmentD the ojet is known to di'rt lightF eprt from this mrosopi point of view of light stteringD we n lso interpret di'rtion in nother wyX wht hppens t mirosopi level tht mkes the tomi loud di'rting lightc e similr question would eX wht is the physil origin t the mirosopi level of the index of refrtionD whih is mrosopi notionD nd mkes n ojet di'rting lightc st is in this hpter tht we study the onnetion etween mirosopi nd mrosopi desription of n ensemle of homogeneously rodened stterersF roweverD the onlusions drwn in this hpter re expeted to e orret s well for inhomogeneously rodened systems like dense ensemle of quntum dotsF eprt from etion RFVD this hpter hs een dpted from our reently pulished pperD lled olritoni modes in dense loud of old toms TUF 4.2 Introduction sn ghpter P we hve seen tht resonnt light sttering from n tomi loud n e modeled in two wysF yn one hnd it n e desried with mirosopi theory for whih set of oupledEdipole equtions is solvedD nd on the other hnd y mens of mrosopi theory where the tomi loud is repled y homogeneous medium desried y dieletri onstntD or equivlently y refrtive indexF okolov et al. TV used oth wys of modeling light sttering nd nlyzed the response of n ensemle of old resonnt stterers in spheril volume nd they ompred modl mirosopi desription with mrosopi desription sed on the e'etive dieletri onstnt under the menE(eld grei RF yvesyxsg wyhi sx e hixi gvyh yp gyvh eyw SV pproximtionY oth pprohes re in good greement for low densitiesF hen the density inresesD devitions etween oth pprohes ourF sn ontexts suh s rndom lsingD light loliztionD nd surdineD mirosopi modes of system of interting toms in the slr pproximtion hve een studied VD WD PWD TW!UPF es opposed to the previously mentioned worksD we study dense @ρ/k 3 1AD disordered nd homogeneously rodened systems onsisting of N resonnt dipoles in smll volume with dimensions omprle to the resonnt wvelengthF e study light sttering y this system usingD on one hndD mirosopi model ounting for ll intertions etween toms see pigF RFI@A ndD on the other hndD mrosopi model sed on homogeneous system with n e'etive dieletri onstnt ε e see pigF RFI@AF he e'etive dieletri onstnt is then derived numerilly y (tting the sttered intensity verged over n ensemle of QHH di'erent reliztions of the systemF his llows us to ompre oth the eigenfrequenies nd sptil struture of the eigenmodes of oth mirosopi nd mrosopi systemsF es we will seeD minly superrdint modes re responsile for oherent light stterE ingF sn this hpterD we oserve tht some of them hve the following propertiesX @iA ll toms ontriute to the modeD @iiA the frequeny nd the sptil struture of the mode re independent of the reliztion ut depend on the geometry nd density of the system so tht they re very roust ginst disorderF reneD despite the ft tht the system is disorderedD these modes re nlogous to the polritons introdued y rop(eld to nlyze olletive exittions in ondensed mtter for ordered systems UQF e thus refer to these superrdint modes s polritoni modesF e further oserve tht the sptil struture of

SW

RFQF wyhiv these modes oinides with the sptil struture of the modes of the e'etive homogeneous prtile with e'etive dieletri onstntF e thus onlude tht there is oneEtoEone orE respondene etween modes of prtile with e'etive dieletri onstnt nd polritoni modes in n tomi systemF he nlysis developed in this hpter is not limited to oldEtom systems ut is lso pplile to ondensedEmtter systems ting like oupled osilltors suh s n ensemle of quntum dots nd lyers of orgni moleules with eletroni trnsitions or virtions in rystlF vet us give two exmples where the polritoni ehvior survives the introdution of disorder nd n e desried in the frmework of n e'etive dieletri onstntF es (rst exmpleD we onsider surfe phonon polritons @hsAF e phonon is olletive exittion whih results from the intertion potentil etween neighorsF st is norml mode of lttieD iFeFD periodilly rrnged set of toms or ionsF por optil phononsD these olletive mehnil exittions re oupled to the eletromgneti (eldD resulting in polriton URF he presene of polritoni mode results in dieletri onstnt tht tkes negtive vlues within frequeny windowF he onept of polritons n e pplied to mny di'erent rystls suh s ig or qurtzF st is now interesting to onsider morphous glssF hereD the onept of phonon disppers s there is no periodiity of the systemF roweverD it turns out tht the dieletri onstnt still hs negtive rel prt so tht hs re still predited y lssil eletrodynmis nd indeed oserved experimentllyF ine phonons formlly do not existD this rises the question how it is possile tht hs still existD despite of the rndomness of the underlying mirosopi strutureF he onepts tht re introdued in this hpter in the ontext of toms tht re oupled vi resonnt dipoleEdipole intertionsD llow to understnd the phenomenon of hs in morphous glss s wellD sine the underlying mehnisms re omprleF he seond exmple is the optil ehvior of disordered ensemle of densely pked quntum dots or orgni moleules suh s t ggregtesF st hs een oserved tht lyers of these resonnt mterils deposited on metlli (lms disply strong oupling with surfe plsmons US!VPF he experimentl results n e explined using resonnt e'etive dieletri onstnt for the ensemle of stterersF st hs further een oserved tht the modes re sptilly oherentF he previous exmples shre the si physil ingredients with our model onsisting of N old toms in loudF hey re dense nd disordered systems of N resonnt stterersF he hpter is orgnized s followsF pirstD using mirosopi modelD we study the olletive eigenmodes of the system onsisting of N old toms in smll volume s deE pited in pigF RFI@AF e (nd tht some of themD lled herefter polritoni modesD re roust ginst disorderD involve ll tomsD nd re superrdintF e then nlyze light sttering y the system using this model nd ompre it with n e'etive dieletri onE stnt s skethed in pigF RFI@AF e show tht polritoni modes n e identi(ed with the mrosopi optil modes of tht systemF e lso disuss the origin of losses in the e'etive dieletri onstntF pinllyD we will disuss the in)uene of polritoni modes on light stteringF 4.3 Model sn this hpterD we onsider loud of toms uniformly distriuted in retngulr ox see pigF RFI@AF he min di'erenes etween this system nd the tomi louds studied in the previous hpter re the ft tht the toms re uniformly distriuted insted of normlly grei RF yvesyxsg wyhi sx e hixi gvyh yp gyvh eyw TH distriuted nd the density is slightly higherX ρ/k 3 = 1F he uniform distriution hs een hosen to filitte the omprison with mrosopi theory sed on dieletri onstntF he dipolesD eing old toms in vuumD do not experiene nonrditive losses nd hve negligile hoppler shift with respet to the rditive linewidth Γ 0 of the tomsF sn the numeril simultions presented elowD we hve tken ruidiumE87 tomsD λ 0 = 780 nmD nd Γ 0 = 2π × (6 MHz)F he numer of tomsD N = 450D nd the dimensions of the oxD 4.8λ 0 × 0.6λ 0 × 0.6λ 0 D orrespond to typil experimentl onditions otined y lser ooling nd trpping tehniques in wvelengthEsize optil dipole trps SIF ith suh prmetersD ρ/k 3 ∼ 1D so reurrent sttering hs to ply signi(nt roleF e lso show some results for retngulr loud with dimensions 2.1λ 0 × 0.4λ 0 × 0.4λ 0 nd density ρ/k 3 ∼ 5F vike in the experimentD the loud is investigted in the wekEexittion limitD where its optil properties n e desried y lssil optis PWD UPF he mirosopi eigenmodes of the tomi loud re otined y serhing for selfE onsistent solution of the oupledEdipole equtions @see iqF PFITA in the sene of driving eletri (eld

p i = ω 2 c 2 N j=1,j =i α(ω) Ḡ(r i , r j ; ω)p j .
@RFIA his set of equtions n e put in the form Ā(ω)P = ωPD whih is nonliner eigenvlue prolemD sine the ω dependene of the equtions is not linerF sn order to rrive t the stndrd form of n eigenvlue prolemD we ssume ω 2 Ḡ(r j , r i , ω) ≈ ω 2 0 Ḡ(r j , r i , ω 0 )F feuse of the nrrow tomi linewidth nd the nonresonnt hrter of the freeEspe qreen tensorD this is n urte ssumptionF he eigenvlue prolem n then e put in the stndrd form of n eigenvlue prolemX Ā(ω 0 )P β = ω β P β D where the mtrix

Ā(ω 0 ) =      (ω 0 -i Γ 0 2 ) 1 -3πcΓ 0 ω 0 Ḡ2→1 . . . -3πcΓ 0 ω 0 Ḡ1→2 (ω 0 -i Γ 0 2 ) 1 F F F F F F     
, of whih ll sumtries re of size 3 × 3F ine the mtrix Ā ontins omplex digonl elements nd Ḡ † i→j = Ḡj→i D it is nonErermitin mtrixF feuse the system loses energy y the rdition in the fr (eld of the dipolesD nd there re resonnt dipoleEdipole interE tionsD the prolem is mthemtilly non rermitinF hereforeD the eigenfrequenies ω β re omplexD nd we write the eigenfrequenies s

ω β = (ω 0 + Ω β ) -i Γ β 2 , @RFPA
where Ω is the olletive frequeny shift nd Γ β the olletive linewidthF hen we sustitute this expression for ω β into the eigenvlue prolemD the eigenvlue prolem eomes

Ā(ω 0 )P β = (ω β -i Γ β 2 )P β . @RFQA
his eigenvlue prolem hs 3N eigenvlues s eh dipole hs three dipole moment omE ponentsF he orresponding eigenvetors P β re omposed of the vetor omponents of ll

N dipole momentsD P β = [p β 1x , p β 1y , p β 1z , • • • p β N x , p β N y , p β N z ] F hey re normlized suh tht |P β | 2 = j |p β j | 2 = 1F
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♥✐ ②❛ ♥ ❞ ❤ ❛ ♣ ❡♦ ❢ ❤ ❡ ❝ • ♦ ✉ ❞ ✳ ❚ ❤ ❡❡ ❢ ❛ ❝ ❝ ♦ ♥ ✜♠ ❤ ❛ ❤ ❡❡ ❢ ❡✇② ♣ ❡✷ ❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡ ❛❡♣ ♦ • ❛✐ ♦ ♥ ✐ ❝❜ ②♥ ❛✉❡ ✳ ❚ ❤ ✐ ✐✐ ♥❝ ♦ ♥❛ ✇ ✐❤ ❤ ❡♦❤ ❡ ② ♣ ❡ ♦ ❢ ♠ ♦ ❞ ❡✱✇ ❤ ✐ ❝ ❤❞ ♦♥ ♦ ❤ ♦ ✇ ❤ ✐ ♣ ❛✐ ❛ •♣ ❡ ✐ ♦ ❞ ✐ ❝ ✐ ②❛ • ♦ ♥ ❣ ❤ ❡③❛ ① ✐ ♥ ♦ ✐ ♥ ✈ ♦ • ✈ ❡❛ • •❤ ❡❞ ✐ ♣ ♦ • ❡ ✐♠ ✉ • ❛ ♥ ❡ ♦ ✉• ② ✳ ❋ ✐ ❣ ✉ ❡✹ ✳ ✹ ✿✭ ❛ ✮❙ ♣ ❛✐ ❛ • ✉ ❝✉❡♦ ❢❛② ♣ ✐ ❝ ❛ •♠ ✐ ❝♦❝ ♦ ♣ ✐ ❝❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡♦ ❢② ♣ ❡✷ ❢ ♦ ρ / k 3 =1✳ ❚ ❤ ❡❞ ✐ ♣ ♦ • ❡♠ ♦♠ ❡ ♥ ✉ ❛❡ ❞| ♣ j | 2 ♦ ❢❡ ❛ ❝ ❤❛♦♠j❤ ❛ ❜ ❡ ❡ ♥♣ • ♦ ❡ ❞ ✱✇ ✐❤❤ ❡❛♦♠ ❜ ❡ ✐ ♥ ❣ ♦ ❡ ❞ ❜ ② ❤ ❡ ✐ ③❝ ♦ ♦❞ ✐ ♥ ❛❡ ✳✭ ❜ ✮❆ ✈ ❡❛ ❣ ❡❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡ ❞ | ♣ | 2 ✭❡ ❞❞ ♦ ✮ ✳ ❚ ❤ ❡❡ ♥❡♠ ❜ • ❡ ❛ ✈ ❡❛ ❣ ❡ ✐ ♣ ❡❢ ♦♠ ❡ ❞♦ ✈ ❡ ❤ ❡♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡ ✐ ♥ ✐ ❞ ❡❤ ❡❜ • ❛ ❝ ❦ ♦ • ✐ ❞ ❡ ❝❛ ♥ ❣ • ❡✭ ❛ ✮ ✐ ♥ ❤ ❡ ✐ ♥ ❡ ♦ ❢ ❋ ✐ ❣ ✳✹ ✳ ✷ ✭❆ ✮ ✳ ❚ ❤ ❡ ♠ ❛ ❝♦❝ ♦ ♣ ✐ ❝ ♠ ♦ ❞ ❡♦ ❢ ❤ ❡❤ ♦♠ ♦ ❣ ❡ ♥ ❡ ♦ ✉ ❝ • ♦ ✉ ❞✭ ❜ • ✉ ❡ ♦ • ✐ ❞• ✐ ♥ ❡ ✮ ❝ ♦ ✐ ♥ ❝ ✐ ❞ ❡ ✈ ❡②✇ ❡ • •✇ ✐❤ ❤ ❡❛ ✈ ❡❛ ❣ ❡♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡ ✱ ❛ ❦ ✐ ♥ ❣✐ ♥♦❛ ❝ ❝ ♦ ✉ ♥ ❛ ♠ ❛ • •♦ ✛ ❡ ❞ ✉ ❡ ♦ ❤ ❡✢ ✉ ❝✉ ❛✐ ♦ ♥♦ ❢ ❤ ❡❞ ✐ ♣ ♦ • ❡ ❛♠ ♣ • ✐✉ ❞ ❡ ❢ ♦♠♦ ♥ ❡ ❡ ❛ • ✐ ③ ❛✐ ♦ ♥♦❛ ♥ ♦❤ ❡✳ | ❡ ❢ ❡ ♦ ❤ ❡ ♠ ❛ ✐ ♥ ❡ ① ❢ ♦❛❞ ✐❝ ✉ ✐ ♦ ♥❛ ❜ ♦ ✉ ❤ ✐ ♦ ✛❡✳ ❆ • • ♠ ♦ ❞ ❡ ✇ ❡❤ ❛ ✈ ❡♥ ♦ ❞ ✐❝ ✉ ❡ ❞ ♦ ❢ ❛ ❜ ❡ • ♦ ♥ ❣ ♦ ② ♣ ❡✸ ✳ ❚ ❤ ❡❡ ♠ ♦ ❞ ❡ ✐ ♥ ✈ ♦ • ✈ ❡❡ ✐ ❤ ❡ ♠ ❛ ♥ ②❞ ✐ ♣ ♦ • ❡ ❞ ✐ ✐ ❜ ✉ ❡ ❞ ❤♦ ✉ ❣ ❤ ♦ ✉ ❤ ❡✇ ❤ ♦ • ❡✈ ♦ • ✉♠ ❡❜ ✉ ✇ ✐❤ ♦ ✉ ❛ ❡ ❣ ✉ • ❛ ♣ ❛✐ ❛ • ✉ ❝ ✲ ✉❡ ❬ ❋ ✐ ❣ ✳✹ ✳ ✸ ✭ ❞ ✮ (❛ ❤ ❡♦ ♥ ❡♦ ❜❡✈ ❡ ❞ ✐ ♥❋ ✐ ❣ ✳✹ ✳ ✸ ✭ ❜ ✮ ✱♦ ❛ • ♦ ❝ ❛ • ✐ ③ ❡ ❞ ♠ ♦ ❞ ❡ ✐ ♥ ✈ ♦ • ✈ ✐ ♥ ❣♦ ♥ • ②❛ ✉ ❜❡♦ ❢❞ ✐ ♣ ♦ • ❡ ❬ ❋ ✐ ❣ ✳ ✹ ✳ ✸ ✭ ❝ ✮ ( ✳■ ♥♦❞ ❡ ♦❞ ❡♠ ♦ ♥ ❛❡❤ ❛ ♥ ❡ ✐❤ ❡ ② ♣ ❡✶♥ ♦ ② ♣ ❡✸♠ ♦ ❞ ❡ ❤ ❛ ✈ ❡❛ ♣ ❛✐ ❛ • ✉ ❝✉❡ ✱❛ ♦ ♣ ♣ ♦❡ ❞ ♦♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡✱✇ ❡❤ ❛ ✈ ❡❝ ❛ • ❝ ✉ • ❛ ❡ ❞ ❤ ❡❛ ✈ ❡❛ ❣ ❡ ❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡❛ • ♦ ♥ ❣ ❤ ❡ • ♦ ♥ ❣❛ ① ✐ ♦ ❢ ❤ ❡❜ ♦ ① ❢ ♦ ❤ ❡❡ ♠ ♦ ❞ ❡✳■✐ ❡ ❡ ♥ ✐ ♥❋ ✐ ❣ ✳✹ ✳ ✺ ❤ ❛ ❤ ❡❡ ♠ ♦ ❞ ❡ ❞ ♦♥ ♦ ♣ ♦ ❡ ❤ ❡♦❝ ✐ • • ❛♦② ✉ ❝✉❡♦ ❜❡✈ ❡ ❞ ❢ ♦ ② ♣ ❡✷ ♠ ♦ ❞ ❡✳ ❚ ❤ ❡♣ ❡✐ ♦ ❞ ✐ ❝❛❛ ♥ ❣ ❡♠ ❡ ♥ ♦ ❢ ❤ ❡❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✱ ✇ ❤ ✐ ❝ ❤♦ ♥ • ②❡ ① ✐ ❢ ♦ ❤ ❡ ② ♣ ❡✷ ❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡✱❛ • • ♦ ✇♣ ❤ ❛❡ ♠ ❛❝ ❤ ✐ ♥ ❣♦ ❢ ❤ ❡ ❛ ❞ ✐ ❛✐ ♦ ♥❜ ②❤ ❡❞ ✐ ♣ ♦ • ❡ ❛ ♥ ❞ ❤ ❡❡ ❢ ♦❡ ✐ ♥ ❣ • ❡ ✲ ♣ ❤ ♦♦ ♥ ✉ ♣ ❡ ❛ ❞ ✐ ❛ ♥ ❝ ❡❛ • ♦ ♥ ❣ ❤ ❡❛ ① ✐ ♦ ❢ ❤ ❡♦ ❜ • ❡ ❝✳ ◆ ♦ ❡ ❤ ❛ ❤ ✐ ♣ ❤ ❛❡ ♠ ❛❝ ❤ ✐ ♥ ❣♦ ♥ • ② ❈❍❆ ❚ ❊|✹ ✳ ❖ ▲❆| ■❚❖◆ ■❈ ▼❖❉ ❊ ❙ ■◆❆❉ ❊◆ ❙ ❊❈ ▲❖❯❉❖ ❋❈❖ ▲❉❆ ❚❖▼ ❙ ✻ ✹ -2 0 2 0 1 2 3 x 1 0 -3 z / λ 0 < | p | 2 > -2 0 2 1 . 5 2 2 . 5 x 1 0 -3 z / λ 0 ( a ) ( b ) ❋ ✐ ❣ ✉ ❡✹ ✳ ✺ ✿ ❚ ❤ ❡❛ ✈ ❡❛ ❣ ❡❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡♦ ❢ ② ♣ ❡✶✭ ❛ ✮❛ ♥ ❞ ② ♣ ❡✸✭ ❜ ✮❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡ ❛ • ♦ ♥ ❣ ❤ ❡ • ♦ ♥ ❣❛ ① ✐ ♦ ❢ ❤ ❡❜ ♦ ①❞ ♦♥ ♦ ❤ ♦ ✇❛ ♥ ② ♣ ❛✐ ❛ • ✉ ❝✉❡ ✱ • ✐ ❦ ❡ ② ♣ ❡✷❞ ♦ ❡✳✭ ❛ ✮ ❆ • • ❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡ ✇ ✐❤| Ω β | >1 5 0 Γ 0 ❤ ❛ ✈ ❡❜ ❡ ❡ ♥ ❛ ❦ ❡ ♥ ✐ ♥♦❛ ❝ ❝ ♦ ✉ ♥ ❢ ♦ ❤ ❡❛ ✈ ❡❛ ❣ ✐ ♥ ❣ ✳✭ ❜ ✮❆ • •♠ ♦ ❞ ❡ ✇ ✐❤ ✐ ♥ ❤ ❡ ❡ ❣ ✐ ♦ ♥❞ ❡ ✜ ♥ ❡ ❞❜ ②| Ω β | <1 9❛ ♥ ❞Γ β <3 Γ 0 ❤ ❛ ✈ ❡❜ ❡ ❡ ♥ ❛ ❦ ❡ ♥ ✐ ♥♦❛ ❝ ❝ ♦ ✉ ♥ ❢ ♦ ❤ ❡ ❛ ✈ ❡❛ ❣ ✐ ♥ ❣ ✳ ♦ ❝ ❝ ✉ ✇ ✐❤ ✐ ♥❛ ❛❤ ❡ • ✐♠ ✐ ❡ ❞ ♦ • ✐ ❞❛ ♥ ❣ • ❡✭ ∆Ω ∼2 π / 1 5 ✮ ✳ ❚ ❤ ✐ ✐✇ ❤ ② ❤ ❡ • ✐ ♥ ❡✇ ✐ ❞❤ ✐ ♥ ♦ ❡ ♥ ❤ ❛ ♥ ❝ ❡ ❞❜ ②❛ ❢ ❛ ❝♦ N ❛ ✐ ❤ ❡❝ ❛❡ ❢ ♦ ✐ ♥ ❣ • ❡ ✲ ♣ ❤ ♦ ♦ ♥ ✉ ♣ ❡ ❛ ❞ ✐ ❛ ♥ ❝ ❡❜ ②N ❡♠ ✐ ❡ ✐ ♥❛ ✉ ❜ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤✈ ♦ • ✉♠ ❡ ✱❜ ✉ • ♦ ✇ ❡ ❡ ❞❜ ②❛ ❢ ❛ ❝♦ ♦ ❢ ♦ ✉ ❣ ❤ • ②∆Ω/ ( 4 π ) ✳ ✹ ✳ ✺ ▲ ✐ ❣ ❤ ❝ ❛ ❡✐ ♥ ❣ ❲ ❡♥ ♦ ✇ ✉ ❞ ② • ✐ ❣ ❤ ❝ ❛ ❡✐ ♥ ❣❜ ②❤ ❡❝ • ♦ ✉ ❞♦ ❢❛♦♠✳❙ ♦ • ✈ ✐ ♥ ❣❤ ❡❝ ♦ ✉ ♣ • ❡ ❞ ✲ ❞ ✐ ♣ ♦ • ❡ ❡✉ ❛✐ ♦ ♥ ✉✐ ♥ ❣❤ ❡ • ❛❡✜ ❡ • ❞❛ ❛ ♥❡ ①❡♥ ❛ •❞✐ ✈ ✐ ♥ ❣♦ ✉❝ ❡❛ • • ♦ ✇✉ ♦❝ ❛ • ❝ ✉ • ❛❡❤ ❡ ❝ ❛ ❡❡ ❞✜ ❡ • ❞❊ ✐ ♥❤ ❡ ❢ ❛ ✜ ❡ • ❞ ❢ ♦ ❛❣ ✐ ✈ ❡ ♥ ❡ ❛ • ✐ ③ ❛✐ ♦ ♥♦ ❢❤ ❡❝ • ♦ ✉ ❞✇ ❤ ❡ ♥ ✐ ✐✐ • • ✉♠ ✐ ♥ ❛ ❡ ❞❜ ②❛ ♥① ✲ ♣ ♦ • ❛✐ ③ ❡ ❞ ♣ • ❛ ♥ ❡✇ ❛ ✈ ❡ ✱❛ ✐❞ ♦ ♥ ❡ ✐ ♥ ❤ ❡✇ ♦❦♦ ❢❈ ❤ ❛ ♣❡ ✸ ✳ ❆ ❢ ❡ ❛ ✈ ❡❛ ❣ ✐ ♥ ❣♦ ✈ ❡ 1 5 0 0 ❡ ❛ • ✐ ③ ❛✐ ♦ ♥ ✱ ❤ ❡ ❝ ❛ ❡✐ ♥ ❣♣ ❛❡♥♦ ❜❛ ✐ ♥ ❡ ❞♥ ❡ ❛ ❡♦ ♥ ❛ ♥ ❝ ❡✭ ω= ω 0 ✮❡ ① ❤ ✐ ❜ ✐ ♦ ♥ ❡ • ♦ ❜ ❡ ✐ ♥ ❤ ❡ ❢ ♦✇ ❛❞ ❞ ✐❡ ❝✐ ♦ ♥ ❬ ❡ ❡❋ ✐ ❣ ✳ ✹ ✳ ✶ ✭ ❛ ✮ (❛ ♥ ❞♦♠ ❡❞ ✐ ✛ ✉❡ • ✐ ❣ ❤✇ ❤ ✐ ❝ ❤ ✐ ♦ ♥❛ ✈ ❡❛ ❣ ❡ ✉ ❛✐ ✲ ✐ ♦ ♦ ♣ ✐ ❝❛ ♥ ❞❤ ❛ ❛ ♠ ❛ • • ❡ ❛♠ ♣ • ✐✉ ❞ ❡ ✳ ❍ ♦ ✇ ❡ ✈ ❡✱✇ ❡✜ ♥ ❞❤ ❛ ❤ ❡❡✇ ♦ ❢ ❡ ❛✉❡❤ ❛ ✈ ❡❝ ♦♠ ♣ ❛❛ ❜ • ❡❝ ♦ ♥ ✐ ❜ ✉ ✐ ♦ ♥ ✇ ❤ ❡ ♥ ✐ ♥❡ ❣❛❡ ❞♦ ✈ ❡ ❤ ❡❢ ✉ • • ♦ • ✐ ❞❛ ♥ ❣ • ❡ ✇ ✐❤5 4%❛ ♥ ❞4 6%♦ ❢ ❤ ❡ ♦❛ • ❝ ❛ ❡❡ ❞ • ✐ ❣ ❤✱ ❡♣ ❡ ❝✐ ✈ ❡ • ② ✳ ❚ ❤ ❡ • ♦ ❜ ❡ ✐ ♥❤ ❡❢ ♦✇ ❛❞❞ ✐❡ ❝✐ ♦ ♥ ✐✈ ❡② ✐♠ ✐ • ❛ ♦ ❤ ❡❞ ✐ ✛❛ ❝✐ ♦ ♥♣ ❛❡♥ ♦✐ ❣ ✐ ♥ ❛ ✐ ♥ ❣ ❢ ♦♠❛❤ ♦♠ ♦ ❣ ❡ ♥ ❡ ♦ ✉ ♣ ❛ ✐ ❝ • ❡ ✱✉ ❣ ❣ ❡ ✐ ♥ ❣❤ ❛ ♦ ♥ ❡❝ ♦ ✉ • ❞ ❡ ♣ • ❛ ❝ ❡ ❤ ❡❝ • ♦ ✉ ❞ ✱✇ ✐❤ ✐ ❛ ♥ ❞ ♦♠❣❛ ✐ ♥ ✐ ♥ ❡ ✱❜ ②❛❤ ♦♠ ♦ ❣ ❡ ♥ ❡ ♦ ✉❝ • ♦ ✉ ❞✇ ✐❤ ❤ ❡ ❛♠ ❡ ❤ ❛ ♣ ❡❛ ♥ ❞ ❡ ① ❛ ❝ ❛ ♥ ❡ ✛ ❡ ❝✐ ✈ ❡ ❞ ✐ ❡ • ❡ ❝ ✐ ❝❝ ♦ ♥❛ ♥✳ ❚ ❤ ❡❡♦ ❜❡✈ ❛✐ ♦ ♥ ❝ ❛ ♥❜ ❡✉ ♥ ❞ ❡ ♦ ♦ ❞ ✇ ✐❤ ❤ ❡❢ ♦♠ ❛ • ✐ ♠♦ ✉ • ✐ ♥ ❡ ❞✐ ♥❙ ❡ ❝ ✐ ♦ ♥✷ ✳ ✷ ✳ ✸ ✳ ❲ ❡❞ ❡ ❝ ♦♠ ♣ ♦ ❡ ❤ ❡❡ • ❡ ❝ ✐ ❝✜ ❡ • ❞❊ ❝ ❛ ❡❡ ❞❜ ② ❤ ❡ ❛ ♥ ❞ ♦♠ ♠ ❡ ❞ ✐ ✉♠✐ ♥♦❛ ♥❡ ♥❡♠ ❜ • ❡ ✲ ❛ ✈ ❡❛ ❣ ❡ ❞✜ ❡ • ❞ ❊ ❛ ♥ ❞❛✢ ✉ ❝✉ ❛✐ ♥ ❣✜ ❡ • ❞ δ ❊✳■✐❦ ♥ ♦ ✇ ♥ ❤ ❛ ❊ ✐ ❤ ❡✜ ❡ • ❞❞ ✐ ✛❛ ❝❡ ❞ ❜ ②❛ ♥❡ ✛ ❡ ❝✐ ✈ ❡ ♠ ❡ ❞ ✐ ✉♠✇ ✐ ❤❡ ✛ ❡ ❝✐ ✈ ❡❞ ✐ ❡ • ❡ ❝✐ ❝❝ ♦ ♥❛ ♥ ε ❡ ✛ ❛ ♥ ❞✇ ✐❤ ❤ ❡ ❛♠ ❡ ❤ ❛ ♣ ❡❛ ❤ ❡❝ • ♦ ✉ ❞ ✳ ❚ ❤ ❡ ❝ ❛ ❡❡ ❞ ✐ ♥❡ ♥✐ ②❝ ❛ ♥❛ •♦❜ ❡❞ ❡ ❝ ♦♠ ♣ ♦❡ ❞ ✐ ♥❛❝ ♦ ❤ ❡❡ ♥ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ❤ ❡❡ ♥ ❝ ♦ ♥ ✐ ❜ ✉ ✐ ♦ ♥| ❊| 2 =| ❊ | 2 + | δ ❊| 2 ∝I ❝ ♦ ❤ +I ✐ ♥ ❝ ♦ ❤ ✳ ❚ ❤ ❡✜ ❡♠❝ ♦ ❡♣ ♦ ♥ ❞ ♦ ❤ ❡ • ♦ ❜ ❡ ✐ ♥❤ ❡ ❢ ♦✇ ❛❞❞ ✐❡ ❝✐ ♦ ♥ ✐ ♥❋ ✐ ❣ ✳ ✹ ✳ ✶ ✭ ❛ ✮ ✱✇ ❤ ✐ • ❡ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ❡♠ ✐ ❤ ❡ ✉ ❛✐ ✲ ✐ ♦ ♦ ♣ ✐ ❝❞ ✐ ✛ ✉❡ • ✐ ❣ ❤ ✳ TS RFTF ippigsi hsivigsg gyxex
4.6 Eective dielectric constant e now proeed to the extrtion of the e'etive dieletri onstnt of the loud s funtion of frequenyF sn textooks suh s tkson VQD the index of refrtion of dense medi is well reprodued y the vorentzEvorenz reltion etween the mrosopi dieletri onstnt of the medi nd the polrizility of the stterersX

ε LL (ω) -1 ε LL (ω) + 2 = ρα(ω) 3 . @RFRA
his formul tkes only prtilly resonnt dipoleEdipole intertions into ount y the lolE(eld orretion PQD PRF eurrent sttering is not inludedD while its impt on the e'etive refrtive index is expeted to e of importne for the dense systems of old toms studied here ITD PUD VRF sn order to derive the e'etive dieletri onstnt y ounting for ll multiple stterE ing eventsD we solve n inverse prolemF e ompre the oherent ontriution | E | 2 of the frE(eld sttering pttern of the tomi loud with tht of n e'etive homogeneous prtile with dieletri onstnt ε e nd the sme geometry s the loudF he ltter is numerilly lulted with n periodi pourier modl method @EpwwA VS minly deE veloped y hilippe vlnne nd tenEul rugoninD using ε e s (tting prmeterF he use of pourier modl method for light sttering of retngulr ox seems ontrditoryF he min ide of using n periodi pourier modl method is tht the strutureD in this se retngulr oxD is periodized in the two trnsverse diretions x nd yF fy doing soD one n use the rigorous oupled wve nlysis @geAF ih ell needs to e optilly isoltedF his n e done y for exmple putting perfetly mthed lyers @wvA t the orders of the ellD or y using omplex oordinte trnsformtions t the orders of eh ell VTF st is the ltter tht is used in this workF e (nd tht the frE(eld sttering pttern omputed using either the e'etive homogeneous prtile or the ensemle verge mirosopi loud gree within ∼ 1% for eh ngle ompre pigsF RFI@A nd @AD showing tht the e'etive dieletri onstnt pproh is vlidF e onlude tht this mrosopi pproh ptures oth the reurrent sttering nd the olletive e'ets of the mirosopi pitureF fy repeting this proedure for di'erent frequeny detunings δω = ω -ω 0 D we otin the spetrum of the e'etive dieletri onstnt @see pigF RFTAF he numerilly retrieved vlues of the dieletri onstnt hve een (tted with the funtion

ε(ω) = 1 + c 1 c 2 -ω -ic 3 , @RFSA
where ll oe0ients re tken relF reneD oth the rel nd imginry prt of the dieletri onstnt were (tted simultneouslyF st is noteworthy tht t resonne the gs is desried y dieletri onstnt with negtive rel prt nd thus optilly ehves s metlF he numerilly retrieved dieletri onstnt is found to e signi(ntly di'erent from the predition y iqF @RFRAF yur results evidene tht the vorentzEvorenz formul is not vlid for dense loud of resonnt stterersY the disrepny etween the simultion nd the vorentzEvorenz theory inreses with the densityD re)eting tht the resonnt dipoleEdipole intertions eome stronger PUDVUF es we desried in ghpter QD this result hs reently een studied experimentlly y tennewein et al. y using loud of old toms with similr shpe nd density UF hey hve shown tht for light sttering from dense loud of old toms menE(eld theory eomes invlidD s expetedF elthough it ws expetedD sine

❈❍❆ ❚ ❊|✹ ✳ ❖ ▲❆| ■❚❖◆ ■❈ ▼❖❉ ❊ ❙ ■◆❆❉ ❊◆ ❙ ❊❈ ▲❖❯❉❖ ❋❈❖ ▲❉❆ ❚❖▼ ❙ ✻ ✻ -9 -6 -3 0 3 -1 0 0 1 0 2 0 δ ω / Γ 0 -3 0 -1 5 5 2 0 -5 0 0 5 0 1 0 0 δ ω / Γ 0 -2 0 -1 0 0 1 0 -2 0 2 4 6 R e ( ε e f f ) Im ( ε e f f ) ( a ) ( b ) ❋ ✐ ❣ ✉ ❡✹ ✳ ✻ ✿❊ ✛ ❡ ❝✐ ✈ ❡❞ ✐ ❡ • ❡ ❝✐ ❝❝ ♦ ♥❛ ♥ ε ❡ ✛ ( ω )❢ ♦✭ ❛ ✮ρ / k 3 =1✱❛ ♥ ❞✭ ❜ ✮ ρ / k 3 =5✳ ❚ ❤ ❡ ❡ ❛ •❛ ♥ ❞ ✐♠ ❛ ❣ ✐ ♥ ❛②♣ ❛ ♦ ❢ ❤ ❡❡ ✛ ❡ ❝✐ ✈ ❡❞ ✐ ❡ • ❡ ❝✐ ❝❝ ♦ ♥❛ ♥ ✭ ❝ ✐❝ • ❡ ❛ ♥ ❞❞ ♦ ✮ • ❛❣ ❡ • ②❞ ✐ ✛ ❡ ❢ ♦♠ ❤ ❡♣❡ ❞ ✐ ❝✐ ♦ ♥ ♦ ❢▲ ♦❡ ♥③ ✲ ▲ ♦❡ ♥ ③ ❤ ❡ ♦②✭ ❞ ❛❤ ❡ ❞❝ ✉✈ ❡✮ ✳ ❙ ♦ • ✐ ❞ • ✐ ♥ ❡✿ ▲ ♦ ❡ ♥③ ✐ ❛ ♥✜ ♦ ❢ ❤ ❡♥ ✉♠ ❡✐ ❝ ❛ •❞ ❛ ❛ ✳ ❡♦ ♥ ❛ ♥ ❞ ✐ ♣ ♦ • ❡ ✲ ❞ ✐ ♣ ♦ • ❡ ✐ ♥❡❛ ❝✐ ♦ ♥❛❡ ♦ ♥ ❣♥ ❡ ❛ ❡♦ ♥ ❛ ♥ ❝ ❡ ✱✇ ❡✇ ❛ ♥ ♦❤ ✐ ❣ ❤ • ✐ ❣ ❤ ❤ ❛ ♦ ✉ ❛♦♠ ✐ ❝ ② ❡♠ ✐ ✜ ✈ ❡♦❞ ❡ ♦ ❢♠ ❛ ❣ ♥ ✐✉ ❞ ❡ ♠ ♦❡❞ ✐ • ✉❡❤ ❛ ♥❛❣ ❛ ❛ ❛♠ ❜ ✐ ❛ ♥ ❝ ♦ ♥ ❞ ✐✐ ♦ ♥ ✱ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❤ ❡▲ ♦❡ ♥③ ✲ ▲ ♦❡ ♥ ③ ❤ ❡ ♦② ✐ ❛ ♥❛ ❝ ❝ ✉❛❡❞ ❡❝✐ ♣ ✐ ♦ ♥ ✳ ✹ ✳ ✻ ✳ ✶ ❆ ♥ ❞ ❡ ♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ■ ♥ ❤ ✐ ❡ ❝✐ ♦ ♥ ✇ ❡ ✇ ✐ • •❜ ✐ ❡ ✢ ②❞ ✐ ❝ ✉ ❛ ❜ ♦ ✉ ❆ ♥ ❞ ❡ ♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥♦ ❢ • ✐ ❣ ❤✐ ♥✸❉ ✳❙ ♦❢ ❛ ✱ ♥ ♦♦ ♥ ❡❤ ❛ ❡ ✈ ❡ ♦ ❜❡✈ ❡ ❞ ❆ ♥ ❞ ❡ ♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥♦ ❢ • ✐ ❣ ❤✐ ♥✸❉ ✳■✐❝ • ❛ ✐♠ ❡ ❞ ❤ ❛✱✉ ♥ ❞ ❡ ❤ ❡❛ ♣ ♣ • ✐ ❝ ❛✐ ♦ ♥♦ ❢❛ ♥❡ ① ❡♥ ❛ • ♠ ❛ ❣ ♥ ❡✐ ❝✜ ❡ • ❞ ✱❝ ♦ • ❞❛ ♦♠ ❛❡❛♣ ♦ ✐ ❜ • ❡❝ ❛ ♥ ❞ ✐ ❞ ❛ ❡ ❢ ♦ ❤ ❡ ♦ ❜❡✈ ❛✐ ♦ ♥ ❬ ✽ ✽ ( ✳ ❆❝ ✐ ❡✐ ♦ ♥♦ ❢❤ ❡♦ ❜❡✈ ❛✐ ♦ ♥♦ ❢❆ ♥ ❞ ❡♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ✐ ❤ ❛ ❤ ❡♠ ❡ ❛ ♥ ❢❡ ❡♣ ❛❤ ❤ ♦ ✉ • ❞ ❜ ❡ ♠ ❛ • • ❡ ❤ ❛ ♥ ❤ ❡✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✿l ♠ ❢ ♣ < 1 / k ✱✇ ❤ ✐ ❝ ❤ ✐❛ •♦❝ ❛ • • ❡ ❞ ❤ ❡■ ♦ ✛ ❡ ✲| ❡ ❣ ❡ •❝✐ ❡✐ ♦ ♥ ❢ ♦✇ ❛ ✈ ❡ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ❬ ✶ ✽ ( ✳ ❋ ♦♠ ❤ ❡❞ ✐ ❡ • ❡ ❝ ✐ ❝❝ ♦ ♥❛ ♥ ❤ ❛ ✇ ❡❤ ❛ ✈ ❡ • ✉ ❞ ❡✐ ✈ ❡ ❞ ✱✇ ❡❝ ❛ ♥ ♦ ❜❛ ✐ ♥ ❤ ❡ ✐♠ ❛ ❣ ✐ ♥ ❛②♣ ❛ ♦ ❢ ❤ ❡ ❡ ❢❛ ❝✐ ✈ ❡ ✐ ♥ ❞ ❡ ① n ❡ ✛ = ε 1 / 2 ❡ ✛ ❛ ♥ ❞ ❤ ✉ ✜ ♥ ❞ ❤ ❡ ♠ ❡ ❛ ♥ ❢❡ ❡ ♣ ❛❤ ✿ l ♠ ❢ ♣ =1/ [ 2 k ■♠ ( n ❡ ✛ ) ] ✳ ❆❛♦♠ ✐ ❝ ❡♦ ♥ ❛ ♥ ❝ ❡ ✱ ✇ ❡✜ ♥ ❞l ♠ ❢ ♣ =1/ ( 3 k ) ✳ ❚ ❤ ✐✐ ♥ ❞ ✐ ❝ ❛ ❡ ❤ ❛ ❤ ❡ ■ ♦ ✛ ❡ ✲| ❡ ❣ ❡ •❝✐ ❡✐ ♦ ♥ ❢ ♦✇ ❛ ✈ ❡ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ✭ k l ♠ ❢ ♣ <1 ✮ ✐ ❛✐ ✜ ❡ ❞ ❬ ✶ ✽ ( ✳ ( ❡ ✱✇ ❡❤ ❛ ✈ ❡ ❢ ♦ ✉ ♥ ❞ ♠ ❛ ♥ ② ♠ ♦ ❞ ❡❡ ①❡ ♥ ❞ ❡ ❞ ❤♦ ✉ ❣ ❤ ♦ ✉ ❤ ❡✇ ❤ ♦ • ❡ ② ❡♠ ✱ ✐ ♥ ❞ ✐ ❝ ❛✐ ♥ ❣❤ ❛ ❤ ❡ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ • ❡ ♥ ❣ ❤ ✱ ✐ ❢❛ ♥ ② ✱ ✐ • ❛ ❣ ❡ ❤ ❛ ♥ ❤ ❡ ✐ ③ ❡♦ ❢❤ ❡ ② ❡♠ ✳ ❲ ❡❜ ❡ • ✐ ❡ ✈ ❡ ❤ ❡❡ ❢ ♦❡ ❤ ❛ ❤ ❡ ♠ ❛ • • ❛♦♠ ✐ ❝❝ • ♦ ✉ ❞ ✇ ❡ ✉ ❞ ②❤ ❡❡ ✱ ✐ ♥❤ ❡❛ ❜❡ ♥ ❝ ❡♦ ❢❛ ♥❡ ①❡♥ ❛ •♠ ❛ ❣ ♥ ❡✐ ❝✜ ❡ • ❞ ✱❛ ❡♥ ♦❝ ❛ ♥ ❞ ✐ ❞ ❛❡ ❢ ♦ ❤ ❡♦ ❜❡✈ ❛✐ ♦ ♥♦ ❢❆ ♥ ❞ ❡♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ✱❞ ❡ ♣ ✐❡♦ ❢❤ ❡ ❢ ❛ ❝ ❤ ❛ ❤ ❡ ■ ♦ ✛ ❡ ✲| ❡ ❣ ❡ • ❝✐ ❡✐ ♦ ♥ ✐ ❛✐ ✜ ❡ ❞ ✳ ❚ ❤ ❡❜ ❡ • ✐ ❡ ❢ ❤ ❛ ❤ ❡ ♠ ❛ • •❛ ♦♠ ✐ ❝❝ • ♦ ✉ ❞ ✇ ❡ ✉ ❞ ②❤ ❡❡❛❡♥ ♦❝ ♦ ♥ ❞ ✐ ❞ ❛❡ ❢ ♦ ❤ ❡♦ ❜❡✲ ✈ ❛✐ ♦ ♥♦ ❢❆ ♥ ❞ ❡♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ✐ ✉ ♣ ♣ ♦ ❡ ❞❜ ②❛ ♥ ♦❤ ❡ ❝✐ ❡✐ ♦ ♥♦ ❢❆ ♥ ❞ ❡♦ ♥ • ♦ ❝ ❛ • ✐ ③ ❛✐ ♦ ♥ ✱ ✇ ❤ ✐ ❝ ❤ ✐ ❝ ❛ • • ❡ ❞❤ ❡❚ ❤ ♦ ✉ • ❡ ❝✐ ❡✐ ♦ ♥ ✳ ❆ ❝ ❝ ♦ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝✐ ❡✐ ♦ ♥ ✱❛ • •♠ ♦ ❞ ❡♥ ❡ ❡ ❞ ♦❜ ❡ ♣ ❡ ❝ ✲ ❛ • • ② ❡ ♣ ❛❛❡ ❞❛ ♥ ❞ ❤ ❡❡ ❢ ♦❡♥ ♦ ♦ ✈ ❡• ❛ ♣ ❬ ✽ ✾ ✱ ✾ ✵ ( ✳▲ ❡✉ ❝ ♦ ♥✐ ❞ ❡ ❤ ❡♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡ ✐ ♥ ♣ ❛ ✐ ❝ ✉ • ❛ ✳❋ ♦♠ ❤ ❡ ✐ ♥❡ ♦ ❢❋ ✐ ❣ ✳✹ ✳ ✷ ✭❆ ✮✇ ❡❝ ❛ ♥♦ ❜❡✈ ❡ ❤ ❛ ❤ ❡❝ ♦ • • ❡ ❝✐ ✈ ❡ • ✐ ♥ ❡✇ ✐ ❞ ❤Γ β ♦ ❢
TU RFUF gyxxigsyx fiiix yvesyxsg wyhi exh wegygysg wyhi polritoni mode is on the order of 20Γ 0 D wheres the typil sping etween polritoni modes is on the order of 5Γ 0 F he houless riterion is thus not stis(ed nd enderson loliztion is not expeted to e oservedF his oservtion is onsistent with reent theoretil work in whih the sene of enderson loliztion for rndom ensemle of lssil dipoles is predited WIF 4.7 Connection between polaritonic modes and macroscopic modes e hve seen tht the frE(eld sttering pttern of the tomi loud is very similr to tht of n e'etive homogeneous prtile with n e'etive dieletri onstntF e now mke expliit the reltion etween the @mrosopiA modes of the homogeneous prtile nd the @mirosopiA polritoni modesF o do thtD we use the (tted e'etive dieletri onstnt ε e (ω) @see solid lines in pigF RFTA nd lulte the mrosopi modes of the homogeneous prtileD whih re poles of the sttering mtrixD y itertively solving wxwell9s equtions in the omplex frequeny plne WPF emrklyD we (nd tht the omplex eigenfrequenies of the mrosopi modes oinideD within error rsD with the omplex eigenfrequenes of the oveEmentioned polritoni modes @see pigF RFPAF he reson tht error rs re given in this theoretil lultion is the ft tht the the oe0ients of the vorentzin (t of the dieletri onstnt hve unertintiesF he oe0ient c 1 of the (t funtion given y iqF RFS hs een vried within the 95% on(dene oundsF es the imginry prt of some resonne frequenies of mrosopi modes flls outside the domin de(ned y the lk oxesD we hve indited y the errorrs in pigF RFP how muh the imginry prt of the resonne frequenies hnges while vrying c 1 F hespite the ft tht the geometril length of the loud is (xed nd the frequeny of the lser is lmost (xed @lose to ω 0 AD it is possile to (nd severl longitudinl modes euse of the strong dispersion of the medium lose to resonneF his provides physil explntion of the results reported y vi et al. UPD in whih they studied the spontneous emission spetrum of oth hike @|D A nd timedEhike @|T k,I A sttes from spheril loud of uniformly distriuted twoElevel systemsF hese two sttes re de(ned s

|D = 1 √ N j |e j , 0 , @RFTA nd |T k,I = 1 √ N j e ik I •r j |e j , 0 , @RFUA
where k I is the wve vetor of the singleEphoton (eld whih prepred the stteD |e j , 0 is omined stte of N tomsD where only tom j is exited nd ll the others re in their ground stteD nd zero photons re presentF hey oserved numerilly tht the spontneous emission spetrum of these sttes were strongly dependent on oth geometry nd densityF xotlyD they oserved tht the sign of the oopertive vm shift n e reversed y suitly modifying the ensemle propertiesF foth the hike nd timedEhike sttes re not eigenmodes of the set of oupledEdipole equtions nd therefore they n e written s liner superposition of eigenmodes of the systemF reneD they prtilly overlp with polritoni modes of the systemF he frequeny position of these polritoni modes depends indeed strongly on oth the density nd the geometryF grei RF yvesyxsg wyhi sx e hixi gvyh yp gyvh eyw TV pigure RFR@A shows tht the frequeny oinidene etween the mrosopi nd miroE sopi modes is not identlX the mirosopi nd mrosopi modes hve not only the sme frequeny nd linewidthD ut lso the sme sptil struture long the z xisF fefore we ontinue with the nlysis of the other mrosopi modesD we mke remrk out how |E| 2 for the mrosopi mode is superposed on |p| 2 F vet us note tht the dipole moment squre for single reliztion see pigF RFR@A orresponds to single eigenmodeF he proedure of otining these eigenmodes hs een explined in etion RFQF prom the derivtionD it follows tht these modes re lulted in the absence of n externl driving (eldD so the dipole moments re known prt from multiplition ftorF his multipliE tion ftor is otined y normlizing the modesD whih hs een done s explined in the min text y imposing j |p β j | 2 = 1F yviouslyD the normliztion issue lso rises for the eletri (eld of the mrosopi modesF he ltter hs een normlized suh tht the intensity pro(le oinides well with the verge dipole moment squreF his wy of omE pring mrosopi nd mirosopi modes is onsistent with the ft tht p ∝ ED whih is the underlying ssumption of the model s we study the lowEexittion limitF vstlyD we note tht |p| 2 = | p | 2 + |δp| 2 F es onsequeneD |p| 2 exhiits n o'setD whih is due to dipole moment )ututionsF his explins the ft tht signi(nt o'set s for |p| 2 is not present for |E| 2 of the mrosopi modeF sn order to superimpose themD we hve dded onstnt o'set to the mrosopi |E| 2 F xowD we ome k to the nlysis of the other mrosopi modesF e note tht the sptil greement s seen in pigF RFR@A is not uniqueF pigure RFU shows the sptil greement for ll polritoni modes for oth ρ/k 3 = 1 nd ρ/k 3 = 5D exept for the mode @A of ρ/k 3 = 5D sine we ould not (nd this mrosopi mode numerillyF ht does not men this mode does not existF e possile reson tht we ould not (nd the mode is tht the (t of ε e (ω) in tht frequeny region is tully not very goodD nd mode @A lies very lose to @A nd @AF he oinidene of most polritoni modes illustrtes tht the oinidene is not restrited to prtiulr vlue of the intertion strength prmeter ρ/k 3 F he sme greement holds for the trnsverse dimensions of the oxF pigure RFV shows n exmple of this greementF ine we hve found oth frequene nd mode pro(le oinidene etween the mrosopi nd mirosopi modesD our nlysis esthlished the onnetion etween the mirosopi nd mrosopi pprohes of light stteringF vet us remrk tht the modes inside the homogeneous retngulr ox re pryEerot resonnesF por toms distriuted inside spheril volumeD the modes would e wie resonnes WD nd in the se of trpped tomi loud with normlly distriuted tomsD polritoni modes lso existD ut they re lolized in the high density regionD like for guided modes in grdedEindex (erF 4.8 Light scattering from polaritonic modes sn this setionD we disuss light sttering from the loud of prtilesF es oserved exE perimentllyD nd extensively disussed in ghpter QD when the density inresesD the onE triution of oherent light sttering inresesF qiven the identi(tion of the oherent mrosopi modes with the polritoni modesD this suggests tht polritoni modes re mostly responsile for the sttering of the loudF his is expeted s they re superrdintF sn this setionD we study diretly their ontriution to stteringF wore generllyD we need to lulte the sttered eletri (eld from suset of eigenmodesF eording to iqF PFITD

✻ ✾ ✹ ✳ ✽ ✳ ▲ ■•❍❚❙❈❆ ❚❚ ❊| ■◆•❋|❖▼ ❖ ▲❆| ■❚❖◆ ■❈ ▼❖❉ ❊ ❙ 0 5 x 1 0 -3 < | p | 2 > -2 0 2 0 5 x 1 0 -3 z / λ 0 < | p | 2 > -2 0 2 z / λ 0 0 5 x 1 0 -3 < | p | 2 > -1 0 1 0 5 x 1 0 -3 z / λ 0 < | p | 2 > -1 0 1 z / λ
❋ ✐ ❣ ✉ ❡✹ ✳ ✼ ✿❚ ❤ ❡❛ ✈ ❡❛ ❣ ❡❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡ ❞♦ ❢♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡ ✭❡ ❞❞ ♦ ✮ ✐ ♣ • ♦ ❡ ❞ ❛ • ♦ ♥ ❣ ❤ ❡ • ♦ ♥ ❣❛ ① ✐ ♦ ❢ ❤ ❡ ❡ ❝❛ ♥ ❣ ✉ • ❛ ❜ ♦ ① ✳❙ ✉ ♣ ❡✐♠ ♣ ♦ ❡ ❞✭ ❜ • ✉ ❡❞ ❛❤ ❡ ❞❝ ✉✈ ❡ ✮ ✐ | ❊| 2 ♦ ❢ ❤ ❡ ♠ ❛ ❝ ♦❝ ♦ ♣ ✐ ❝ ♠ ♦ ❞ ❡♦ ❢ ❤ ❡❤ ♦♠ ♦ ❣ ❡ ♥ ❡ ♦ ✉ ♦ ❜ • ❡ ❝✳ ❊ ❛ ❝ ❤♣ ❛ ♥ ❡ •❜ ❡ • ♦ ♥ ❣ ♦❛❞ ✐ ✛ ❡❡ ♥ ♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡ ✳ | ❡ ❢ ❡ ♦❋ ✐ ❣ ✳✹ ✳ ✷❢ ♦ ❤ ❡❡ ✐ ❣ ❡ ♥ ❢❡✉ ❡ ♥ ❝ ✐ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ❡✳❋ ♦ ❤ ❡❛ ✈ ❡❛ ❣ ✐ ♥ ❣♣♦ ❝ ❡ ❞ ✉❡ ✱ ✇ ❡❞ ✐ ✈ ✐ ❞ ❡ ❞ ❤ ❡ ② ❡♠❛ • ♦ ♥ ❣✐ • ♦ ♥ ❣❛ ① ✐ ✐ ♥❤ ✐ ♥ • ✐ ❝ ❡ ✳ ❚ ❤ ❡❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡ ❞✐ ❛ ✈ ❡❛ ❣ ❡ ❞♣ ❡ • ✐ ❝ ❡❛ ♥ ❞❤ ❡ ♥❛ ✈ ❡❛ ❣ ❡ ❞♦ ✈ ❡ ✸ ✵ ✵ ❡ ❛ • ✐ ③ ❛✐ ♦ ♥ ✳✭ ❛ ✮ ρ / k 3 =1 ❛ ♥ ❞✭ ❜ ✮ρ / k 3 =5 -0 . 2 -0 . 1 0 0 . 1 0 . 2 1 1 . 5 2 2 . 5 x 1 0 -3 x / λ 0 < | p | 2 > ✳ ❋ ✐ ❣ ✉ ❡✹ ✳ ✽ ✿ ❚❛ ♥✈ ❡ ❡♣♦ ✜ • ❡♦ ❢ ❤ ❡♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡✭ ❛ ✮♦ ❢❋ ✐ ❣ ✳✹ ✳ ✷ ✭❆ ✮✭❡ ❞❞ ♦ ✮❛ ♥ ❞ ♦ ❢ ❤ ❡ ♠ ❛ ❝♦❝ ♦ ♣ ✐ ❝ ♠ ♦ ❞ ❡✭ ❜ • ✉ ❡ ♦ • ✐ ❞❝ ✉✈ ❡✮ ✳ ❚ ❤ ❡♣♦ ✜ • ❡❤ ❛ ❜ ❡ ❡ ♥♦ ❜❛ ✐ ♥ ❡ ❞❜ ②❛ ✈ ❡❛ ❣ ✐ ♥ ❣ ❤ ❡ ❞ ✐ ♣ ♦ • ❡ ♠ ♦♠ ❡ ♥ ✉ ❛❡ ❞ ✐ ♥✐ ❞ ❡❛❤ ✐ ♥ • ✐ ❝ ❡❛z -1 . 3 λ 0 ❛ ♥ ❞❛ • ♦ ♥ ❣ ❤ ❡②❛ ① ✐✳
grei RF yvesyxsg wyhi sx e hixi gvyh yp gyvh eyw UH the set of oupledEdipole equtions re given y p j (r j ) = ε 0 α(ω) j =j µ 0 ω 2 Ḡ(r j , r j ; ω)p j + E inc (r j ) .

@RFVA he dipole moments n e lulted y solving this set of equtionsF he otined dipole moments re then written in terms of the eigenvetors P β D whih re de(ned in iqF RFQX

           p 1,x p 1,y p 1,z F F F p N,x p N,y p N,z            = β c β P β .
@RFWA fy inverting this liner system of equtionsD one n dedue the oe0ients c β F sn order to lulte the sttering pttern whih origintes from suset of eigenmodesD it is su0ient to lulte ll the dipole moments pi due to the exittion of the wnted polritoni modes

SD S ∈ βX            p1,x p1,y p1,z F F F pN,x pN,y pN,z            = S c S P S .
@RFIHA he sttered eletri (eld due to these dipoles n e lulted E sc,S (r) = µ 0 ω 2 j Ḡ(r, r j ; ω)p j .

@RFIIA elthough the di'erent eigenmodes P β re not orthogonlD sine the mtrix Ā whih de(nes them is nonErermitinD the sttered eletri (eld due to single eigenmode or suset of eigenmodes n lwys e lultedF st is understood tht the totl sttered eletri (eld is sum of oth the (elds lulted from the suset S ∈ β nd the reminder of sttesX E sc = E sc,S + E sc, not S F hen we lulte the sttering pttern tht is due to the modes S onlyD it is understood tht the interferenes etween E sc,S nd E sc, not S re not inludedF e now pply this formlism to light sttering from the tomi loud de(ned in pigF RFI@AF e hve lulted light sttering from the eigenmodes inside the retngles de(ned in pigF RFP@eAY the polritoni modesF he result is presented in pigF RFW y the lue urveD together with the oherent sttering pttern of ll the modes togetherF st is seen tht light sttering from the modes within the retnglesD the numer of modes within the retngles is only 0.54% of ll the modesD reprodues n importnt prt of the totl sttering ptternF roweverD it is lso seen tht in the kwrds diretionD the polritoni modes give rise to too lrge intensityF he ft tht the kwrd sttering is so dominnt n e explined y symmetry rgumentsF olritoni modes re modes of the verge system nd rdite therey symmetrillyF ell modes re needed to reover the preise sttering ptternF xotlyD nonpolritoni modes re needed to interfere destrutively with the kwrds sttered (elds of the polritoni modesF he ft tht suh smll mount of modes is le to reprodue signi(nt prt of the sttering pttern is remrkleF roweverD we know from lssil eletrodynmis tht light sttering t frequeny lose to resonne is dominted y the resonnes of the systemF his notion seems to e on(rmed y the ft tht light sttering is dominted y only few modesX the polritoni modesF 4.9 Absorption sn the mrosopi modelD the pperne of n imginry prt of the e'etive dieletri onstnt @see pigF RFTA ounts for lossesF sn our sitution of elsti light stteringD sorption orresponds to energy trnsfer from the oherent (eld to the inoherent (eldF rereD we demonstrte tht the totl sorption in the system is equl to the totl mount of inoherent light stteringF pirstD we disuss the mirosopi systemF prom n energy lneD we know tht for system without sorption P micro s,coh + P micro s,incoh -P micro ext = 0D where P micro s,coh is the oherent sttered power nd P micro s,incoh the inoherent sttered power ISF sn the mrosopi systemD there is oth sorption nd light stteringX P macro ext = P macro s +P macro abs F es the ensemleEverged eletri (eld follows the relmholtz eqution with n e'etive dieletri onstntD P micro s,coh = P macro s F feuse of the equivlene etween the mirosopi nd mrosopi systemsD their extintion is the smeF prom this simple resoningD it follows tht P micro s,incoh = P macro abs . @RFIPA hile losses re generlly onsidered s eing irreversile s result of dephsing proessesD eFgFD oupling to phononsD we note tht this nnot e the se hereD s there is no loss mehnism in the mirosopi modelF fefore vergingD the (eld sttered y single grei RF yvesyxsg wyhi sx e hixi gvyh yp gyvh eyw UP reliztion of the loud n e time reversedD provided tht the stterers9 positions hve not hngedF efter vergingD only the oherent (eld hs wellEde(ned phse nd n e time reversedF sn summryD in the presene of dephsing proessesD or when the positions of the toms re hnged rndomly from one reliztion to nother @s is the se when we ensemle vergeAD the oupling of the inident light to dipole )ututions leds to n irreversile rdition of inoherent lightF e re now le to expliitly ddress the question t the eginning of this hpterX wht is the physil origin t the mirosopi level of the index of refrtionc ine the index of refrtion inludes oth the di'rtive properties of n ojet nd the sorption of lightD it is understood tht the polritoni modes give n importnt ontriution to itD ut not the unique ontriutionF he surdint modes tht give rise to inoherent light stteringD whih orresponds to sorption in the homogeneous mediumD due to the rndom orienttion of the dipole moments hve lso n importnt ontriution to the index of refrtionF his mens tht ll modes re importntD ut the polritoni modes dominte the di'rtive properties of the homogeneous ojetF 4.10 Conclusion sn onlusionD in this hpter we hve shown the existene of polritoni modes in dense tomi systemF hese polritoni modes do not depend on the tomi positions ut only on the shpeD nd size of the loudD nd on the tomi densityY they re sptilly delolized nd strongly superrdintF e hve shown tht they n e identi(ed to the mrosopi modes of homogeneous ojet with n e'etive dieletri onstntF hese results pply not only to old tomi louds ut lso to ny dense system of resonnt stterers suh s moleules or quntum dotsF his hpter thus provides uni(ed vision of sttering y dense systems of resonnt stterersF Chapter 5

Revisiting homogenization for interacting resonant scatterers 5.1 Motivation hen light impinges on liquid or solidD it gets refrtedX the optil response is uniquely given y the index of refrtionF his is s opposed to light sttering from gsD for whih inoherent light sttering due to density )ututions domintesF yne wy to homogenize the system is to inrese the densityX moving from wter vporD for whih inoherent light sttering domintesD towrds liquid wterD where oherent light sttering domintesF sn ghpter Q we studied the experimentl results otined y ellegrino nd tenneweinF hey studied nerEresonne light sttering from dense loud of resonntD old tomsF ellegrino mesured inoherent light stteringF es the numer of toms ugmentedD the mount of inoherent light sttering inresed untill it sturtedF por similr tomi densityD tennewein hs lso oserved sturtion of oherent light stteringF st seems tht even though the loud eomes denserD inoherent light sttering keeps eing signi(nt prt of light stteringF his is s opposed to wht we know from light sttering y liquids nd solidsD where despite the ft they re denser thn loud of old tomsD there is no inoherent light stteringF his rises the question if resonnt tomi systemsD suh s those studied in the group of froweysD n tully reh the homogeniztion regimeD regime in whih the optil response is uniquely determined y n e'etive refrtive index nd thus not y )ututionsF e will see tht resonnt light sttering from loud of old toms is more sutle thn light sttering from liquids nd solidsF e will show tht inoherent light sttering nnot e eliminted lose to tomi resonneF 5.2 Introduction sn this setionD we desrie wht homogeniztion mounts toF efter giving some exmples of homogeniztion prolems for nonEresonnt light stteringD we will pose the prolem of homogeniztion for nerEresonne light stteringF 5.2.1 What is homogenization? vet us onsider system tht is omposed of smll onstituentsD for exmple loud of tomsD loud ontining mirodropletsD or solid tht hs rystlline struture with n tomi lttieF yne wy of lulting the response of ompound system to n externl grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii UR exittion is y rigorously lulting the response y mens of mirosopi theory tht tkes eh onstituent into ountF enother wy of lulting the response of ompound system is y pproximting the system s homogeneous mediumD nd y studying the response of the system for n externl exittionF he former pproh is often used for light sttering from the skyD wheres the ltter pproh is very ommon for light sttering y liquidF vet us onsider some exmples of the ltter pproh where the ompound system is deE sried s homogeneous mediumF he reltion etween exittion nd response is in mny situtions given y liner response funtionF ome exmples of liner response funtions re the eletri suseptiility χ e D the eletril ondutivity σD the therml ondutivity kD nd oung9s modulus EF ithin the frmework of liner response theoryD the following liner reltions re some exmples of the reltion etween n exittion @rightEhnd sideA nd the response @leftEhnd sideAX

J p = ε 0 χ e E, @SFIA J q = σE, @SFPA J φ = -k∇T, @SFQA ∆L L 0 = Eσ strain , @SFRA
where J p is the polriztion urrent densityD χ e = ε -1D with ε eing the dieletri onstntD J q eing the freeEhrge urrent densityD J φ eing the het )uxD T eing the tempertureD nd L 0 eing the originl length of n ojet tht vries y n mount ∆L under the pplition of n extensionl strin σ strain F ypillyD these macroscopic liner response funtions n e otined y n e'etive medium theoryF sn the se of eletromgnetismD there re severl models whih relte mirosopi properties like the tomi polrizility IU or the dieletri onstnts of spheril prtiles in omposite dieletri rndom medium WQ!WS to the mrosopi dieletri onstntF hen we stik to the exmple of light sttering from system ontining dieletri prtiles dispersed in host mediumD we relize tht there re two di'erent points of view to ssess if medium n e homogenizedX IF yne my e interested in the propgtion of the ensemleEverged (eld in the homogeneous e'etive mediumF rereD the question is whether the e'etive refrtive index desries accurately the ehvior of the mediumD PF he seond possile point of view is to require tht there is no incoherent light scat-teringF his is more strit formultion of homogeniztion thn the (rst formultionD sine the (rst formultion llows to hve inoherent lightD lthough its (eld needs to e smll ompred to the oherent (eldF he (rst point of view is generlly used in mny reserh (eldsF por exmpleD when n eletril urrent )ows through n eletril ondutor nd one mesures the totl urrentD the experimentl results n e (tted y yhm9s lw @iqF SFPAF yne does not hve seprtely ess to )ututing urrent due to possile inhomogeneities of the mediumF rowever in optisD inoherent sttered light is n ovious mnifesttion of the )ututions of the mediumF he ility of seprtely mesuring the verge response of system nd its US SFPF sxyhgsyx

)ututing response is prtiulr property of optis UD SIF ine we hve seprtely ess to oth oherent nd inoherent lightD we will dopt the more stringent formultion of homogeniztionX there should e no inoherent lightF sn other wordsD we dopt de(nition whih requires tht not only the e'etive dieletri onstnt existsD ut its imginry prt ssoited to sttering losses hs to e negligileF

Homogenization in optics

por the se of di'rtion from periodi mediD the seond formultion of homogeniztion @P incoh = 0A n e usedF he zerothEorder di'rted mode ontinues propgting in the sme diretion s the inident (eldD wheres higher order di'rted modes get di'rted under n ngleF hen the period Λ of the periodi medium is smller thn λD where λ is the wvelengthD only the zerothEorder di'rted mode is propgtiveY ll higherEorder modes re evnesent nd therefore do not reh the fr (eldF st is only under the ondition Λ < λ tht the optil response of the periodi medium n e desried y n e'etive refrtive indexD sine no propgting modes exist with ngles di'erent thn the ngle of inideneF sn this simple seD the homogeniztion vlidity ondition is thus given y the requirement tht the wvelength should e lrger thn length sle given y the period WTF st is often ssumed tht this ondition n e extended to rndom medi IVF sf we onsider for exmple inident light on rndom dilute medium like gsD there re density )ututions on the wvelength sle whih give rise to inoherent light stteringF hen the system typil length sle l is hosen s the men interEprtile distne r = ρ -1 3 D where ρ is the numer densityD nd r λD the medium is expeted to e self verged nd inoherent sttering is expeted to e negligile IVF his is ertinly the se for dense mediF es n exmpleD onsider light inident on pure wter or morphous siliF hese medi only refrtD re)et ndGor di'rt lightY they do not di'use lightF roweverD mny oservtions for near-resonance light sttering from optilly thik louds of old toms RHD RPD RTD SI do show tht inoherent light sttering ontinues eing signi(ntD even though the tomi density of the louds inresesF his result is surprising euse the e'etive dieletri onstnt n e s lrge s SD s we hve seen in pigF RFTD so tht the medium optilly ehves s dense mediumD ut nevertheless exhiits lot of inoherent light sttering SIF st is euse of this experimentl result tht we wnt to revisit the onept of homogeniztion in light sttering from denseD resonnt nd old tomi systemsD sine the ondition r λ seems to e insu0ient for this kind of systemsF elthough homogeniztion in light sttering is textook mterilD there re few reports regrding the prtiulr se of resonnt light sttering from dense nd old tomi gsesF

Contents

his hpter is orgnized s followsF pirstD we desrie the system tht hs een modeled nd we formlly introdue the quntities of interestF e then summrize the known mehE nisms of sttering suppression for nonEresonnt light stteringF his summry will enle us to pinpoint the di'erenes with resonnt light sttering y dense tomi systems whih is treted therefterF efter the summry on nonEresonnt light stteringD we will onsider resonnt light sttering for whih we study the reltive importne of oherent nd inoE herent light stteringF pinllyD we study prtiulr lss of smll systems omposed of grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii UT mny tomsD whihD surprisinglyD stter less light thn single tom nd whih n never e homogenizedD lthough it stis(es the ondition r λF

System description

e study light sttering from disordered wvelengthEsize loud of old tomsD s ilE lustrted in pigF SFIF es in ghpter RD we tke uoidEshpe loud with dimensions 4.8λ 0 × 0.6λ 0 × 0.6λ 0 D where λ 0 = 780 nm is the resonne wvelength of the studied trnsition of ruidiumEVU tomsF he toms re uniformly distriuted nd re modeled s lssil dipolesF glssil dipoles re modeled y n isotropi eletri polrizility mtrix ᾱ(ω) = α(ω) 1D where

α(ω) = 3πΓ r /k 3 ω 0 -ω -i Γr+Γnr+Γ * 2 , @SFSA
with ω 0 = 2πc/λ 0 the trnsition frequenyD c the speed of light in vuumD Γ r D Γ nr nd Γ * respetively the rditiveD nonrditive nd dephsing rteF hen Γ nr = Γ * = 0D this polrizility model orresponds to lssil J = 0 → J = 1 tomD s used in the previous hpterF his model n lso inlude nonrditive dey hnnels @Γ nr A nd dephsing proesses @Γ * AF nless stted di'erentlyD we ssume elsti sttering nd no dephsing proesses to ourD iFeFD Γ nr = Γ * = 0D whih is good model for old gs of identil lssil dipolesD where no ollisions tke pleF e tke Γ r = Γ 0 = 2π × 6 wrzF es we disuss dense tomi systemsD iFeF k r 1D the full qreen9s tensor is used to desrie resonnt dipoleEdipole intertionsD inluding the terms proportionl to 1/(kr) 2 nd 1/(kr) 3 UU SFRF xyxEiyxex vsqr geisxq the other diretions there re mny spikes visileF he ig loe in the forwrd diretion indites tht light sttering is indeed dominted y the ensemleEverged eletri (eldD even for single reliztionF he presene of severl peks in the other diretions indites inoherent light stteringF sn order to quntittively hrterize homogeniztionD we will ompre the scattered powers of oth oherent nd inoherent light sttering tht re lulted fter verged over mny reliztionsF he powers tht will enle us to disuss homogeniztion re de(ned s

P coh = ε 0 c 2 ‹ | E sc,far eld | 2 dS, @SFTA nd P incoh = ε 0 c 2 ‹ |δE sc,far eld | 2 dS, @SFUA
where the integrl is evluted over losed spheril surfe in the frE(eld regionD s is indited y the susripts fr (eldF e de(ne the totl sttered power s P tot = P coh + P incoh D whih is de(ned in this hpter fter the ensemle vergingF 5.4 Non-resonant light scattering sn this setion we summrize wellEknown physis out nonEresonnt light sttering y dilute ensemles of prtiles IUD WVF sn prtiulrD we illustrte the ftors leding to suppression of light stteringF sn other wordsD we review the usul onditions to otin homogeniztion in the nonEresonnt light sttering regimeF e illustrte these onditions y keeping the sme system of old toms nd the sme geometry s desried in etion SFQF he illuminting frequeny is fr detuned from tomi resonne @-10 4 Γ 0 AD so tht we onsider nonEresonnt light stteringF e lulte the oherent nd inoherent sttered powersD s we de(ned them in the previous setionF he powers re lulted s funtion of the numer of toms inside the loudD nd normlized y the sttered power of single tomF pigure SFP shows the numeril results in log!log sleF e see tht inoherent light sttering sles linerly with the numer of toms inside the (xed volumeX P incoh ∝ N D nd tht oherent light sttering sles qudrtilly with the numer of tomsX P coh ∝ N 2 D where N is the numer of toms in the loudF his result is in greement with theory on single light sttering @forn pproximtionAD s we will demonstrte nextF he wvelengthE size tomi loud is within the single sttering regimeD sine the sttering men free pthD whih is given y l sc = 1/[ρσ sc (ω)] = 3 m for N = 450 toms inside the loudD is muh lrger thn the size of the tomi loudF st is the smll sttering ross setion fr from resonne tht leds to lrge sttering men free pthF vight stters therefore t most one inside the loudF sn the single sttering regimeD eh stterer is uniquely illuminted y the inident eletri (eld WWF hen linerly polrized plne wveD E(r) = E 0,inc e ik L •r D is inident upon suh systemD the totl sttered (eld in the fr (eld is given y ptternD iFeFD the integrnd of the surfe integrlsD is proportionl to |E sc | 2 in the fr (eldF prom iqF SFVD it follows tht the sttering pttern is therefore given y

E sc (r) = ω 2 c 2 α(ω) e ikr 4πr [E 0,inc -(E 0,inc • u)u]
|E sc | 2 = ω 4 c 4 |α(ω)| 2 |E 0,inc | 2 (4πr) 2 sin 2 (θ)S(q), @SFWA
where θ is the ngle etween the oservtion diretion nd the polriztion diretionD nd S(q) is given y the soElled struture ftor

S(q) = | N j=1 e iq•r j | 2 .
@SFIHA e see tht in the single sttering regimeD the sttering pttern is determined y the struture ftorF he struture ftor n e deomposed s follows

S(q) = N j=1 e iq•(r j -r j ) + N j,l,j =l e iq•(r j -r l ) = N + N j,l,j =l
e iq•(r j -r l ) . @SFIIA he ensemleEverged struture ftor gives the ensemleEverged sttering ptternX

S(q) = N + N (N -1) e iq•(r-r ) , @SFIPA
whih n e rewritten s

S(q) = N [1 -e iq•(r-r ) ] + N 2 e iq•(r-r ) , @SFIQA
where the verge of the exponentil funtion is this time given y e iq•(r-r ) = ¨(r, r )e iq•(r-r ) d 3 rd 3 r , @SFIRA UW SFRF xyxEiyxex vsqr geisxq with (r, r ) eing the joint proility distriution of (nding one tom t position r nd one tom t position r D so tht ˜(r, r )d 3 rd 3 r = 1F prom iqF SFWD nd iqF SFIQ it follows tht for systems tht re muh smller thn the sttering men free pth nd therefore in the single sttering regimeD the inoherent sttering spetrum sles linerly with N F he oherent sttering spetrum is the seond term in the struture ftor nd sles qudrtilly with N F e oserve tht the frtion P incoh /P coh ∝ 1/N → 0 for inresing numer of toms in (xed volumeF etD inoherent light sttering keeps inresing linerly when N inresesF sn view of our strit formultion of homogeniztion @numer PX P incoh = 0AD the result tht P incoh ∝ N hs s onsequene tht fr from resonneD dense louds of toms re not in the homogeniztion regimeF he (nding tht system whih is illuminted fr from resonne nnot e homogenized is indeed wht is oserved in the skyD where yleigh sttering oursF he moleulr numer density of the sky is 10 7 moleules/µm 3 F elthough the ir is very dense for visile wvelengths @k r ∼ 0.003AD the sky ppers lue in ll oservtion diretionsF st is inoherent light sttering tht we oserve in the sky1 F he strong ondition tht P incoh should e negligile is thus not stis(edF he ondition to reh homogeniztionX r λD is lerly not su0ient for this systemF

Eect of position correlations on non-resonant light scattering

xextD we introdue position orreltions in the loud of stterersF hen position orreE ltions re presentD the joint proility density funtion (r 1 , r 2 ) of hving stterer I t position r 1 nd stterer P t position r 2 nnot e ftorized s (r 1 )(r 2 ) nymoreF snstedD the joint proility density n e written s (r 1 , r 2 ) = 1

V 2 [1 + g(r 1 -r 2 )],
@SFISA where V is the volume of the system nd g is the pir orreltion funtionD for whih n exmple @hrd sphere pproximtionA is given in pigF SFQF his is numerilly implemented y introduing spheril exlusion volume with dimeter d round eh sttererF his dimeter sets the minimum distne etween nerestEneighor stterersX d = r nn,min D so they nnot pproh one nother within distne dF he exlusion volume is illustrted in pigF SFR@AD where the dshed irles orrespond to the virtul oundriesF pigure SFR@A shows gin the normlized oherent nd inoherent sttered powerD ut this time s funtion of the nerestEneighor distneF pigure SFR@A shows tht inoherent light sttering gets redued y introduing order in the systemF en evlution of the integrl given in iqF SFIR for n in(nite volume leds to

S(q) ≈ N 1 -f V + N (N -1)δ(q), @SFITA
where f V is the volume frtion tht is oupied y the exlusion volumesF e oserve tht s the exlusion volume of eh stterer inresesD inoherent light sttering diminishesF wore generllyD iFeFD eyond the single sttering regimeD it n n e understood tht reE ting order in system redues density )ututionsF he redution of density )ututions grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii VH redues )ututions in the polriztion density when the system is illumintedD nd therey inoherent light sttering gets reduedF reneD it is possile to reh homogeniztion when introduing sptil orreltionsF his illustrtes wht we know out light sttering from pure liquidX it stters less light thn gsD lthough there re more stterersF his e'et where inoherent light sttering gets redued y orreltions is lso responsile for the trnspreny of the orne despite its heterogeneous struture IHHF

VI SFSF iyxex vsqr geisxq

Elastic mean free path he regime of suppression of light sttering n e hrE terized y n elsti men free pth l mfp whih n e onneted to the imginry prt of the e'etive refrtive index 1/l sc = 4πsm(n e )/λD s we hve seen in etion PFPFQF he ondition for suppressing sttering is given y L l sc where L stnds for typil size of the systemD so tht photons do not stter multiple timesF osition orreltions n inrese the elsti men free pth nd therey ring system of given size nd given numer of toms into the homogeniztion regimeF 5.4.2 Eect of nonradiative decay on non-resonant light scattering pinllyD we note tht nother mehnism whih leds to homogeniztion is losses in systemF sf each scatterer hs n sorption ross setion σ abs tht is muh lrger thn its sttering ross setion σ sc D the stterers do not stter lot of lightF snoherent light sttering is sid to e suppressedD sine light sorption is preferleF roweverD n ensemble of absorbing scatterers n still stter light oherentlyF en exmple is droplet of sndin ink @lso known s ghin ink in fritish inglishAD whih onsists of suspension of olloidl ron nnoprtilesF hile eh nnoprtile sors lightD the droplet ontining the nnoprtiles stters light oherentlyX light tht is inident upon droplet of sndin ink gets for exmple re)etedF he ink droplet n e desried s homogeneous droplet with omplex refrtive index to ount for the role of the nnoprtilesF uh system is in the homogeniztion regime s the inoherent omponent vnishesF he ondition for this regime n e written in terms of dey length l sc/abs = 1/(ρσ sc/abs )F he homogeniztion ondition is thus l sc l abs to ensure tht sorption is the leding ontriution to extintionF 5.5 Resonant light scattering e now turn to the resonnt sttering seF por the tomi systems we study hereD it is possile to (nd dieletri onstnt with vlues similr to the ones enountered in ondensed mtter physisF sn ghpter RD we hve found tht loud with density of ρk 3 = 1 hs n e'etive permittivity with rel prt vrying in the rnge [-2, 2] lose to tomi resonneF elthough the system is more dilute thn irD it is the resonnt nture of exittion tht leds to this lrge vlueF pigure SFS@A shows the oherent nd inoherent sttered power s funtion of the numer of toms in loud with (xed volumeD for n inident plne wve on tomi resonneF sn strk ontrst with the o'Eresonne se @dshed linesAD oth oherent nd inoherent powers do not keep inresing with the numer of tomsF he sturtion of the inoherent power when N inreses hs een oserved experimentlly y ellegrino et al. in efF SI nd the sturtion of the oherent power hs een oserved y tennewein et al. in efF UY oth experiments hve een disussed in ghpter Q of this thesisF heres oherent light sttering @di'rtionA gives rise to lrge pek in the forwrd diretionD inoherent light sttering ours in ll diretionsF efter n ngulr integrtion over the full solid ngleD the oherent nd inoherent ontriutions turn out to e of similar importanceD s n e seen in pigF SFS@A 2 F his is s opposed to wht is expeted for light sttering from grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii VP system for whih the interEtomi distne is smllX oherent light sttering might e expeted to dominteF eording to the seond formultion of homogeniztionD P incoh = 0D this tomi system n therefore not reh the homogeniztion regimeF smportntlyD this system nnot reh the homogeniztion regime ording to the (rst formultion eitherD s inoherent light sttering is s importnt s oherent light stteringF

Scattering spectra for coherent and incoherent light

sn order to further nlyze the resonnt sttering regimeD we study the spetrum of the sttered lightD iFeFD the mount of light sttered oherently or inoherently s funtion of the detuning of the lserF e plot in pigF SFS@A four spetr ofX @IA ensemleEverged oherent sttered lightD @PA ensemleEverged inoherent sttered lightD @QA the sum of the ensemleEverged oherent nd inoherent sttered lightD nd @RA the sttered light for single reliztionF yne feture is tht the spetrum is signi(ntly di'erent from vorentzin spetrumF he peks tht re visile for single reliztion orrespond to exited olletive modesD whih were introdued in ghpter RF purthermoreD it is oserved tht the oherent nd inoherent spetr show some similritiesD notle oth seem to hve seond pek for negtive detuningF xextD we will interpret the oserved oherent nd inoherent spetr from oth miE rosopi nd mrosopi perspetiveF e strt with the mirosopi point of viewF Microscopic view on spectrum pollowing the ide developed in ghpter RD we dopt the olletive modes point of view to nlyze the spetrF st ws shown tht there exist few polritoni modesD whih hve two key feturesX @iA they re roust ginst disorder @iFeFD they depend on density nd geometry ut not on the detiled position of the stterersAD nd @iiA they re superrdint nd therefore responsile for most of the oupling etween VQ SFSF iyxex vsqr geisxq the inident (eld nd the systemF rereD the term superrdint is used to de(ne sttes with dey rte lrger thn 8Γ 0 D nd these modes n e found in the loseup of pigF RFP@eAF e limit the study to the lowEexittion regime where lssil model is vlidF hese olletive modes were shown to e identil to the pryEerotElike modes of the e'etive homogeneous prtile @see ghpter RAF st ws expeted tht the light sttered y these modes ounts for lrge prt of oherent light stteringF his is indeed wht hs een oserved in pigF RFW where omprison of the the overll oherent sttering pttern with the sttering pttern of suset of modes @polritoni modesA is givenF he spetrum of oherent light sttering is therefore expeted to e 'eted y polritoni modesF xotlyD the seond pek t red detuning n e ttriuted to polritoni modesF e hve shown tht the position of this pek hnges ording to the size of the loudY indeed polritoni modes depend on the size of the loud s wellF xext we explin why the inoherent light sttering spetrum shows similrities with the oherent light sttering spetrumF snoherent light sttering is due to the modes whih re not superrdintF hey re the mjority of the modesF hese modes re exited vi the oupling with polritoni modesD euse the modes re not orthogonlF sndeedD when the set of oupledEdipole equtions is put in the form of n eigenvlue prolemD we n see in etion PFP tht the orresponding mtrix is not rermitinF hereforeD the eigenvetors re not orthogonlF ogether with the ft tht surdint modes exist over lrge frequeny rngeD s we hve seen in ghpter RD the oupling etween polritoni modes nd surdint modes lies t the origin of the similrities etween the oherent @∼polritoni modesA nd inoherent @∼surdint modesA light sttering spetrF Macroscopic view on spectrum he similrities etween oth spetr n e underE stood from mrosopi view from the ft tht the e'etive homogeneous medium hs n imginry dieletri onstnt ounting for losses @see ghpter RAF sn the sene of rel losses @onversion into hetAD these losses ount for the energy trnsfer from the oherent @vergeA (eld to the inoherent @)ututingA (eldF es the exittion of the system is medited y the resonnes of the homogeneous systemD oth oherent nd inoherent power re expeted to hve similr spetrumF 5.5.2 Eect of nonradiative decay and dephasing on resonant light scattering he ft tht lose to resonne lossless system onsisting of old toms nnot reh the homogeniztion regime mkes us wonder if there exists ny system of resonnt stterers tht n reh the homogeniztion regimeF sn etion SFRFP we hve mentioned the role of losses for nonresonnt stterersF e report in pigF SFT the results otined for resonnt light sttering from tomsD when dding nonrditive losses ndGor dephsing following iqF SFSF xonrditive losses exist in quntum dots nd dephsing might exist in the se of n tomi system due to motionl e'etsF prom pigF SFT it is seen tht the inoherent power gets signi(ntly redued wheres the oherent power is only wekly 'etedF his n esily e understood y using the mode piture to nlyze stteringF st n e seen from iqF RFQ tht the only hnge of the @omplexA eigenfrequeny of olletive eigenmode β y introduing nonrditive nd dephsing rtes isX ωβ = ω 0 -i Γr 2 → ωβ -i Γnr+Γ *

2

F his formul gives ler piture of the ompetition etween rditive nd nonrditive dey for eh modeX it follows tht light whih gets sttered y surdint modes is (ltered

❈❍❆ ❚ ❊|✺ ✳ | ❊❱ ■ ❙ ■❚ ■◆•❍❖▼❖• ❊◆ ■ )❆ ❚ ■❖◆❋❖| ■◆❚ ❊|❆ ❈❚ ■◆•| ❊ ❙❖◆❆◆❚ ❙❈❆ ❚❚ ❊| ❊| ❙ ✽ ✹ ♦ ✉ ❜ ② ❤ ❡♣❡❡ ♥ ❝ ❡♦ ❢♥ ♦ ♥❛ ❞ ✐ ❛✐ ✈ ❡• ♦❡ ❛ ♥ ❞❞ ❡ ♣ ❤ ❛✐ ♥ ❣ ✳ ❚ ❤ ✐ ❝ ❛ ♥❜ ❡✉ ♥ ❞ ❡ ♦ ♦ ❞❛ ❢ ♦ • • ♦ ✇ ✿ ✉ ❜❛ ❞ ✐ ❛ ♥ ♠ ♦ ❞ ❡ ❡♠ ✐ • ✐ ❣ ❤♦ ✈ ❡ ❛✈ ❡②• ♦ ♥ ❣ ✐♠ ❡♣ ❡ ✐ ♦ ❞ ✳ ❲ ❤ ❡ ♥♥ ♦ ♥ ❛ ❞ ✐ ❛✐ ✈ ❡ ❞ ❡ ❝ ❛ ②❝ ❤ ❛ ♥ ♥ ❡ • ❡ ① ✐ ✱ ❤ ❡ ②• ❡ ❤ ❡ ♠ ♦ ❞ ❡❞ ❡ ❝ ❛ ②❢ ❛ ❡ ✈ ✐ ❛ ❤ ✐ ♥ ♦ ♥❛ ❞ ✐ ❛✐ ✈ ❡❝ ❤ ❛ ♥ ♥ ❡ • ✱✐ ✳ ❡ ✳ ✱ ✐ ❛ ❞ ✐ ❛✐ ✈ ❡❡✣ ❝ ✐ ❡ ♥ ②❣ ❡ ❡ ❞ ✉ ❝ ❡ ❞ ✐ ❣ ♥ ✐ ✜ ❝ ❛ ♥ • ② ✳ ❆ ♦ ♣ ♣ ♦❡ ❞ ♦ ❤ ❡ ♦ ♥ ❣✐ ♥ ✢ ✉ ❡ ♥ ❝ ❡♦ ❢ ♥ ♦ ♥❛ ❞ ✐ ❛✐ ✈ ❡❞ ❡ ❝ ❛ ②♦ ♥ ✉ ❜❛ ❞ ✐ ❛ ♥ ♠ ♦ ❞ ❡ ✱• ✐ ❣ ❤ ❝ ❛ ❡❡ ❞❜ ②❛♣ ♦ • ❛✐ ♦ ♥ ✐ ❝ ♠ ♦ ❞ ❡✐ ♦ ♥ • ② ✇ ❡ ❛ ❦ • ②❛ ✛ ❡ ❝❡ ❞❜ ②♥ ♦ ♥❛ ❞ ✐ ❛✐ ✈ ❡❞ ❡ ❝ ❛ ②❛• ♦ ♥ ❣❛ ❤ ❡ ❛ ❞ ✐ ❛✐ ✈ ❡❞ ❡ ❝ ❛ ②❛❡❞ ♦♠ ✐ ♥ ❛❡ ❛ ♥ ❞ ❤ ❡❡ ❢ ♦❡❤ ❡ ❛ ❞ ✐ ❛✐ ✈ ❡ ❡✣ ❝ ✐ ❡ ♥ ❝ ② ✐❤ ✐ ❣ ❤ ✳❋ ✐ ❣ ✉❡ ✺ ✳ ✻ ❤ ♦ ✇ ❤ ❛ ❛ •♦ ❝ ♦ ❤ ❡❡ ♥• ✐ ❣ ❤ ❝ ❛ ❡✐ ♥ ❣ ❛ ♦❞ ❡ ❝ ❛ ②✇ ❤ ❡ ♥Γ ♥ + Γ * ❜ ❡ ❝ ♦♠ ❡ ❝ ♦♠ ♣ ❛❛ ❜ • ❡♦❤ ❡ ❛ ❞ ✐ ❛✐ ✈ ❡❞ ❡ ❝ ❛ ②❛❡♦ ❢✉ ♣ ❡ ❛ ❞ ✐ ❛ ♥ ♠ ♦ ❞ ❡ ✳ ❚ ❤ ✐❤ ❛ ♣ ♣ ❡ ♥ ❢ ♦ ❤ ❡ ❛♠ ❡ ❡ ❛♦ ♥❛ ❢ ♦ ✉ ❜❛ ❞ ✐ ❛ ♥ ♠ ♦ ❞ ❡ ✳ ❋ ✐ ♥ ❛ • • ② ✱ ❢ ♦♠❋ ✐ ❣ ✳✺ ✳ ✻ 1 0 -1 1 0 0 1 0 1 1 0 0 ( Γ n r + Γ * ) / Γ 0 P c o h / P 1 a t om P i n c o h / P 1 a t om ❤ ❡ ✐ ♥ ♦ ❞ ✉ ❝✐ ♦ ♥♦ ❢ • ♦❡❛ ♣ ♣ ❡ ❛ ❛ ❛ ♥❡✣ ❝ ✐ ❡ ♥ ✇ ❛ ② ♦❛ ❝ ❤ ✐ ❡ ✈ ❡❤ ♦♠ ♦ ❣ ❡ ♥ ✐ ③ ❛✐ ♦ ♥ ✐ ♥❤ ❡ ❡ ♥❡ ♦ ❢ ✉ ♣ ♣❡ ✐ ♥ ❣ ✐ ♥ ❝ ♦ ❤ ❡ ❡ ♥ ❝ ❛ ❡❡ ❞ • ✐ ❣ ❤✳ ❋ ✐ ❣ ✉ ❡✺ ✳ ✻ ✿❈ ♦ ❤ ❡❡ ♥❛ ♥ ❞ ✐ ♥ ❝ ♦ ❤ ❡❡ ♥ ❝ ❛❡❡ ❞♣ ♦ ✇ ❡ ❛❡❣ ✐ ✈ ❡ ♥ ❢ ♦ N =4 5 0❛♦♠ ✉ ♥ ❞ ❡ ✈ ❛✐ ❛ ✐ ♦ ♥♦ ❢❤ ❡♥ ♦ ♥❛ ❞ ✐ ❛✐ ✈ ❡❛ ♥ ❞❞ ❡ ♣ ❤ ❛ ✐ ♥ ❣❛❡ ✳ ❆ • •♣ ♦ ✇ ❡ ❛❡♥ ♦♠ ❛ • ✐ ③ ❡ ❞❜ ② ❤ ❡ ❝ ❛❡❡ ❞ ♣ ♦ ✇ ❡ ❢ ♦❛ ✐ ♥ ❣ • ❡❛ ♦♠❛ ❡♦ ♥ ❛ ♥ ❝ ❡ ✳ ✺ ✳ ✻ • ✐ ❛ ♥ ✢ ✉ ❝✉ ❛✐ ♦ ♥ ♦ ❢ ❤ ❡ ❝ ❛ ❡❡ ❞♣ ♦✇ ❡ ❜ ② ❡♦ ✲ ♥ ❛ ♥ ❛♦♠ ✐ ♥❛✉ ❜✇ ❛ ✈ ❡ • ❡ ♥ ❣❤✈ ♦ • ✉♠ ❡ ■ ♥❤ ✐ • ❛ ❡ ❝✐ ♦ ♥✇ ❡❞ ✐ ❝ ✉ • ✐ ❣ ❤ ❝ ❛ ❡✐ ♥ ❣ ✐ ♥❤ ❡❡ ① ❡♠ ❡❝ ❛❡♦ ❢❛❞ ❡ ♥❡ ✉ ❜ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✲ ✐ ③ ❡❝ • ♦ ✉ ❞♦ ❢❛♦♠✳ ❚ ❤ ✐ ✉ ❜ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ② ❡♠❝ ♦ ❡♣ ♦ ♥ ❞ ♦ ❤ ❡ ② ❡♠ ✐ ♥ ✐✐ ❛ • • ②❡ ♥ ✈ ✐ ✲ ✐ ♦ ♥ ❡ ❞❜ ②❉ ✐ ❝ ❦ ❡♦❞ ✐❝ ✉ ✉ ♣ ❡ ❛ ❞ ✐ ❛ ♥ ❝ ❡❡ ✛ ❡ ❝ ❬ ✶ ✵ ✶ ( ✳ ❚ ❤ ❡ ❡ ❤ ❡ ❢ ♦ ❝ ✉ ✇ ❛ ♦ ♥ ❤ ❡❜ ✉ ✐ • ❞ ✐ ♥ ❣ ♦ ❢❝ ♦ ❤ ❡❡ ♥ ❝ ❡❜ ❡✇ ❡ ❡ ♥❛ ♥❡ ♥❡♠ ❜ • ❡♦ ❢ ✐ ❞ ❡ ♥✐ ❝ ❛ •❛ ♦♠ ❛ • • ✐ ♥ ✐✐ ❛ • • ②❡ ① ❝ ✐ ❡ ❞ ✳ ❆♦ ♣ ♣ ♦❡ ❞♦ ❤ ❡ ❡ ❣ ✐♠ ❡ ❤ ❛ ✇ ❛ ❡ ♥ ✈ ✐✐ ♦ ♥ ❡ ❞❜ ② ❉ ✐ ❝ ❦ ❡ ✱✇ ❡❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ • ♦ ✇ ✲ ❡ ① ❝ ✐❛✐ ♦ ♥❡ ❣ ✐♠ ❡ ✱ ✐ ✳ ❡ ✳ ✱❛ ✐ ♥ ❣ • ❡ ❡ ① ❝ ✐❛✐ ♦ ♥ ✐♣❡❡ ♥ ✐ ♥❤ ❡❛♦♠ ✐ ❝ ② ❡♠ ✳ ❆ ✇ ❡❤ ❛ ✈ ❡❞ ♦ ♥ ❡ ✐ ♥❈ ❤ ❛ ♣❡ ✷ ❤♦ ✉ ❣ ❤✺ ✱✇ ❡ ✐ ♥ ❝ • ✉ ❞ ❡❤ ❡ ✐ ♥❡❛ ❝✐ ♦ ♥❜ ❡ ✇ ❡ ❡ ♥❛♦♠ ✐ ♥♦ ❞ ❡ ♦❞ ❡✐ ✈ ❡❤ ❡❝ ♦ • • ❡ ❝✐ ✈ ❡❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡♦ ❢ ❤ ❡ ② ❡♠ ✱ ♦♠ ❡❤ ✐ ♥ ❣ ❤ ❛ ✇ ❛ ♥ ♦ ❞ ♦ ♥ ❡ ✐ ♥ ❉ ✐ ❝ ❦ ❡ ✬ ❢ ♦ ♠ ❛ • ✐ ♠ ✱✇ ❤ ♦❛ ✉♠ ❡ ❞♥ ♦ ♥ ✲ ✐ ♥❡❛ ❝✐ ♥ ❣ ❛♦♠✳ ❇ ②❞ ✐ ❝ ✉ ✐ ♥ ❣❤ ❡ ② ❡♠ ✐ ♥ ❡♠ ♦ ❢❝ ♦ • • ❡ ❝✐ ✈ ❡ ♠ ♦ ❞ ❡ ✱ ✇ ❡✇ ✐ • •♦ ❜ ❡✈ ❡ ❤ ❛ ✈ ❡② ✽ ✺ ✺ ✳ ✻ ✳ • ■❆◆❚❋ ▲❯❈❚❯❆ ❚ ■❖◆ ❙❖ ❋❚❍ ❊❙❈❆ ❚❚ ❊| ❊❉ ❖❲ ❊|❇(| ❊ ❙❖◆❆◆❚ ❆ ❚❖▼ ❙ ■◆❆❙❯ ❇❲ ❆ ❱ ❊ ▲ ❊◆•❚❍❱❖ ▲❯▼ ❊ ❞ ❡ ♥❡ ✱✉ ❜ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✱❞ ✐♦❞ ❡❡ ❞❛♦♠ ✐ ❝ ② ❡♠ ❝ ❛ ❡• ❡ • ✐ ❣ ❤ ❤ ❛ ♥❛ ✐ ♥ ❣ • ❡❛ ♦♠❛ ♥ ❞ ✐ ♥ ❛✈ ❡②✢ ✉ ❝✉ ❛✐ ✈ ❡♠ ❛ ♥ ♥ ❡ ✳ ❖ ❤ ❡ ✇ ♦❦ ❤ ❛ ✈ ❡❞ ✐❝ ✉ ❡ ❞❛✐♠ ✐ • ❛ ② ❡♠ ✱❜ ✉ ❤ ❡ ②❞ ✐❝ ✉ ❡ ❞ ❤ ❡❛♦♠ ✐ ❝ ② ❡♠❛❛ ♥❡ ♥❡♠ ❜ • ❡♦ ❢❛♦♠ ✐ ♥❡ ❛ ❞♦ ❢❛ ♥❡ ♥❡♠ ❜ • ❡♦ ❢❝ ♦ • • ❡ ❝✐ ✈ ❡♠ ♦ ❞ ❡❬ ✺ ✱ ✻ ( ✳ ❆ • ❤ ♦ ✉ ❣ ❤ ❤ ❡ ✐ ❝ ♦ ♥ ❝ • ✉✐ ♦ ♥❛❡ ❤ ❡ ❛♠ ❡❛ ❤ ❡❝ ♦ ♥ ❝ • ✉✐ ♦ ♥♣❡❡ ♥❡ ❞ ✐ ♥ ❤ ✐ ❡ ❝✐ ♦ ♥ ✱ ✇ ❡ ❜ ❡ • ✐ ❡ ✈ ❡ ❤ ❛ ❞ ✉ ❡ ♦ ❤ ❡ ♦ ♥ ❣ ✐ ♥❡❛ ❝✐ ♦ ♥ ✱❞ ❡ ♥ ❡❛ ♥ ❞ ❝ ♦ • ❞❛♦♠ ✐ ❝ ② ❡♠ ❤ ♦ ✉ • ❞❜ ❡ ❡ ❛❡ ❞ ✐ ♥❡♠ ♦ ❢❝ ♦ • • ❡ ❝✐ ✈ ❡❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡✐ ♥❡ ❛ ❞♦ ❢❛ ♥❡ ♥❡♠ ❜ • ❡♦ ❢ ✐ ♥ ❞ ✐ ✈ ✐ ❞ ✉ ❛ •❛♦♠✳ ✺ ✳ ✻ ✳ ✶ ❙ ❝ ❛ ❡✐ ♥ ❣ ♣ ❡ ❝ ✉♠❢ ♦ ✉ ❜✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✲✐ ③ ❡❛♦♠ ✐ ❝❝ • ♦ ✉ ❞ ❲ ❡❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ✐ ✉ ❛✐ ♦ ♥✇ ❤ ❡ ❡❛ ❡♦ ♥ ❛ ♥ ♣ • ❛ ♥ ❡✇ ❛ ✈ ❡ ✐ ✐ ♥ ❝ ✐ ❞ ❡ ♥✉ ♣ ♦ ♥❛ ✉ ❜ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✲ ✐ ③ ❡❝ • ♦ ✉ ❞♦ ❢❛ ♦♠✳ ❚ ❤ ❡❝ • ♦ ✉ ❞❤ ❛❛ ❤ ❛ ♣ ❡♦ ❢❛❜ ♦ ①✇ ✐❤ ✐ ❞ ❡λ 0 / ( 2 π ) ✱♦ ❤ ❛ ❡ ❛ ❝ ❤❛♦♠ ✐✐ ♥❡ ❛ ❝ ❤♦ ❤ ❡✬♥ ❡ ❛✲ ✜ ❡ • ❞❡ ❣ ✐ ♦ ♥ ✳ ❲ ❤ ✐ • ❡❝ ❤ ❛ ♥ ❣ ✐ ♥ ❣ ❤ ❡♥ ✉♠ ❜ ❡ ♦ ❢❛♦♠ ✐ ♥ ✐ ❞ ❡❤ ✐ ♠ ❛ • • ❝ • ♦ ✉ ❞ ✱✇ ❡❝ ❛ • ❝ ✉ • ❛❡❤ ❡❡ ♥❡♠ ❜ • ❡ ✲ ❛ ✈ ❡❛ ❣ ❡ ❞ ♦❛ • ❝ ❛ ❡❡ ❞♣ ♦ ✇ ❡✳ ❚ ❤ ❡♦❛ • ❝ ❛ ❡❡ ❞♣ ♦ ✇ ❡ ✐♥ ♦♠ ❛ • ✐ ③ ❡ ❞❜ ② ❤ ❡ ♦❛ • ❝ ❛ ❡❡ ❞♣ ♦ ✇ ❡ ❢ ♦❛ ✐ ♥ ❣ • ❡❛ ♦♠ ✳ ❚ ❤ ❡ ❡✉ • ❛❡♣ • ♦ ❡ ❞ ✐ ♥ ❋ ✐ ❣ ✳✺ ✳ ✼ ✳■✐ ❡ ❡ ♥ ❤ ❛ ❛ ② ❡♠✇ ✐❤ ❡ ✈ ❡❛ •❛♦♠ ✐ ♥ ✐ ❞ ❡❤ ❡❝ ✉ ❜ ✐ ❝❜ ♦ ① ❝ ❛ ❡ • ❡ ❤ ❛ ♥ ❛ ✐ ♥ ❣ • ❡❛ ♦♠ ✳■ ✐❛ •♦ ❡ ❡ ♥ ❤ ❛ ❤ ❡ ❝ ❛ ❡❡ ❞♣ ♦ ✇ ❡ ✐❛ •♠ ♦ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥♦ ❢ ❤ ❡♥ ✉♠ ❜ ❡ ♦ ❢❛♦♠✳ ❚ ♦❛ ♥ ❛ • ② ③ ❡❤ ✐ ❜ ❡ ❤ ❛ ✈ ✐ ♦✱✇ ❡♣ • ♦ ❤ ❡ ♣ ❡ ❝ ✉♠♦ ❢ ❤ ❡ ❝ ❛ ❡❡ ❞ • ✐ ❣ ❤✐ ♥❤ ❡ ✐ ♥❡ ❢ ♦❛ ✐ ♥ ❣ • ❡❡ ❛ • ✐ ③ ❛✐ ♦ ♥❛ ♥ ❞✶ ✵❛ ♦♠✳ ❲ ❡♦ ❜ ❡✈ ❡ ❤ ❛ ❜ ②✈ ❛② ✐ ♥ ❣ ❤ ❡ ✐ ♥ ❝ ✐ ❞ ❡ ♥ ❢ ❡✉ ❡ ♥ ❝ ② ✱ ❡ ✈ ❡❛ • ♠ ♦ ❞ ❡ ❛❡❡ ① ❝ ✐❡ ❞ ✳ ❚ ❤ ❡❡ ♠ ♦ ❞ ❡ ❛❡ ♣ ❡ ❝ ❛ • • ② ❡ ♣ ❛❛❡ ❞ ✳ ❚ ❤ ❡❛♠ ♦ ✉ ♥ ♦ ❢• ✐ ❣ ❤ ❝ ❛ ❡❡ ❞ ❢ ♦ ❡ ❛ ❝ ❤ ♠ ♦ ❞ ❡❞ ♦ ❡ ♥ ♦ ✈ ❛② ♠ ✉ ❝ ❤❛ ♥ ❞ ✐ ❝ • ♦❡ ♦ ❤ ❛ ♦ ❢❛ ✐ ♥ ❣ • ❡❛ ♦♠ ✳ ❚ ❤ ✐ ✐ ❡♠ ❛❦ ❛ ❜ • ❡ ✱ ✐ ♥ ❝ ❡ ✇ ❡♥ ❡ ✈ ❡✜ ♥ ❞❛ ♠ ♦ ❞ ❡ ❤ ❛ ❝ ❛ ❡ ♠ ♦ ❡ ❤ ❛ ♥❛ ✐ ♥ ❣ • ❡❛ ♦♠ ✇ ♦ ✉ • ❞ ❀ ✇ ❤ ✐ ❝ ❤ ✐ ❝ ❤ ❛❛ ❝❡✐✐ ❝ ❢ ♦❛ ✉ ♣ ❡ ❛ ❞ ✐ ❛ ♥ ♠ ♦ ❞ ❡ ✳■ ♥ ❡ ❛ ❞ ✱ ✇ ❡♦ ❜❡✈ ❡❛ ♣ ❡ ❝ ❛ • ❡ ♣ ✉ •✐ ♦ ♥
♦ ❢ ❤ ❡ ♠ ♦ ❞ ❡ ❞ ✉ ❡ ♦ ❡♦ ♥ ❛ ♥ ❞ ✐ ♣ ♦ • ❡ ✲ ❞ ✐ ♣ ♦ • ❡ ✐ ♥❡❛ ❝✐ ♦ ♥ ✳❋ ♦❛ ♥❛♦♠ ✐ ❝ ② ❡♠✇ ✐❤ ♦ ✉ ❛ ♥ ② ❋ ✐ ❣ ✉ ❡✺ ✳ ✼ ✿❆ ❡♦ ♥ ❛ ♥ ♣ • ❛ ♥ ❡✇ ❛ ✈ ❡❣ ❡ ❝ ❛❡❡ ❞❜ ②❛♦♠ ✐ ♥ ✐ ❞ ❡❛❝ ✉ ❜ ✐ ❝❜ ♦ ①✇ ✐ ❤ ✐ ❞ ❡ λ 0 / ( 2 π ) ✳ ❚ ❤ ❡♦❛ • ❝ ❛❡❡ ❞♣ ♦ ✇ ❡ ✐♥ ♦♠ ❛ • ✐ ③ ❡ ❞❜ ② ❤ ❡ ❝ ❛❡❡ ❞♣ ♦ ✇ ❡ ❢ ♦❛ ✐ ♥ ❣ • ❡❛ ♦♠ ❛ ❡♦ ♥ ❛ ♥ ❝ ❡ ✳■ ✐ ❡ ❡ ♥ ❤ ❛ ❤ ❡ ② ❡♠♦ ❢N ❛♦♠ ❝ ❛❡ • ❡ ❤ ❛ ♥❛ ✐ ♥ ❣ • ❡❛ ♦♠ ✳■ ✐ ❛ •♦ ❡ ❡ ♥ ❤ ❛ ❤ ❡ ♦❛ • ❝ ❛❡❡ ❞♣ ♦ ✇ ❡ ✐❛ •♠ ♦ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥♦ ♥ ❤ ❡♥ ✉♠ ❜ ❡ ♦ ❢❛♦♠✳■ ♥ ❤ ❡ ✐ ♥❡✱❤ ❡ ♣ ❡ ❝ ✉♠ ❢ ♦ ❛ ✐ ♥ ❣ • ❡❡ ❛ • ✐ ③ ❛✐ ♦ ♥♦ ❢❤ ❡❜ ♦ ①✇ ✐❤N =1 0❛♦♠ ✐ ♥ ✐ ❞ ❡ ✐ ❤ ♦ ✇ ♥ ❜ ② ❤ ❡ ❡ ❞ ✱ ♦ • ✐ ❞❝ ✉✈ ❡✳ ❚ ❤ ❡❝ ♦ • • ❡ ❝ ✐ ✈ ❡❡ ✐ ❣ ❡ ♥♠ ♦ ❞ ❡❛❡ ♣ ❡ ❝ ❛ • • ② ❡ ♣ ❛❛❡ ❞ ✳
grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii VT intertionD ll modes re degenerteF st is the resonnt dipoleEdipole intertion tht lifts the degenery of these modesF o explin why on verge the system stters muh less thn single tomD s we n see from the min pnel of pigF SFUD we use the spetrum s presented in the inset of pigF SFUD in whih the modes re spetrlly seprtedF vet us denote δω the typil sping etween two peks nd tke Γ 0 s the verge width of eh pekF he proility tht given lser line exites mode is thus Γ 0 /δωF es δω > Γ 0 D most of the reliztions the system does not get exitedF hereforeD the verge sttered power is smller thn tht of single tomF his is on(rmed in the min pnel of pigF SFUF 5.6.2 Average mode spacing sn n tul experimentD toms re moving so tht the spetrl positions of the modes will )utute over timeF por monohromti inident lserD it follows tht the sttering signl is expeted to disply gint )ututions in timeF his will e ler signture of this intertion regimeF vet us disuss the onditions for oservtion of this prtiulr sttering regime with low verge sttering ross setion nd gint )ututionsF e use s de(nition of this sttering regime system where the sttering peks re well seprted s displyed in the inset of pigF SFUF he ondition to e in this regime is given y δω > Γ 0 F xextD we study this ondition for the oservtion of gint )ututions nd see under whih experimentl onditions this regime n e ttinedF snsted of lulting the sttering spetrD let us lulte the eigenfrequenies of the systemF he totl spred of the eigenfrequenies ∆Ω is divided y the totl numer of modesD so tht we otin the verge distne etween modes δωF sn order to determine ∆ΩD we ritrrily selet WH7 of the modesF e repet this proedure for ues with vrying volume nd vrying numer of toms insideF pigure SFV@A shows the typil sping δω/Γ 0 of the modes s funtion of the volume for N = {20, 40, 80} tomsF he results re (tted with the funtionX δω/Γ 0 = c 1 {V /[λ 0 /(2π)]} c 2 D where c 1 nd c 2 re the free (t prmetersF st is oserved tht the mode sping is independent of the numer of toms nd tht it is inversely proportionl to the volumeF hese results re onsistent with kEofEtheEenvelope lultionF sn order to estimte the verge sping etween two modes δωD we (rst estimte the totl spred ∆ΩF por this estimteD we ssume the frequeny spred is ounded y the shift whih is due to the intertion etween two toms tht re in eh other9s nerE(eld regionX Γ 0 /(kr) 3 D where we tke r to e the typil interEtomi distneF por r we hve the reltionship N r 3 = V D where V is the volume of the tomi loudF he typil sping etween modes is estimted y dividing the totl frequeny spred y the totl numer of modes @3N AF st follows tht the typil sping

δω/Γ 0 ∝ 1 V k 3 pull qreen9s tensorF @SFIUA @SFIVA
he verge mode sping is therefore indeed independent of the numer of toms nd inversely proportionl to the volumeF his explins wht we hve seen in pigF SFUD where the totl sttered power is independent of the numer of toms inside the loudF Importance of near-eld interaction e now ssess the importne of the nerE(eld intertion whih vries s 1/r 3 F por thisD we ssume tht only the frE(eld intertion The entire Green's tensor has been taken, i.e., also the 1/r 3 term. It is observed that the typical spacing is independent of the number of atoms and δω ∝ V -1 . (b) Only the 1/r term of the Green's tensor has been used in the calculation. It can be seen here that the typical spacing depends on the number of atoms and that δω ∝ V -1/3 . term proportional to 1/r is present. The average mode separation as a function of the volume for dierent numbers of atoms is shown in Fig. 5.8(b), where the same t function δω/Γ 0 = c 1 {V /[λ 0 /(2π)]} c 2 is used. We observe that the typical spacing depends on the number of atoms and is roughly proportional to V -1/3 . By using the same formalism as described above, but by replacing the typical interaction energy by Γ 0 /(kr), the typical spacing can be shown to be

δω/Γ 0 ∝ 1 N 2/3 (V k 3 ) 1/3
Only the 1/r contribution of Green's tensor.

(5.19) Figure 5.9 shows the t parameter c 1 as a function of the number of atoms. The t function for the c 1 (N ) plot is also a power function and the t results in δω/Γ 0 ∝ N -α , with the power α = 0.68 ± 0.07 with 95% condence bounds. This result coincides with the predicted dependence of the mode spacing δω/Γ 0 ∝ N -2/3 . With this we proved the theoretical estimate of the mode spacing:

δω/Γ 0 ∝ 1/[N 2/3 (V k 3 ) 1/3
] for the situation where we only take the far-eld term of the Green's tensor into account.

Condition for nonobservation of single-photon superradiance

Now we study a consequence of this result. It follows that the condition δω/Γ 0 > 1, for which no single-photon superradiance can occur, within a constant factor, can be put in the form

V k 3 < 1 Full Green's tensor, (5.20) V k 3 < 1 N 2
Only the 1/r contribution of Green's tensor. his shows tht for systems tht do not strongly intertD iFeFD only the frE(eld term is present in the qreen9s tensorD one nnot resonly reh the regime where singleE photon superrdine is sentD s it would require V < λ 3 /[(2π) 3 N 2 ]D whih is hrd to hieve experimentlly in the visile regimeF etD for systems where nerE(eld dipoleE dipole intertions re inluded @iqF SFPHAD this regime n e rehed for any number of atomsF sn other wordsD in the regime of N toms in suwvelengthEsize volume nd the full qreen9s tensor is onsideredD the system is driven y olletive modes spetrlly seprted nd displys gint )ututionsF es soon s the volume inreses eyond λ/(2π) 3 D the modes strt overlpping spetrlly therey llowing singleEphoton superrdine to show upY n exmple is given in pigF SFS@AD where we oserve tht the sttering pttern rehes vlues tht re superior to IF sn summryD the ondition V k 3 < 1D whih is ondition for singleEphoton superrdine for noninterting stterersD eomes ondition of nonobservation of singleEphoton superrdine for interting resonnt stterersF e explin the nonoservtion of singleEphoton superrdine in terms of modes repulsionF 5.6.4 Thouless criterion for Anderson localization pinllyD we mke smll remrk on enderson loliztionF e riterion to oserve enderson loliztion is tht ll modes re spetrlly seprtedD so tht δω > Γ 0 VWD WHF his ondition is known s the houless riterion for enderson loliztionF e hve seen tht suwvelengthEsize tomi loud stis(es the houless riterionF en interprettion of the houless riterion is tht n exited lolized mode nnot hop to nother lolized mode nd eventully to the other side of the systemD sine the modes do not overlp spetrllyD thus ssuring the loliztion of lightF roweverD the suwvelengthEsize loud is very smllF xote however tht to oserve enderson loliztionD the system should e smller thn the loliztion lengthD ondition tht my not e veri(ed for the very smll systems we hve studied hereF ine we do not know the loliztion length of this systemD we nnot onlude on the possiility of oserving enderson loliztion in these smll loudsF VW SFUF iehsexgi pyw yhiihD hixi eywsg gvyh e note tht kipetrov nd okolov hve theoretilly studied enderson loliztion in rndom ensemles of resonnt point stterers WIF sn tht workD the houless riterion is never met under the vetoril desription of the resonnt dipoleEdipole intertionsF he oservtion tht the houless riterion is met for suwvelengthEsize loud is not in ontrdition with efF WID sine efF WI only onsiders tomi louds tht re lrger thn [λ/(2π)] 3 F 5.7 Superradiance from ordered, dense atomic clouds he onlusions we hve drwn so fr re sed on disordered systemsF por ompletenessD we rie)y disuss how the imge of olletive modes n e used to show tht it might e possile to oserve superrdine in orderedD not movingD suwvelengthEsize loudsF en ordered system of N prtiles hsD like disordered systemD lso 3N olletive eigenE modesF roweverD due to the order of the systemD it is expeted tht there is ertin level of degenery of the olletive modesF hen severl modes spetrlly overlpD n inE ident ontinuousEwve lser will exite severl modes t the sme timeD s opposed to the se of disorderedD suwvelength loudsD therey mking the oservtion of superrdine possileF e prtiulr orderedD (niteEsize system is one for whih primitive unit ell n e de(nedF en exmple of sitution for whih this is possile is one where toms re regulrly pled on suwvelength irle @see efF TAF uh system hs high level of degeneryD sine there re only 3N unit cell di'erent olletive eigenmodesD where N unit cell is the numer of toms inside the primitive unit ellF st would e interesting to experimentlly study superrdine from these kind of systemsD lthough it is omplited tsk due to geometril onstrintsF

Conclusion

he initil motivtion of this study ws the numeril oservtion tht inoherent light sttering y n ensemle of toms does not derese s the numer density inreses for system with (xed volumeF wotivted y this numeril oservtionD we hve revisited the onditions of homogeniztionF e dopt strit formultion of homogeniztionX the power ssoited to the )ututing (eld should e negligileF eording to this de(nitionD it is understood tht for nonEresonnt light sttering the ondition r λD where r is typil length sle of the prolemD is not su0ient onditionF etD y introduing lol order or losses in systemD the inoherent power n eome negligileF he former inreses the typil sttering lengthD nd when L < l sc D the riterium of homogeniztion is stis(edY for the ltter l sc l abs needs to e rehedF hen studying system of interting resonnt tomsD we nnot nlyze the results with the sme onepts s we re used to for noninterting prtilesF he spetrl nlysis of the sttered light demonstrtes the onset of olletive modes with spetrl signtures tht re mrkedly di'erent from the tomi vorentzin pro(lesF por lrge systemsD we found in ghpter R tht inoherent light sttering is mostly due to the nonpolritoni modes wheres the oherent sttered power ssoited to the ensemleEverged (eld is due to polritoni modesF st is found in this hpter tht when the numer of toms inresesD inoherent light sttering does not vnish nd is tully eqully importnt s oherent light stteringD so tht this is lss of system tht nnot reh the homogeniztion grei SF isssxq rywyqixsesyx py sxiegsxq iyxex geii WH regimeF e (nd tht here ginD the introdution of losses or dephsing mehnisms suh s motion of the stterersD my e used to suppress inoherent light stteringF pinllyD we hve oserved prtiulr lss of systemsX ensemles of toms in volume V smller thn 1/k 3 F hen this ondition is ful(lledD the olletive modes re spetrlly seprtedF st is found tht when inresing the numer of tomsD the olletive modes sty spetrlly seprted nd stter less thn single tomF roweverD fter n ensemle vergeD the system stters less thn single tom doesF hese systems nnot reh the homogeniztion regime y inresing the numer of tomsF pinllyD we (nd tht for interting resonnt tomsD the ondition V < 1/k 3 is ondition tht ensures tht no singleEphoton superrdine is oservedF Chapter 6

Theory of electroluminescence by quantum-dots based metasurface light-emitting devices 6.1 Introduction sn the previous hptersD we disussed olletive e'ets in light scattering from dense tomi systemsF e hve seen tht the oherent optil response is determined y the e'etive index of refrtion whih is found to e eyond the vorentzEvorenz model for the e'etive index of refrtionD nd tht )ututions in either density or spetrum re responsile for inoherent light stteringF he oherent optil response is dominted y polritoni modes @ghpter RAF he inoherent optil response stys importnt s ompred to the oherent optil response s the density inreses @ghpter SAF sn this hpter we study the emission properties of dense ensemle of emitters lose to resonnt nnostruturesF wore spei(llyD we study the emission properties of dense ensemle of olloidl quntum dots @hsA lose to plsmoni nnontennsF he interest in the emission properties of dense ensemle of hs within this thesis omes from the ft tht quntum dots hve very similr properties s tomsF ome exmples of similrities re the presene of quntized energy levelsD therey mking them resonnt entitiesD resonnt dipoleEdipole intertions exist etween the individul prtiles of whih the ensemle is omposedD dense ensemle n e retedD whih is espeilly true for olloidl quntum dotsF untum dots re onsidered to e very importnt for mny industril pplitionsD n exmple eing quntumEdot sed lightEemitting soures IHPF e key property for lighting pplitions is their tunle ndgpF fy slightly vrying the size of the quntum dotD the ndgp nd therey the wvelength of the emitted light n e hosenF roweverD s quntum dots nnot e perfetly identil due to the fritionD they ll hve slightly di'erent emission nd sorption spetrF his inhomogeneous rodening is ig di'erene etween old toms nd quntum dotsF sn this hpter we study the eletroluminesene @ivA properties of dense ensemle of hsF sn the next setionD we will explin the physil mehnism of eletroluE minesene nd see wht is the stte of the rt in the (eld of quntumEdot sed eletroE grei TF riy yp ivigyvwsxigixgi f exwEhy feih wiepegi vsqrEiwssxq hisgi WP lumineseneF sn etion TFQ we desrie the experimentl results otined in the group of drF eloyse hegiron t snstitut d9Életronique pondmentle @sipA in niversité ris udF his will llow us to onsider if the quntum dots in their experiments should e modeled s independent entities or s n ensemleF fsed on this setionD we will disuss in etion TFR the stndrd eletroluminesene model for vihF he eletroluminesene model for vih will e generlized to the sitution of quntum dots in the ner viinity of nnostrutures in etion TFSF sn etion TFT we will pply the proposed generlized eleE troluminesene model to experiments tht hve een onduted in the group of hegiron on two di'erent nnontennsF 6.2 Electroluminescence from quantum dots efter we introdued the ide of eletroluminesene from quntum dotsD we will summrize wht hs lredy een done on eletroluminesene from quntum dotsF 6.2.1 Quantum-dot based electroluminescence o explin eletroluminesene from quntum dotsD we use the shemti presented in pigF TFIF sn order to oserve n eletroluminesene signlD voltge is pplied ross multilyer system tht ontins hsF he pplied voltge will mke tht holes nd eletrons move towrds the lightEemitting lyerD whih is in this drwing single lyer of hsF roles re trnsported vi soElled hole trnsport lyer @rvA nd eletrons vi n eletron trnsport lyer @ivA towrds the h lyerF here re three possile omintions of the hrge trnsport lyersX @IA ll inorgniD @PA ll orgniD or @QA hyridD where typilly the iv is hosen to e inorgni nd the rv to e orgniF he devie tht is treted in prtiulr in this hpter hs n llEinorgni rhitetureD with the iv eing iy 2 nd the rv eing woy x F he eletron nd hole form n exiton tht n reomine t the level of the quntum dots under emission of photonF he energy of the photon is de(ned y the ndgp energy of the hF he photon tht is reted needs to Inside the quantum dots, the hole and electron form an exciton that can recombine under emission of a photon. The anode is chosen to be optically transparent, so that light can escape.

WQ

TFQF exwEhy feih wiepegi vsqrEiwssxq hisgi espe the devieF his is why in prtie one of the eletrodes is hosen to e optilly trnsprentF 6.2.2 History of quantum-dot based light-emitting devices e give here shortD nonEexhustiveD historil outline of the development of quntumEdot sed lightEemitting devies @hEvihAF st ws in IWWR tht similr ide s presented in pigF TFI ws used to rete the (rst quntumEdot sed lightEemitting devie IHQF por thisD golvin et al. used gde olloidl quntum dot (lm of few tens of nnometersF his lyer of hs served oth s iv nd emissive lyerF por the trnsport of holesD they used n orgni lyerF e very importnt prmeter for lightEemitting diodes @vihsA is the externl quntum e0ieny @iiAD whih is de(ned s the numer of photons tht espe from the devie per eletron tht psses through the devieF he very (rst devie of golvin et al. hd ii = 0.01% nd turnEon voltge of R IHQF his ii ws t tht time three orders of mgnitude smller thn the ii tht ws rehed for onventionl p!n juntion sed vihsX thereD y texturing the surfe in order to extrt more lightD ii = 30% ws rehed in IWWQ IHRF he next importnt improvement for hEvihs ws y reting orgnilly pped gdeGgd oreEshell hsF his llowed to improve the ii y ftor PP to ii = 0.22%D while keeping the sme turnEon voltge of R IHSF st ws in PHHP tht goe et al. relized tht y using single monolyer of hs insted of multilyerD the poor ondution through the multilyer hs ould e voidedF hey used n orgni lyer s n ivD therey inresing the performne of the devie IHTF hey reported n ii of 0.52% nd PSEfold improvement in luminesent power e0ieny whih orresponds to the luminous intensity per unit of surfe IHTF hree yers lterD it ws relized tht y pling the hs inside qx p!n juntionD therey reting n ll inEorgni devieD the eletril injetion of rriers ould e improved IHUF roweverD the ii of this system ws only 0.01%F enother type of hEvih ws introdued soon fter for whih the iv ws n inorgni lyer nd the rv n orgni lyerF h ision hs worked on suh kind of hEvih nd reported in PHIP n IV7Eii hEvih IHVY n ii tht ws muh higher thn ll previously pulished externl quntum e0ienies IHPF elthough there is lot of progress for hEvihsD their ii is still fr ehind the ii of onventionl vihs for whih there exist devies with ii > 60%F sn order to improve the iiD ttention ws not uniquely drwn on optimizing the eletril injetion of rriers nymoreF st ws understood tht it is very importnt to optimize the optis of the devie s wellF wo hllenges re for exmple to extrt most of the emitted light from the devie nd to enhne the rditive reomintion rte of exitonsF st is only very reently tht one introdued metls in the viinity of the hs so tht plsmoni modes re used to enhne the rditive dey rte IHW!IIPF 6.3 Quantum-dot based metasurface light-emitting device eentlyD new lss of hEvihs hs een introdued y the group of hegironX quntumE dot sed metasurface lightEemitting devies @hEwvihA IHD IIF en exmple of suh devie is shown in pigF TFPF e will introdue the ide ehind the introdution of metsurfe nd disuss qulittively the experimentl results otined y veEn et al. he min ojetive of this hpter is to propose model to desrie eletroluminesene grei TF riy yp ivigyvwsxigixgi f exwEhy feih wiepegi vsqrEiwssxq hisgi WR from hEwvih devies where the quntum dots re densely pkedF e elieve tht the model will e importnt for future design of hEwvihsF show some experimentl results tht will motivte the study of eletroluminesene from this kind of deviesF pigure TFQ@A shows the iv intensity s funtion of the pplied voltge for three di'erent deviesF he presene of the gold nnorings redues the turnEon voltge from T to only IFQ F yne mehnism ehind the redution of the turnEon voltge is the presene of pth of lest resistne tht filittes the eletril injetion IIF his result of redution of the turnEon voltge is in itself very promising for hEwvihs s ndidte in the mrket of lightEemitting deviesF pigure TFQ@A shows the eletroluminesene signl tht omes from the system where the hs touh the gold nnoringsF st is seen tht light only ppers from the region where nnorings re presentF he system presented in pigF TFQ@A ontins SEnmEthik dieletri sper etween the hs nd the gold nnoringsF por this on(gurtionD it seems tht the eletroluminesene signl is not enhned y the gold nnoringsF he presene of dieletri sper of only S nm prevents eletroluminesene to ourF his result is surprisingD s one typilly introdues thin dieletri sper in order to prevent for )uoresene quenhingD iFeFD nonrditive energy trnsfer from the emitter to the metlD therey reduing the rdition e0ieny of the quntum dotsF In (c), no electroluminescence is observed from these regions. The dierences between the patches and the symbols M8 and M9 are irrelevant for this thesis. Courtesy of Quynh Le-Van.

Description of a dense ensemble of colloidal quantum dots

prom the experimentl results presented in pigF TFQ@DAD it follows tht hs touhing plsmoni nnontenns is fvorle for the eletroluminesene signlF roweverD it is known tht individual quntum emitters lose to metlli ojet ontining plsmoni resonne hve redued rdition e0ieny due to )uoresene quenhing IIS!IPIF sn order to prevent )uoresene quenhing to tke pleD sper ws introdued etween the emitters nd the metl in eletroluminesene experiments IHW!IIPF imilr to themD veEn et al. did the sme thingD ut the experimentl results show tht enhned eletroluminesene is only visile in the sene of dieletri sper IHF his result hllenges desription of the ensemle of hs s olletion of individul entitiesF sn this setionD we theoretilly study if the ensemle of hs should e onsidered s n ensemle of independent emitters or s n ensemle of olletive modesF he ltter grei TF riy yp ivigyvwsxigixgi f exwEhy feih wiepegi vsqrEiwssxq hisgi WT se hs een thoroughly disussed for dense loud of old toms in ghpter RF he nswer to this prolem will hve n importnt onsequene on the wy eletrolumine from hEwvihs should e modeledF Quantum dots: are they independent or not? he question of whether quntum dots n e onsidered to e dependent or independent entities omes down to the question of the resonnt dipoleEdipole intertions etween the hs eing strong or notD iFeFD is energy trnsfer etween hs fster proess thn the radiative deyc o ddress this questionD we show n iw imge of the h (lm used y veEn et al. is displyed in pigF TFR@AF st shows tht the hs re lmost touhing eh otherF hen emitters re so lose to eh otherD nonrditive energy trnsfer n tke ple etween n exited h nd h in its ground stteF his proess is lled pörster resonne energy trnsfer @piAF e shemti of pi proess is given in pigF TFR@AF pi tkes ple due to resonnt dipoleEdipole intertionsF pi eing pure nerE(eld e'etD the rte of the energy trnsfer sles s Γ D→A ∝ 1/r 6 D where D nd A stnd for donor nd eptorD respetivelyD nd r is the interEprtile distne IPF his trnsfer rte n e ompred to the single prtile spontneous emission rte nd it n e shown @efF IPA tht within the eletri dipole pproximtion

Γ D→A Γ r = r 0 r 6 = 1 r 6 × 3c 4 4π ˆ∞ 0 f D (ω)σ A (ω) n 4 (ω)ω 4 dω, @TFIA
where c is the speed of light in vuumD n is the refrtive index of the host mediumD r 0 is cQD 1 cQD 2

σ A (ω) = ksm[α exciton (ω)], @TFRA
where the exiton is modeled s lssil dipoleX

α exciton (ω) = 6πc 3 Γ r ω 2 0 (ω 2 0 -ω 2 -iω(Γ r + Γ nr + Γ * ) , @TFSA
with ω 0 eing the resonne frequeny nd Γ * eing the pure dephsing rteF prom iqF TFR nd iqF TFS it follows tht the sorption ross setion of the exiton is vorentzinF sn order to otin f D (ω)D we hve tken the sme funtion s the sorption ross setionD ut with di'erent resonne frequeny ω 0 nd we normlized this funtion ording to iqF TFQF hereforeD in our lultions we ssume f D (ω) to e vorentzin s wellF e proeed to the lultion of the pörster rdius of the olloidl quntum dots in homogeneous medium within the eletri dipole pproximtionF por thisD we ssume the host medium to hve n index n = 1F he full width t hlf mximum @prwA of the frequeny spred due to inhomogeneous rodening is ITH me @∼ 180 nmA for the hs studied y veEn et al. t room temperture IPPF pigure TFS@A shows the two sorption spetr together of two quntum dots for sitution where the quntum dots hve resonne wvelengths λ 1 = 1.11 µm nd λ 2 = 1.29 µmF he result of the lultion of the pörster rdius r 0 is presented in pigF TFS@AD where ∆λ = λ 2 -λ 1 nd indites the distne etween the two resonnt wvelengths whih re symmetrilly positioned round λ = 1.2 µmF he lk @redA urve orresponds to the sitution tht h I @PA is the eptor nd h P @IA the donorF he lk dshed lines orrespond to the pörster rdius for two hs tht re spetrlly seprted y the prw of the inhomogeneous rodeningD whih is IVH nmF st is seen tht the typil pörster rdius is out V!IH nmF ine the interEprtile distne dependene of pi is inversely proportionl to r 6 D nonrditive energy trnsfer domintes rditive dey s soon s the interEprtile distne is smller thn the pörster rdiusF work et al. hve reently shown tht the tul pörster rdius of densely pked gdeGgdn oreEshell olloidl quntum dots is lrger thn the pörster rdius tht is predited under the eletri dipole pproximtion IPQF prom experiments they derived tht the gdeGgdn quntum dots under studyD hving resonne wvelength round SUH nmD show pörster rdius of V!W nm insted of the predited S!T nmF sn efF IPQ it is reported tht one of the possile resons for the lrger pörster rdius is the presene of dipole!qudrupole intertions tht re nonnegligile when the surfeEtoEsurfe distne is smll ompred to the enterEtoEenter distne etween hsF imilrlyD fose et al. hve infered pörster rdius of IP!IQ nm t room temperture for olloidl quntum

❈❍❆ ❚ ❊|✻ ✳ ❚❍ ❊❖| (❖ ❋❊ ▲ ❊❈❚|❖ ▲❯▼ ■◆ ❊ ❙❈ ❊◆❈ ❊❇(◗❯❆◆❚❯▼ ✲❉❖❚ ❙ ❇❆ ❙ ❊❉ ▼ ❊❚ ❆ ❙❯| ❋ ❆ ❈ ❊▲ ■•❍❚ ✲ ❊▼ ■❚❚ ■◆•❉ ❊❱ ■❈ ❊ ❙ ✾ ✽ 1 1 . 1 1 . 2 1 . 3 1 . 4 0 1 2 3 4 5 x 1 0 -1 9 λ ( µ m ) σ A (m 2 ) 0 1 0 0 2 0 0 3 0 0 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 ∆ λ ( nm ) r 0 ( nm ) ( b ) ( a ) σ A , 1 σ A , 2
WW TFRF ivigyvwsxigixgi wyhiv py e vih 6.4 Electroluminescence model for a LED e hve suggested tht the ensemle of hs n e modeled s ulk semiondutorF he stndrd model of eletroluminesene y semiondutor t thermodynamic equilibrium predits tht the emitted power of vih sles exponentilly with the pplied voltgeF his model is sed on modi(ed theory of thermal radiationF e will desrie the model used to lulte the emitted power of vih nd emphsize the hypotheses tht re done nd re not vlid for hEwvihsF st is in the next setion tht this disussion will llow us to generlize the vih model to hEwvihsF he emitted power of vih is lulted y model tht is derived from uirhho'9s lwF his lwD whih ws pulished in IVTH y uirhho'D sttes tht ny mteril t thermodynamic equilibriumD sors s muh of energy s it emitsX

I e (ω, T ) = ε(ω, T )I BB (ω, T ), @TFTA I a (ω, T ) = α(ω, T )I BB (ω, T ), @TFUA
where I e aI a D nd where oth I e nd I a hve units •m -2 •rz -1 •sr -1 F prom this it follows tht the emissivity ε nd sorptivity α re the smeX ε = αF hey re oth unitless nd nnot exeed IF he spetrl irrdine of lk ody follows from lnk9s lw 2 X

I BB (ω, T ) = ω 2 4π 2 c 2 ω exp[ ω/(k B T ) -1]
. @TFVA uirhho'9s lw pplies to therml rditionF his lw ws generlized to eletroluminesE ene from semiondutor mterilsD eletroluminesene eing n exmple of nontherml rditionD y hokley et al. in IWTI IPTF sn order to do soD lnk9s lw needs to e modi(ed under the introdution of n eletrohemil potentil in lnk9s funtion IPUF por vihD this generliztion of lnk9s lw reds

I BB (ω, T ) = ω 2 4π 2 c 2 ω exp[ ω/(k B T )] -1 → ω 2 4π 2 c 2 ω exp[( ω -eφ)/(k B T )] -1 , @TFWA
where e = |q|D where q is the hrge of n eletronD nd φ is the voltgeF hen we ssume the voltge drop in the enter of the vih to e smller thn the gp energy of the vihD nd exp[( ω -eφ)/(k B T )] 1D we n simplify the generlized uirhho'9s lwX

I e (ω, T, φ) = α(ω)I BB (ω, T ) exp[eφ/(k B T )].
@TFIHA 2 For historical reasons, we note that Kirchho 's law can be written in a similar notation as was in- troduced by Kirchho [125]: I e (ω, T )/α(ω) = I BB (ω, T ). grei TF riy yp ivigyvwsxigixgi f exwEhy feih wiepegi vsqrEiwssxq hisgi IHH e reover the exponentil dependene of the emitted power from vih on the pplied voltgeF st is oserved tht the eletroluminesene spetrum is the sme s tht of therE ml soure omposed of the sme mterilsF roweverD the intensity inreses exponentilly with the pplied voltgeF he physil origin lies in the ft tht y pplying voltge the numer of eletrons in the ondution nd inresesD therey lso inresing the numer of holes in the vlene ndF he onsequene of this is tht the eletrohemil potentil of eletrons in the ondution nd nd holes in the vlene nd re not the sme nymoreF he system is out of equilirium nd the inresed numer of eletronEhole pirs mkes the system rditing moreF ine t room temperture k B T = 26 meD hnge of voltge ross the p!n juntion will drstilly hnge the emitted powerF e ntiipte tht this e'et explins the oservtion of eletroluminesene from the smple without dieletri sper etween the hs nd goldD see pigF TFQ@DAF prom n eletril perspetiveD the hsD sper nd gold form series iruitF en dditionl sper will redue the voltge drop ross the hs (lmD euse of the voltge divider property of series iruitF 6.5 Electroluminescence model in the presence of nanoantennas por p!n juntion vihD oth the sorptivity nd eletrohemil potentils re uniform in the lightEemitting regionF hereforeD the generlized uirhho'9s lw s given in iqF TFIH models eletroluminesene orretlyF rowever in hEwvihsD the olloidl quntum dots re put in oth nonEuniform eletromgneti environment nd nonEuniform eletril environmentD due to the presene of plsmoni nnontennsF he nonEuniform eletroE mgneti environment of the hs is expeted to mke light sorption not uniform inside the h (lm nd the nonEuniform eletril environment is expeted to give rise to nonEuniform eletrohemil potentilF Non-uniform light absorption o illustrte the nonuniform light sorptionD we hve lulted the eletri (eld distriution lose to the metlli nnoring for n xEpolrized @horizontlly polrizedA inident plne wveF he inEhouse numeril ode hs een develE oped y tenEul rugonin t the snstitut d9yptiqueF he method ws (rst introdued y fonod et al. IPVF he eletri (eld is lulted y mens of modl method for whih the eletri (eld is written in the pourierEfessel sisD respeting in this wy the ylindril symmetry of the prolemF he ngulr dependene of the eletri (eld is written in the pourier sis nd the rdil dependene is expnded in fessel funtionsF pigure TFT@A shows the rel prt of the z omponent of the eletri (eldD whih origintes from di'rtion of light y the nnostrutureF he ontours of the di'erent mterils tht re displyed in pigF TFT@A re drwn y dshed linesF xote tht the sles of pigF TFT@A nd pigF TFT@A re the smeF st is oserved in pigF TFT@A tht the z omponent of the eletri (eld is the strongest ove nd elow the metlli nnoringF hereforeD the quntum dots tht re positioned ove the gold nnorings hve the highest sorption ross setion for plne wve tht is inident from oveF untum dots tht re positioned in the inside of the nnoring or outside the nnoring hve smller sorption ross setion for the sme inident plne wveF prom this lultionD we onlude tht the sorptivity of hEwvih system is not uniform nd the stndrd theory of eletroluminesene from onventionl vihs is therefore not vlid for this kind of systemsF Non-uniform electrochemical potential feuse of the nonEuniform eletril enviE ronmentD the eletrohemil potentil is expeted to hnge lollyF his di'erene lies in the ft tht the voltge drop φ is not uniform in the hs lyerF e voltge is pplied ross sy nd luminiumD s indited in pigF TFT@AF e onsider the multilyer s series iruitF es gold is good eletril ondutorD the lol urrent density is higher in the region vertil to the gold nnoringF feuse the urrent density lose to gold is higher thn outside the gold regionD the voltge drop t the level of the quntum dots ove the gold nnorings is higher thn for the one tht re pled outside the nnoring regionF ine oth the sorptivity @optisA nd the voltge drop @eletronisA is the highest for the hs right ove the gold nnoringsD nd ording to iqF TFIH oth re relevnt for the eletroluminesene signlD it is expeted tht most of the eletroluminesene signl omes from the region ove goldF his is on(rmed y pigF TFQ@A whih shows tht eletroluminesene signl omes only from the regions with gold nnoringsF abs (u, ω) = ˆVobj α (l) (u, r , ω)d 3 r , @TFIPA so tht α(u, r , ω) n e onsidered n sorption ross setion densityY not to e onfused with the sorptivity s introdued in etion TFQD V obj is the volume of the ojet tht rditesD nd l indites the polriztionF ine there re two polriztions possileD the lkEody spetrl irrdine hs een divided y twoF st is importnt to highlight tht the lol sorption is represented y the sorption ross setion density αF iletrolumiE nesene origintes predominntly from the exitons inside the lyer of hsD whih limits the sptil extent of the volume integrl V obj to the h (lm onlyF sn hEwvihs t room tempertureD we n ssume tht ll prts of the smple re t the sme tempertureF ine the h (lm is very thinD we n ssume the voltge to e onstnt inside the (lmD so tht fter the volume integrl we end up with

P (l) e = exp eφ k B T ˆ∞ 0 dω ˆ4π σ (l) abs (u, ω) I BB [ω, T ] 2 dΩ, @TFIQA
for whih it is importnt to remind tht φ is the voltge drop t the level of the hsD whih is not equl to the pplied voltgeF sn order to lulte the sorption ross setion of the quntum dots tht re inside the region indited y the stripes in pigF TFT@A @s explined eforeD eletroluminesene origintes from this regionAD we lulte the eletri (eld inside the struture for n inident plne wve long the u diretionF he reson the eletri (eld is needed is euse the sored power n e lulted from it ording to

P abs (ω) = 1 2 ˆsm[ε(ω)]|E(r)| 2 d 3 r. @TFIRA
prom the sored powerD we n derive the sorption ross setion y mens of the following de(nition of the sorption ross setionX

P abs (ω) = σ abs (u, ω) cε 0 |E inc | 2 2 . @TFISA
ith this generliztion of uirhho'9s lwD we hve model tht llows to lulte the eletroluminesene signl of systems s the one presented y veEn et al. in efF IIF 6.5.2 Eective refractive index calculation prom now onD we will present numeril results of lultions tht re sed on the system tht is shemtilly presented in pigF TFTF o do soD we need to explin how we lulted the sorption ross setionF he lyer of hs is optilly modeled y n e'etive refrE tive indexD ording to the results presented in etion TFQFQF sn this setion we disuss how to lulte the e'etive refrtive indexF IHQ TFSF ivigyvwsxigixgi wyhiv sx ri iixgi yp xexyexixxe

Lorentz-Lorenz formula yne model to lulte the e'etive refrtive index is the vorentzEvorenz formulF es we hve seen in ghpter PD we need to verify the vlidity of this formul for our systemD sine we will lulte resonnt light sorption y dense ensemle of resonnt stterersF e know from iqF PFPS tht in order to hve rek down of the vorentzEvorenz formul for inhomogeneously rodened systemD the system needs to stisfy the ondition

C = 9π 8 ρ k 3 Γ r ∆ω 2 > 1. @TFITA
sn the prtiulr se of the system of veEnD one n show tht C = 9π 8 ρ k 3

Γr ∆ω 2 ∼ 10 -10 D y using the prmetersX ∆ω = 160 meD τ r = 2π/Γ r = 0.5 µsD ρ = 10 18 hsGm 3 D nd λ = 2π/k = 1.2 µmF es C

1D the vorentzEvorenz formul is expeted to e vlidY reurrent sttering is not n importnt proess due to the very strong inhomogeneous rodening present in this systemF Lorentz-Lorenz formula for a core-shell quantum dot ih quntum dot of veEn is modeled s oreEshell prtile with inner rdius b nd outer rdius aF he ore is mde of with dieletri onstnt ε c nd the shell is lignd with dieletri onstnt ε s F he quntum dots re emedded in host medium with dieletri onstnt ε h F e deompose the totl polrizility α(ω) of quntum dot in oreEshell prt nd n exiton prtD ording to α(ω) = α core-shell (ω) + α exciton (ω)D whih is sed on the ft tht the dipole moment nd the inident eletri (eld re linerly reltedX p = εα(ω)EF fy tking into ount the inhomogeneous rodening for the exitoni prtD the vorentzEvorenz formul eomesX

ε LL (ω) -ε h ε LL (ω) + 2ε h = 1 3 ˆρ(ω )α exciton (ω; ω )dω + ρ 0 α core-shell (ω) 3 , @TFIUA
where ρ 0 is the homogeneous h densityF he inhomogeneous rodening ρ(ω ) is ssumed to follow qussin distriutionD nd ´∞ 0 ρ(ω )dω = ρ 0 F he polrizility of the exiton is given y

α exciton (ω; ω ) = 6πc 3 Γ r ω 2 (ω 2 -ω 2 -iω(Γ r + Γ nr + Γ * )
, @TFIVA nd the polrizility of the oreEshell is given y IQP

α core-shell (ω) = 4πa 3 a 3 (ε c + 2ε s )(ε s -ε h ) + b 3 (ε c -ε s )(2ε s + ε h ) 2b 3 (ε c -ε s )(ε s -ε h ) + a 3 (ε c + 2ε s )(ε s + 2ε h ) . @TFIWA
por the prmeters of the qussin distriution of resonne frequeniesD we hve tken the entrl wvelength λ central = 1.2 µmD nd prw inhomogeneous rodening of ITH me @∼ 180 nmAF he other prmeters re tken s followsX Γ r = 2π 0.5 µs D Γ * = 2π 300 fs @whih orresponds to IT nmAD a = 4.3 nmD nd b = 2.3 nmF por h density ρ = 1 × 10 20 hsGm 3 D we hve found the dieletri onstnt tht is shown in pigF TFU whih shows in semilog sle oth the rel @red dshed urveA nd imginry @lue solid urveA prt of the dieletri onstnt of the h (lmF e ssume to e lossless itselfD so tht the sorption tht is lulted uniquely origintes from the sorption y the exitonsF e only wnt to know the sorption y the exitonD sine we re only interested in emission from the exitonsF st is due to reiproity tht oth quntities re reltedF he vorentzE vorenz formul is herefter used in order to lulte the sorption ross setion of the exitons ove the gold nnoringF

( µ m ) ❋ ✐ ❣ ✉ ❡✻ ✳ ✼ ✿ ❚ ❤ ❡❞ ✐ ❡ • ❡ ❝ ✐ ❝❝ ♦ ♥❛ ♥ ♦ ❢❛• ❛ ② ❡ ✇ ✐❤ρ=1×1 0 2 0 ❝◗❉✴ ❝♠ 3 ✳ ✭ ❛ ✮ ❚ ❤ ❡ ❡ ❛ •♣ ❛ ♦ ❢ ❤ ❡❞ ✐ ❡ • ❡ ❝ ✐ ❝❝ ♦ ♥❛ ♥✳✭ ❜ ✮❚ ❤ ❡ ✐♠ ❛ ❣ ✐ ♥ ❛②♣ ❛ ♦ ❢ ❤ ❡❞ ✐ ❡ • ❡ ❝ ✐ ❝❝ ♦ ♥❛ ♥✳ ❚ ❤ ❡ ✐♠ ❛ ❣ ✐ ♥ ❛②♣ ❛ ❝ ♦ ♥❛ ✐ ♥ ❤ ❡❡ ① ❝ ✐♦ ♥ ✐ ❝❝ ♦ ♥ ✐ ❜ ✉ ✐ ♦ ♥♦ ♥ • ② ✳ ✻ ✳ ✻ ❆ ♣ ♣ • ✐ ❝ ❛✐ ♦ ♥♦ ❢❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ♠ ♦ ❞ ❡ • ♦❡ ① ♣ ❡✲ ✐♠ ❡ ♥ ■ ♥❤ ✐ ❡ ❝✐ ♦ ♥✇ ❡♣ ❡❡ ♥ ❡✉ • ❢ ♦♠♥ ✉♠ ❡✐ ❝ ❛ •❝ ❛ • ❝ ✉ • ❛ ✐ ♦ ♥♦ ♥ ✇ ♦❞ ✐ ✛ ❡❡ ♥ ② ❡♠ ❤ ❛ ✇ ❡❡ ✉ ❞ ✐ ❡ ❞ ✐ ♥ ❤ ❡❣♦ ✉ ♣♦ ❢❉ ❡ ❣ ✐♦ ♥ ✳ ✻ ✳ ✻ ✳ ✶ ❊ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥ ❜ ②♥ ❛ ♥ ♦✐ ♥ ❣ ❲ ❡♣ ❡❡ ♥ ❤ ❡♥ ✉♠ ❡✐ ❝ ❛ •❝ ❛ • ❝ ✉ • ❛ ✐ ♦ ♥♦ ♥ ❤ ❡❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥ ♦ ❢ ❤ ❡❛ ❜♦♣✐ ♦ ♥❝ ♦ ❡ ❝✐ ♦ ♥ ❢ ♦♠❛ ♥❡ ♥❡♠ ❜ • ❡♦ ❢ ❜ ❙ ❈◗❉ ✐ ♥❤ ❡♥ ❡ ❛ ✈ ✐ ❝ ✐ ♥ ✐②♦ ❢❛❣ ♦ • ❞♥ ❛ ♥ ♦✐ ♥ ❣❛♣❡❡ ♥❡ ❞✐ ♥ ❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❜ ✮❛ ♥ ❞❋ ✐ ❣ ✳ ✻ ✳ ✸ ✭ ❝ ✮ ✳ ❲ ❡❝ ♦ ♥✐ ❞ ❡❜ ♦❤ ② ❡♠ ✇ ✐❤❛ ♥ ❞✇ ✐❤ ♦ ✉ ❛✺ ✲ ♥♠ ✲❤ ✐ ❝ ❦ • ❛ ② ❡ ♦ ❢ ❚ ✐❖ ① ✳ ❋ ✐ ❣ ✉ ❡✻ ✳ ✽ ❤ ♦ ✇ ❤ ❡❛ ❜♦♣✐ ♦ ♥❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥❛ ❛ ❢ ✉ ♥ ❝✐ ♦ ♥♦ ❢❤ ❡ ✇ ❛ ✈ ❡ • ❡ ♥ ❣❤ ✱ ✇ ❤ ❡❡ ❤ ❡ ❡ ❞❞ ♦ ❝ ♦ ❡♣ ♦ ♥ ❞ ♦ ❤ ❡ ✐ ✉ ❛✐ ♦ ♥✇ ❤ ❡ ❡ ❤ ❡❝◗❉ ♦ ✉ ❝ ❤ ❡ ❞ ❤ ❡❣ ♦ • ❞♥ ❛ ♥ ♦✐ ♥ ❣ ❛ ♥ ❞ ❤ ❡❜ • ✉ ❡ ❛ ♦ ❤ ❡ ✐ ✉ ❛✐ ♦ ♥✇ ✐ ❤❛✺ ✲ ♥♠ ✲❤ ✐ ❝ ❦ ♣ ❛ ❝ ❡✳■✐♦ ❜❡✈ ❡ ❞ ❢♦♠ ❤ ❡ ❡ ❞ ❞ ♦ ❤ ❛ ❛ ❜♦♣✐ ♦ ♥❣ ❡ ❡ ♥ ❤ ❛ ♥ ❝ ❡ ❞✉ ♣ ♦❛ ❢ ❛ ❝♦ ✺ ✵✇ ❤ ❡ ♥ ❤ ❡❝◗❉ ♦ ✉ ❝ ❤❣ ♦ • ❞ ✳❊ ♥ ❤ ❛ ♥ ❝ ❡ ❞ ❛ ❜♦♣✐ ♦ ♥ ✐❛ ✐ ❜ ✉ ❡ ❞♦❤ ❡ • ♦ ❝ ❛ • ❡ • ❡ ❝ ✐ ❝✜ ❡ • ❞ ❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥ ❤ ❛ ✇ ❛ ❤ ♦ ✇ ♥ ✐ ♥❋ ✐ ❣ ✳✻ ✳ ✻ ✭ ❜ ✮ ✳ ❚ ❤ ❡♣❡❡ ♥ ❝ ❡♦ ❢❛✺ ✲ ♥♠ ✲❤ ✐ ❝ ❦ ♣ ❛ ❝ ❡ ❡ ❞ ✉ ❝ ❡ ❤ ❡❝ ❛ • ❝ ✉ • ❛❡ ❞❛ ❜♦♣✐ ♦ ♥❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥❜ ② ❛ ❜ ♦ ✉ 2 5% ✳ ❋ ✐ ❣ ✉ ❡✻ ✳ ✸ ✭ ❜ ✮❛ ♥ ❞❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❝ ✮ ❤ ♦ ✇ ✐♠ ❛ ❣ ❡ ♦ ❢ ❤ ❡❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ❣ ♥ ❛ • ❢ ♦ ❤ ❡ ② ❡♠✇ ✐❤❛ ♥ ❞✇ ✐❤ ♦ ✉ ❛✺ ✲ ♥♠ ✲❤ ✐ ❝ ❦ ♣ ❛ ❝ ❡✳ ❋ ♦♠❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❜ ✮❛ ♥ ❞❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❝ ✮ ✐ ❡ ❡♠ ❡ ❛♦ ♥ ❛ ❜ • ❡ ♦ ❛ ② ❤ ❛ ❤ ❡ ❡ ❞ ✉ ❝✐ ♦ ♥♦ ❢❡ • ❡ ❝♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ❣ ♥ ❛ • ✐♠ ♦ ❡ ❤ ❛ ♥2 5% ✳ ❋♦♠ ❋ ✐ ❣ ✳✻ ✳ ✽✇ ❡❤ ❛ ✈ ❡♦ ❜❡✈ ❡ ❞ ❤ ❛ ❤ ❡♣❡❡ ♥ ❝ ❡♦ ❢ ❚ ✐❖ x ♣ ❛ ❝ ❡ ❤ ❛ ❛ ♥❡ ✛ ❡ ❝ ♦ ♥ ❤ ❡❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ❣ ♥ ❛ • ✱❜ ✉♦ ♥ • ②✇ ❡ ❛ ❦ • ② ✳ ❆ ♦ ♣ ♣ ♦❡ ❞ ♦ ❤ ✐ ♥ ✉♠ ❡✐ ❝ ❛ •♦ ❜ ❡✈ ❛✐ ♦ ♥ ✱ ❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡❡ ❡♠ ♦❜ ❡ ♦ ♥ ❣ • ②❛♠ ♣ • ✐ ✜ ❡ ❞ ✐ ♥❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❜ ✮❛❝ ♦♠ ♣ ❛❡ ❞♦❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❝ ✮ ✱ ❛ •❤ ♦ ✉ ❣ ❤♥ ♦ ✉ ❛ ♥✐ ❛✐ ✈ ❡ ❝ ♦♠ ♣ ❛ ✐ ♦ ♥ ❝ ❛ ♥❜ ❡❞ ♦ ♥ ❡ ✳ ❲ ❡❜ ❡ • ✐ ❡ ✈ ❡ ❤ ❡❡ ❢ ♦❡❤ ❛ ❤ ❡♦ ❜❡✈ ❛✐ ♦ ♥ ♦ ❢❛❜ ✐ ❣ ❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥♦ ❢ ❤ ❡ ❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡✐ ❣ ♥ ❛ • ✐ ♥❤ ❡ ❡ ① ♣ ❡✐♠ ❡ ♥✇ ✐❤ ♦ ✉ ❤ ❡ ♣ ❛ ❝ ❡ ✐♠ ❛ ✐ ♥ • ②❞ ✉ ❡ ♦❛❤ ✐ ❣ ❤ ❡ ❡ • ❡ ❝ ✐ ❝ ❛ •✈ ♦ • ❛ ❣ ❡❞♦ ♣❛ ❝♦ ❤ ❡❝◗❉✜ •♠ ✐ ♥ ❤ ❡❛ ❜❡ ♥ ❝ ❡♦ ❢ ❤ ❡ ♣ ❛ ❝ ❡✳ ✶ ✵ ✺ ✻ ✳ ✻ ✳ ❆ ▲ ■❈❆ ❚ ■❖◆❖ ❋❊ ▲ ❊❈❚|❖ ▲❯▼ ■◆ ❊ ❙❈ ❊◆❈ ❊ ▼❖❉ ❊ ▲❚❖❊❳ ❊| ■▼ ❊◆❚ ❙ 1 1 . 1 1 . 2 1 . 3 1 . 4 0 1 0 2 0 3 0 4 0 5 0 6 0 λ ( µ m ) σ a b s ,w i t h g o l d / σ a b s ,w i t h o u t g o l d ❋ ✐ ❣ ✉ ❡✻ ✳ ✽ ✿❋ ♦ ❛♥ ♦♠ ❛ • ✐ ♥ ❝ ✐ ❞ ❡ ♥ ❝ ❡♣ • ❛ ♥ ❡✇ ❛ ✈ ❡ ✱ ❤ ❡❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥❞ ✉ ❡ ♦ ❤ ❡♣❡❡ ♥ ❝ ❡♦ ❢❣ ♦ • ❞♥ ❛ ♥ ♦✐ ♥ ❣❤ ❛ ❜ ❡ ❡ ♥❝ ❛ • ❝ ✉ • ❛❡ ❞ ❢ ♦ ❞ ✐ ✛ ❡❡ ♥✇ ❛ ✈ ❡ • ❡ ♥ ❣❤✳ ❚ ❤ ❡❡ ❞❞ ♦ ❝ ♦ ❡♣ ♦ ♥ ❞ ♦ ❤ ❡ ✐ ✉ ❛✐ ♦ ♥✇ ❤ ❡ ❡ ❤ ❡❝◗❉ ♦ ✉ ❝ ❤ ❤ ❡❣ ♦ • ❞♥ ❛ ♥ ♦✐ ♥ ❣ ✳ ❚ ❤ ❡❡ • ❡ ❝♦ • ✉♠ ✐ ♥ ❡✲ ❝ ❡ ♥ ❝ ❡❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥ ❝ ❛ ♥ ❡ ❛ ❝ ❤✈ ❛ • ✉ ❡ ✉ ♣ ♦✺ ✵ ✳ ❚ ❤ ❡❜ • ✉ ❡ ❛ ❝ ♦ ❡♣ ♦ ♥ ❞ ♦ ❤ ❡ ✐ ✉ ❛✐ ♦ ♥ ♣❡❡ ♥❡ ❞✐ ♥❋ ✐ ❣ ✳✻ ✳ ✸ ✭ ❝ ✮ ✱ ✇ ❤ ❡❡❛• ❛ ② ❡ ♦ ❢✺♥♠ ❚ ✐❖ ① ✐ ♣ ✉ ❜ ❡✇ ❡ ❡ ♥ ❤ ❡❝◗❉ ❛ ♥ ❞ ❤ ❡ ♥ ❛ ♥ ♦✐ ♥ ❣ ✳ ❚ ❤ ❡❡ • ❡ ❝♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡❡ ♥ ❤ ❛ ♥ ❝ ❡♠ ❡ ♥ ❞♦ ♣ • ✐ ❣ ❤ • ② ✳ ✻ ✳ ✻ ✳ ✷ ♦ • ❛✐ ③ ❡ ❞❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ■ ♥ ❤ ✐ ❡ ❝✐ ♦ ♥ ✇ ❡ ✇ ✐ • • ✉ ❞ ② ♦♠ ❡❡ ① ♣ ❡✐♠ ❡ ♥ ❤ ❛ ❤ ❛ ✈ ❡❜ ❡ ❡ ♥❝ ♦ ♥ ❞ ✉ ❝❡ ❞❜ ②▲ ❡ ✲❱ ❛ ♥❡ ❛ • ✳ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❤ ❡❣ ♦ • ❞♥ ❛ ♥ ♦✐ ♥ ❣✇ ❡❡ ❡ ♣ • ❛ ❝ ❡ ❞❜ ②❣ ♦ • ❞♥ ❛ ♥ ♦♦ ❞ ❤ ❛ ❤ ❛ ✈ ❡❞ ✐♠ ❡ ♥✐ ♦ ♥ 1 5 0×7 0×7 0♥♠ 3 ✳ ◆ ❛ ♥ ♦ ♦ ❞ ❜❡ ❛ ❦ ❤ ❡❝ ② • ✐ ♥ ❞✐ ❝ ❛ •②♠♠ ❡ ②❛ ♥ ❞ ❤ ❡❡ ❢ ♦❡❝❡ ❛❡❛♣ ♦ ✲ • ❛ ✐ ③ ❛ ✐ ♦ ♥❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❝ ❡♦ ❢❡ • ❡ ❝♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✱❛ ✇ ❛ ♣❡❡ ♥❡ ❞ ✐ ♥ | ❡ ❢ ✳ ❬ ✶ ✶ ( ✳ ❚ ❤ ❡ ✐ ♥ ❡ ♦ ❢ ❋ ✐ ❣ ✳✻ ✳ ✾ ✭ ❛ ✮❤ ♦ ✇ ❤ ❡♣ ❡✐ ♦ ❞ ✐ ❝❛❛ ②♦ ❢♥ ❛ ♥ ♦♦ ❞ ❛ ♥ ❞ ❤ ❡♠ ❛ ✐ ♥♣ ❛ ♥ ❡ • ❤ ♦ ✇ ❤ ❡ ❡ ① ♣ ❡✐♠ ❡ ♥ ❛ • ❡✉ • ✳ ❚ ❤ ❡❡ • ❡ ❝♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ♥❡ ♥✐ ②❤ ❛ ❜ ❡ ❡ ♥ ❤ ♦ ✇ ♥ ❢ ♦ ✇ ♦❞ ✐ ✛ ❡❡ ♥♣ ♦ • ❛✐ ③ ❛ ✐ ♦ ♥♦ ❢ • ✐ ❣ ❤ ✳ ❚ ❤ ❡❤ ✐ ❣ ❤ ❡ ❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ♥❡ ♥✐ ②❝ ♦♠ ❡ ❢ ♦♠ • ✐ ❣ ❤ ✇ ❤ ✐ ❝ ❤ ✐ ♣ ♦ • ❛✐ ③ ❡ ❞❛ • ♦ ♥ ❣ ❤ ❡ • ♦ ♥ ❣❛ ① ✐ ♦ ❢ ❤ ❡♥ ❛ ♥ ♦♦ ❞✳ ❚ ❤ ❡♣ • ❛ ♠ ♦ ♥ ✐ ❝ ❡♦ ♥ ❛ ♥ ❝ ❡❜ ❡ ✐ ♥ ❣ ♦ ♥ ❣ • ②❛ ♥ ✐♦ ♦ ♣ ✐ ❝ ✱ ❤ ❡ ❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ❣ ♥ ❛ •❢ ♦• ✐ ❣ ❤♣ ♦ • ❛✐ ③ ❡ ❞❛ • ♦ ♥ ❣❤ ❡ ❤ ♦ ❛ ① ✐ ✐♠ ✉ ❝ ❤ ♠ ❛ • • ❡ ✳ ❚ ❤ ❡ ❡ ① ♣ ❡✐♠ ❡ ♥ ❛ •❞ ❛❛❤ ❛ ♦♠ ❡ ❛ ♣ ✐ ❞✢ ✉ ❝✉ ❛✐ ♦ ♥❛♦ ✉ ♥ ❞ ❤ ❡ ♠ ❛ ① ✐♠ ✉♠❡ • ❡ ❝ ♦ • ✉♠ ✐ ♥ ❡❝ ❡ ♥ ❝ ❡ ✐ ♥ ❡ ♥✐ ② ✳❙ ✐ ♥ ❝ ❡✇ ❛❡ ✈ ❛ ♣ ♦ ❤ ❛ ❛ ♥❛ ❜♦♣✐ ♦ ♥❜ ❛ ♥ ❞❛ ♦ ✉ ♥ ❞λ=1. 3 8µ ♠ ✱ ✐ ✐• ✐ ❦ ❡ • ②❤ ❛ ❤ ❡♥ ♦ ✐❡ ✐ ♥ ❤ ❡ ♣ ❡ ❝ ✉♠ ✐ ❞ ✉ ❡ ♦ ❤ ❡♣❡❡ ♥ ❝ ❡♦ ❢ ✇ ❛❡ ♠ ♦ • ❡ ❝ ✉ • ❡ ✐ ♥❤ ❡❡ ♥ ✈ ✐♦ ♥♠ ❡ ♥ ✇ ❤ ❡❡ ❤ ❡❛♦♠ ❛❡♣ • ❛ ❝ ❡ ❞ ✳ ❙ ✐♠ ✐ • ❛ ❛ ❜ ❡ ❢ ♦❡ ✱ ✇ ❡❝ ❛ • ❝ ✉ • ❛❡ ❤ ❡❛ ❜♦♣✐ ♦ ♥❜ ②❤ ❡❡ ① ❝ ✐♦ ♥✱❜ ✉ ❤ ✐ ✐♠ ❡❢ ♦ ✇ ♦ ❞ ✐ ✛ ❡❡ ♥ ♣ ♦ • ❛✐ ③ ❛ ✐ ♦ ♥♦ ❢ ❤ ❡ ✐ ♥ ❝ ✐ ❞ ❡ ♥ ♣ • ❛ ♥ ❡ ✇ ❛ ✈ ❡ ✳ ❲ ❡♦ ♥ • ②❝ ♦ ♥✐ ❞ ❡❛ ✐ ♥ ❣ • ❡✉ ♥ ✐❝ ❡ • •♦ ❢ ❤ ❡♣ • ❛♠ ♦ ♥ ✐ ❝ ♠ ❡ ❛✉❢ ❛ ❝ ❡ ✳■ ♥♦ ❞ ❡ ♦❝ ❛ • ❝ ✉ • ❛❡ ❤ ❡❡ • ❡ ❝ ✐ ❝✜ ❡ • ❞ ✐ ♥ ✐ ❞ ❡❤ ❡ ✉ ❝✉❡ ✱❛ ♥ ❛ ♣ ❡✐ ♦ ❞ ✐ ❝ ✲ ❋ ♦ ✉ ✐ ❡♠ ♦ ❞ ❛ •♠ ❡ ❤ ♦ ❞ ✐ ✉❡ ❞ ✳ ❚ ❤ ❡❡❛❡❛• ❡ ❛ ✇ ♦❞ ✐ ✛ ❡❡ ♥✇ ❛ ② ♦ ✐♦ • ❛❡ ❡ ❛ ❝ ❤ ❝ ❡ • •♦ ♣✐ ❝ ❛ • • ② ✳ ❚ ❤ ❡ ✐ ♦ • ❛✐ ♦ ♥❝ ❛ ♥❡ ✐ ❤ ❡ ❜ ❡❞ ♦ ♥ ❡❜ ② ❢ ♦ ❡ ① ❛♠ ♣ • ❡♣ ✉ ✐ ♥ ❣♣ ❡ ❢ ❡ ❝ • ② ♠ ❛ ❝ ❤ ❡ ❞ • ❛ ② ❡ ✭▼ ▲ ✮❛ ❤ ❡❜ ♦❞ ❡ ♦ ❢ ❤ ❡❝ ❡ • •✱♦❜ ②✉✐ ♥ ❣❝ ♦♠ ♣ • ❡ ①❝ ♦ ♦ ❞ ✐ ♥ ❛❡ ❛ ♥❢ ♦ ♠ ❛ ✐ ♦ ♥ ❛ ❤ ❡❜ ♦❞ ❡ ♦ ❢❡ ❛ ❝ ❤❝ ❡ • • ❬ ✽ ✻ ( ✳■✐ ❤ ❡ • ❛ ❡ ❤ ❛ ✐✉❡ ❞ ✐ ♥ ❤ ✐ ✇ ♦❦ ✳ ❋ ✐ ❣ ✉❡ ✻ ✳ ✾ ✭ ❜ ✮❤ ♦ ✇ ❤ ❡ ❝ ❛ • ❝ ✉ • ❛❡ ❞❛ ❜♦♣✐ ♦ ♥ ❝ ♦ ❡ ❝✐ ♦ ♥ ❢ ♦ ✇ ♦ ✐ ♥ ❝ ✐ ❞ ❡ ♥ ♣ ♦ • ❛✐ ③ ❛ ✐ ♦ ♥ ✭ ♦❛ ♥ ❣ ❡ ♦ • ✐ ❞ ❝ ✉✈ ❡ ✿♣ ❛❛ • • ❡ • ♦ • ♦ ♥ ❣❛ ① ✐ ♦ ❢♥ ❛ ♥ ♦♦ ❞ ✱❛ ♥ ❞❜ • ❛ ❝ ❦❞ ❛❤ ❡ ❞ ❝ ✉✈ ❡ ✿♣ ❡♣ ❡ ♥ ❞ ✐ ❝ ✉ • ❛ ♦ • ♦ ♥ ❣❛ ① ✐ ♦ ❢♥ ❛ ♥ ♦♦ ❞ ✮ ✳❋♦♠ ❤ ❡♥ ✉♠ ❡✐ ❝ ❛ • ❝ ❛ • ❝ ✉ • ❛ ✐ ♦ ♥✇ ❡ ❝ ❛ ♥♦ ❜ ❡✈ ❡ ❤ ❛ ❤ ❡❛ ❜♦♣✐ ♦ ♥

Conclusion

fsed on wht we hve disussed in this hpterD we elieve tht denselyEpked hs in the ner viinity of metlli nnostrutures nnot e modeled s independent entitiesF hue to strong nerE(eld intertionsD ny exittion of the system would e delolized throughout the h (lmF snsted of studying eletroluminesene y onsidering the hs s eing independentD we elieve tht the ensemle forms homogeneous system tht onsists of eletroni ndsD like ulk semiondutor doesF hereforeD the h (lm is optilly desried y n e'etive refrtive indexF iletroluminesene from h (lm lose to plsmoni nnostruture does however not our in the sme wy s it does for onventionl p!n juntion vihF he origin of this lies in the ft tht the eletronEhole pir density is not uniform throughout the h (lm nd lso light emission does not our homogeneously due to the presene of the gold nnostrutureF e hve therefore proposed to model eletroluminesene from this kind of systems y mens of the generlized uirhho' tht hs reently een reported in efF IPWF e hve seen tht the model qulittively sueeds in modeling eletroluminesene from the hEwvihsD ut more quntittive study needs to e done in order to hek the vlidity of the proposed generlized uirhho' lwF

IHU

TFUF gyxgvsyx e hve lso studied olletive e'ets in eletrolumineseneF fy modeling the h lyer s n e'etive mediumD we did not oserve enhned sorption y the (lmD sine the sorption inreses linerly with the h densityF prom the generlized uirhho' lw we n onlude tht the eletroluminesene signl is therefore linerly dependent on the h densityF Chapter 7

Conclusion and outlook

sn this hFhF thesis we hve studied theoretilly light sttering nd light emission y dense ensemles of resonnt dipolesF he mjority of this mnusript onerns light sttering from dense louds of old tomsF sn the lst hpterD ghpter TD we studied eletroluminesene from dense (lm of olloidl quntum dots pled in the ner viinity of plsmoni metmterilF ghpter P e hve shown tht in the wekEexittion regimeD iFeFD the inident intensity is so low tht t most one tom is exitedD lssil nd quntum models of light sttering led to the sme set of oupledEdipole equtionsF his set of oupledEdipole equtions desries the resonnt dipoleEdipole intertions etween the dipolesF ine oth models re equivlent in the regime we re interested inD whih is the regime tht is relevnt for the experiments onduted in the group of froweysD we studied light sttering from dense loud of old toms lssillyF sn this hpter we hve tried to mke link etween the terminology used y vrious ommunitiesF pinllyD we studied the reltionship etween evnesent wves nd virtul photonsF st follows tht n evnesent wve is uilt up of mny virtul photonsF ghpter Q hen the numer of stterers in dilute gs inresesD the mount of sttered light inreses ordinglyF roweverD reently it hs experimentlly een oserved tht the inoherent power sttered orthogonlly to the inident em from denseD wvelengthEsize loud of old toms does not inrese linerly with the numer of toms SIF e ttriute the oservtion of this suppression to strong resonnt dipoleEdipole intertions etween the tomsF feuse of the strong resonnt dipoleEdipole intertionsD the loud lso prtilly di'rts light in the forwrd diretionF e understood tht ellegrino et al. were studying inoherent light sttering SIF sn order to study oherent stteringD we lso stuied light sttering in the forwrd diretionF heoretilly nd experimentlly otined spetr do not greeF he preise nture of this disgreement is not fully understood yetF he omined experimentl nd theoretil work of oherent stteringD where the experimentl results were ompred to results from the oupledEdipole equtionsD hs reently een pulished y tennewein et al. in the journl hysil eview vetters UF ghpter R e hve shown tht strong resonnt dipoleEdipole intertions in wvelengthEsize loud grei UF gyxgvsyx exh yvyyu IIH of old toms give rise to the onset of olletive modes tht re extended throughout the loudF hese modesD whih re superrdint nd therefore ouple very e0iently to lightD re sensitive to the shpe of the ojet nd to the tomi density distriutionY not to the preise position of the tomsF es these modes re similr to polritons tht were introdued y rop(eld to nlyze olletive exittions in ordered ondensed mtter systemsD we ll the superrdint modes tht re extended throughout the smple polritoni modesF e hve numerilly retrieved the e'etive refrtive indexD whih does not follow vorentzEvorenz theory due to the strong resonnt dipoleEdipole intertionsF he speE trum of the e'etive refrtive index llowed us to lulte the modes of the mrosopi wxwell equtions for n ojet with the sme shpe s the tomi loud nd n e'etive refrtive indexF st turns out tht the modes of the mrosopi wxwell equtions re identil to the polritoni modes found in the tomi loudF imilr to the importne of the modes of homogeneous ojet on light di'rtionD the few polritoni modes tht exist hve signi(nt impt on oherent light sttering y the tomi loudF he ft tht polritoni modes re importnt for light sttering illustrtes tht we nnot onsider the toms in denseD wvelengthEsize loud of old toms s eing independent entitiesX we need to onsider the tomi loud s eing n ensemle of olletive eigenmodesF his work on polritoni modes in dense loud of old toms hs een pulished in the journl hysil eview e TUF ghpter S he oherent response of the tomi loud is determined y its refrtive indexD whih nturlly tkes the polritoni modes from mirosopi theory into ountF roweverD there re mny other modes present in the tomi loud tht re not neessrily tken into ount y the refrtive indexF he question whether the optil response is uniquely determined y the index of refrtionD elongs to homogeniztion prolemF e hve shown tht the mount of inoherent light sttering is omprle to the mount of oherent light sttering for denseD wvelengthEsize loudF his oservtion hs shown tht the riterion for homogeniztion whih is sed on n interEtomi distne tht should e smll s ompred to the wvelength is invlidF his rings us to revision homogeniztion in resonnt tomi systemsF trikinglyD we hve numerilly shown tht resonnt rndom medium nnot reh the homogeniztion regimeF e show tht s is the se of light sttering from nonEresonnt systemsD the presene of nonrditive losses nd pure dephsing re e'etive mens to rrive t homogeneous mediumF purthermoreD we hve demonstrted y mens of olletive mode point of view tht suwvelengthEsize loud of resonnt toms lwys stters less light thn single tom due to olletive modes tht re spetrlly seprtedF he hike regime for the oservtion of superrdineX suwvelengthEsize loudD is shown to e the ondition for nonoservtion of singleEphoton superrdine e'etsF ghpter T sn this hpter we studied the emission of light y dense ensemle of quntum dotsF he dense (lm of olloidl quntum dots @hsA ws pled in ner viinity of plsmoni metsurfe nd emitted light under the pplition of voltge @eletrolumineseneAF he theoretil study is sed upon experimentl reserh onduted in the tem of eloyse hegiron t snstitut d9Életronique pondmentle IIF sn ghpter T we propose model to understnd the physis of eletroluminesene from dense (lm of olloidl quntum dots in the ner viinity of metlli metsurfesF e propose model for eletroluminesene III from this kind of system tht is sed upon generlized model for light emission from p!n juntion vihF he model of onventionl vih hs een dpted to systems where the emission of light is strongly position dependentF his is done y invoking the generlized uirhho' lw of therml rdition tht ws introdued in efF IPWF yutlook Superradiant pulse in strongly interacting atomic system sn IWTVD irnst nd tehle hve theoretilly shown tht light emission from n ensemle of exited toms is hrterized y the emission of superrdint pulse IQQF he pulse durtion ∆t ∝ 1/(N Γ)D where N is the numer of toms nd Γ is the singleEtom spontneous emission rte TD IQQD IQRF his onlusion follows from model where the toms do not intert with eh otherF roweverD we hve understood tht dense loud of old tomsD in whih resonnt dipoleEdipole re very strongD needs to e studied from olletive modes point of viewF st would e interesting to study oth theoretilly nd experimentlly how the emergene of olletive modes modi(es the superrdint pulse formtion for n initlly fully inverted dense loud of old tomsF Observing superradiance from ordered ensemble of Rydberg atoms xo superrdine is expeted from disordered and subwavelength-size tomi loudsD s we hve seen in ghpter S nd s it ws proposed with di'erent point of view y qross nd rrohe TF prom modl perspetiveD we drew the onlusion tht superrdine ould e expeted for ordered systemsD like regulrly pled toms on suwvelength irleF he geometril onstrint eing strong for optil trnsitionsD this onstrint would e strongly relxed y using yderg toms tht hve trnsitions in the mirowve regimeF ith the ility of positioning individul yderg toms y mens of optil mirotrpsD it is now experimentlly possile to study superrdine from regulrly pled toms on suwvelengthEsize irle IQSF Validation of proposed generalized electroluminescence model e propose model for eletroluminesene experiments tht re sed on dense ensemle of quntum dots in the ner viinity of plsmoni metsurfeF he study presented in this work stys very qulittiveF st would e interesting to quntittively ompre the eletroluminesene signl s funtion of the wvelengthD in order to vlidte the proposed modelF Appendices Appendix A Quantization of evanescent waves in absorbing media e present the quntiztion of evnesent wves in soring mediF he gol of this hpE ter is to formlly show the orrespondne etween evnesent wves nd virtul photonsF ivnesent wves exist in severl situtionsF e few of them reX @IA point soure with evnesent wves surrounding itD @PA n inident wve inident eyond the ritil ngle t n interfe etween two dieletrisD nd @QA )ututions in urrent densities in lossy mediF e hve tried to quntize the evnesent (eld for sitution @IAD ut we did not sueed s the eletri (eld hs singulrity t the position of the soureD y whih the eletri (eld ontins n in(nite mount of energyF he seond sitution is the sitution tht hs een onsidered y grnigli nd wndel QQF he inonveniene of this sitution is tht one nnot desrie the evnesent wve independently of the inident nd re)eted (eldsF rereD we onsider the quntiztion of evnesent wves in lossy dieletriF untiztion of eletromgneti modes in free spe n e found in mny textooksD like efs IQD IQTF sn free speD one typilly quntizes plne wvesF omething tht is less known is the quntiztion of evnesent wvesF vet us disuss wht hs een done on this topi in the pstF sn IWUID grnigli nd wndel pulished the work untiztion of evnesent eleE tromgneti wves QQF sn their workD they onsidered single interfe etween two dieletri mterilsF hen n inident (eld is inident with n ngle eyond the ritil ngleD there is re)etion of light nd n evnesent til in the other dieletriF he triplets of inidentD re)eted nd trnsmitted wves form orthogonl modes nd it is these modes tht hve suessfully een quntizedF sn IWUQD wreque eli hs pulished the pper lled ivnesent wves in quntum eletrodynmis with unquntized soures QRF fy using lssil souresD he hs shown tht evnesent wves re relted to virtul photonsF roweverD the evnesent wves were pure cEnumers nd therefore not written in quntum optis formlismF o the est of our knowledgeD no theory exists so fr tht quntizes evnesent (elds onlyF rereD we propose sheme to quntize evnesent wvesF his quntiztion will llow us to show the reltionship etween evnesent wves nd virtul photons in more forml setting thn ws done in ghpter I of this thesisF eixhs eF exsesyx yp iexigix ei sx efyfsxq wihse IIT

A.1 Quantization of the electromagnetic eld in dispersive and absorbing media ruttner nd frnett were the (rst to quntize rdition in dispersive nd sorptive liner ulk dieletris IQUF qruner nd elsh generlized the quntiztion sheme IQVF yne element of this generliztion ws tht the system did not need to e ulk mediumF qruner nd elsh relized tht the introdution of noise urrent opertor in the wxwell equtions ws neessry to ount for the dissiptionEssisted quntum noise in lossy systemF hen soure term is dded in the wxwell equtionsD the qreen9s tensor nturlly rises when the eletri (eld is lultedF st is the introdution of the qreen9s tensor tht led to the generliztion of the quntiztion sheme to nonEuniform mediF he eletri (eld ws shown to eX

E + (r) = i ˆ∞ 0 dω πε 0 ω 2 c 2 ˆd3 r ε (r , ω) Ḡ(r, r ; ω) f(r , ω), @eFIA
where ε is the imginry prt of the dieletri onstntD nd f(r, ω) is lled in the words of qruner nd elsh the opertor si (eld nd is liner superposition of opertors for the eletri (eld nd mtter (eldF his opertor is suh tht the rmiltonin of the prolem is

Ĥ = ˆd3 r ˆ∞ 0 dω ω f † (r, ω) • f(r, ω). @eFPA
ith the expression of the eletri (eld s given in iqF eFID we hve mens for quntizing evnesent wves tht exist due to the presene of quntum noise urrentsF A.2 Evanescent wave operator sn this setion we de(ne the evnesent wve opertorF e onsider homogeneousD E soring mediumD so tht ε (r, ω) = ε (ω) nd Ḡ(r, r ; ω) = Ḡ(rr ; ω)F e reognize tht the sptil integrl in iqF eFI is onvolution integrlF hnks to the onvolution theorem we n rewrite the eletri (eld in terms of n inverse pourier trnsform over the wvevetor kX

E + (r) = i ˆ∞ 0 dω πε 0 ω 2 c 2 ε (ω) ˆd3 k Ḡ(k; ω) f(k, ω)e ik•r @eFQA with Ḡ(k; ω) = 1 -kk k 2 k 2 -ε ω 2 c 2
. @eFRA es next step we write the noise opertor f in its two polriztion sttes

f(k, ω) = 2 λ=1 e λ (k) Ĉλ (k, ω), @eFSA
where Ĉλ (k, ω) is liner omintion of the osoni destrution nd retion opertors of the mtter (elds nd the photon destrution nd retion opertors nd for whih the following ommuttion reltions hold IQWX

[ Ĉλ (k, ω), Ĉ † λ (k , ω ) = δ λ,λ δ(k -k )δ(ω -ω ), @eFTA IIU eFQF gywwesyx ivesyx py iexigix ei yiey nd [ Ĉλ (k, ω), Ĉλ (k , ω ) = 0.
@eFUA fy sustituting oth the expression of the qreen9s tensor nd the noise opertorD the eletri (eld turns into the following form

E + (r) = i ˆ∞ 0 dω πε 0 ω 2 c 2 ε (ω) 2 λ=1 ˆd3 k 1 -kk k 2 k 2 -ε ω 2 c 2 e ik•r e λ (k) Ĉλ (k, ω).
@eFVA he key step of this setion follows nowF e de(ne the quntum opertor for evnesent wvesX

â(k , z; ω) = 2 λ=1 âλ (k , z; ω)e λ (k), @eFWA where âλ (k , z; ω) is mtrixX âλ (k , z; ω) = √ ε ω c ˆ∞ -∞ dk z 2π 1 -kk k 2 k 2 z -ε ω 2 c 2 -k 2 e ikzz
Ĉλ (k, ω). @eFIHA st is t this moment tht we see the forml reltion etween evnesent wves nd virtul photonsF vet us onsider for the moment system for whih ε is relF wodes with k < ( √ ε)ω/c re propgtive modesF e oserve tht modes for whih k 2 z = ε ω 2 c 2 -k 2 re domintingF his is s opposed for modes with k > √ εω/cD whih re evnesent wvesF por these modesD there is no resonne in k z nd therefore ll modes re importntX oth modes for whih k 2

x + k 2 y + k 2 z = εω/c nd k 2 x + k 2 y + k 2 z = εω/cF he ltter modes re modes tht n e exited y virtul photonsF prom this oservtionD nd from iqF eFIHD it follows tht n evnesent wve is uilt up of mny modes tht re desried quntum optilly y the opertor Ĉλ (k, ω)D whih is n opertor tht nnihiltes virtul photonF nder the introdution of this opertorD the eletri (eld opertor redues to

E + (r) = i ˆ∞ 0 dω πε 0 ω c ˆd2 k (2π) 2 â(k , z; ω )e ik •r .
@eFIIA e proeed with the lultion of the ommuttion reltion [ âλ (k , z; ω), â † λ (k , z ; ω )]D sine this quntity is losely relted to the orreltion funtion of the eletri (eld opertorsD nd therey losely relted to the power spetrl density funtion IQWF A.3 Commutation relation for evanescent wave operators he ommuttion reltion for the evnesent wve opertors is derived mthemtillyF es (rst stepD we (ll in the expression of the quntum opertors nd use the ommuttion reltion for the Ĉ opertorX

[ âλ (k , z; ω), â † λ (k , z ; ω )] = ε ω 2 c 2 ˆ∞ -∞ dk z 2π ˆ∞ -∞ dk z 2π 1 -kk k 2 k 2 -ε ω 2 c 2 × 1 -k k k 2 k 2 -ε * ω 2 c 2 e i(kzz-k z z ) ×δ λ,λ δ(k -k )δ(ω -ω ).
@eFIPA eixhs eF exsesyx yp iexigix ei sx efyfsxq wihse IIV he integrl over k z is strightforwrd to evluteD euse of the delt funtion δ(k -k )F

efter the integrtion we otinX

[ âλ (k , z; ω), â † λ (k , z ; ω )] = ε ω 2 c 2 ˆ∞ -∞ dk z 2π 1 -kk k 2 |k 2 -ε ω 2 c 2 | 2 e ikz(z-z ) ×δ λ,λ δ(k -k )δ(ω -ω ).
@eFIQA he integrl n e evluted y mens of omplex ontour integrl nd the residue theoryF his results in

[â λ (k , z; ω), â † λ (k , z ; ω )] = π 2 1 γ 1 - k 1 k 1 k 2 1 e iγ|z-z | + 1 γ * 1 - k 2 k 2 k 2 2 e -iγ * |z-z | ×δ λ,λ δ(ω -ω )δ(k -k ), @eFIRA
where we hve de(ned the prmeters

γ 2 = ε ω 2 c 2 -k 2 , @eFISA k 1 = k x , k y , sign(z -z )γ , @eFITA k 2 = k x , k y , sign(z -z ) × -γ * .
@eFIUA prom this ommuttion reltion we oserve tht the ommuttor deys exponentilly with the distne etween z nd z F xextD we tke the limit of nonsoring mediumD iFeFD ε = 0D nd ssume tht k √ εω/cF e de(ne γ = iκD so tht κ > 0F he ommuttion reltion simpli(es to

[â λ (k , z; ω), â † λ (k , z ; ω )] = 0. @eFIVA
ivnesent wves re osoni of ntureD nd therefore this ommuttion reltionship should not e stritly zeroF he oservtion tht this ommuttor is zeroD leds therefore to the onlusion tht the opertor elongs to (eld tht physilly does not existF sndeedD from lssil optis we know tht evnesent wves do not exist in uniformD nonsoring medium without n eletri soureF hen we tke propgting wveD iFeFD k √ εω/c the ommuttor eomes

[â λ (k , z; ω), â † λ (k , z ; ω )] = π 2γ 1 - k 1 k 1 k 2 1 e iγ(z-z ) + 1 - k 2 k 2 k 2 2 e -iγ(z-z ) ×δ λ,λ δ(ω -ω )δ(k -k ), @eFIWA
where illustrted in pigF fFP@AD so tht pourier modl method n e usedF xextD eh ell should e optilly isoltedF his n e done y for exmple putting perfetly mthed lyers @wvA t the orders of the ell long the z diretionD or y using omplex oordiE nte trnsformtions t the orders of eh ell VTF st is the ltter tht is used in this workF e iLθ e iβmz FL m (r), @fFIA where F n stnd for oth the eletri nd mgneti (eld nd β m = β 0 + m 2π d D d eing the rti(il period tht hs een introdued while periodilly repeting the periodi strutureF ih sis element is hrterized y the the zimuthl numer L nd the order m of ploquet modeF he zimuthl numer L is nturl to the physil prolemD whih is in ontrst with the order mD tht is purely mthemtil nd does not hve ny physil meningD s we hve rti(illy reted periodiity long the z xisF por numeril nlysisD not ll terms of the sums n e inludedF herefore the sums over L nd m re truntedX

k 1 = k x , k y , γ , @eFPHA k 2 = k x ,
F t (r, z, θ) = Lmax L=-Lmax mmax m=-mmax
e iLθ e iβmz FL m (r), @fFPA so tht we keep 2L max + 1 nd 2m max + 1 terms of eh sumF he susript t indites tht the solution follows from truntion of the sumsF figourdn et al. hve shown tht for eh mode de(ned y the pir (m, L)D the tngentil (eld omponents Fθ (r) nd Fz (r) t the orders of eh slie see pigF fFP@A for the de(nition of slie re relted y so lled sttering mtrix @ mtrixA IRH

    Ẽz (r j ) Ẽθ (r j ) Hz (r i ) Hθ (r i )     = S(r j -r i , r i )     Ẽz (r i ) Ẽθ (r i ) Hz (r j ) Hθ (r j )     .
@fFQA he Emtrix formlism is typilly used for multilyer systems in rtesin oordintesY here this formlism is used for multilyer system long the rdil diretionF eprt from lulting the eletri nd mgneti (elds t the order of eh sliesD the sme formlism llows to lulte the eletri nd mgneti (elds t ny position inside nd outside the nnostrutureF B.2.2 Source decomposition o fr we hve only disussed the eletri nd mgneti (eldsF hey need to e exitedD whih is numerilly done y point soureF his point soure t position r 0 X Jδ(rr 0 )D

needs to e written in the sme sis s the eletromgneti (eldsF efter periodilly repeting the point soureD the soure term is

+∞ m=-∞ Jδ[r -(r 0 + m • dẑ)] = J 2πr 0 d +∞ L=-∞ +∞ m=-∞
e iL(θ-θ 0 ) e iβm(z-zs) δ(r -r 0 )δ(z -z 0 ). @fFRA e point soure is therefore expnded s sum of ring soures see pigF fFP@AD so tht eh term respets the symmetry of the prolemF e liner superposition of ring souresD eixhs fF xwisgev gevgvesyx yx yppEes ygi sx e gvsxhsgevv wwisg xexygi IPP with pproprite @omplexA oe0ients in front of eh termD pprohes point soure y summing over ll vlues of LF vet us rie)y disuss two distint situtionsX @IA e point soure is loted t the enter @r 0 = 0AD nd @PA point soure is loted o' enter @r 0 = 0AF st n e shown tht for point soure t the enterD the sum over L is limited to only few termsF sf the soure is polrized long the z xisD only L = 0 should e onsideredD nd for ny retiliner polriztion only L = {-1, 1} re neededF es soon s the soure is o' enteredD one theoretilly needs to tke ll terms into ountF imilrly s for the eletri (eldD this is not possile nd we therefore trunte this sum to the sme order s the (eldsF he onsequene of this trunture of the soure is tht we do not hve true point soureX the soure is spred out on ringF en exmple of this is given in pigF fFQ@AD where we lulted the eletri (eld in the plne z QD @see pigF fFIAF he trunture prmeters reX m max = 100 nd L max = 3F yne n redily see tht determining the eletri (eld somewhere lose to the ring is prolemti nd therefore lso the lultion of the totl emitted powerD sine we lulte the totl emitted power y

P = 1 2 e J • E * (r QD ) .
@fFSA his e'et hs potentilly ig in)uene on our lultions in prtiulrF ine eh quntum dot is positioned out ISH nm from the enterD they lie on eh other9s ringF hereforeD the onvergene for the eletri (eld t the position of h P is therefore very dF

B.2.3 Improving electric eld calculation

e me up with tehnique to improve the (eld lultionF his tehnique is illustrted in pigF fFRF prom the eletri (eld lulted with the periodi pwwD we sustrt the IPQ fFPF xwisgev wiryh eletri (eld lulted with the sme method ut with the gold disk tht hs een mde sptilly extendedF he strong eletri (eld on the irle whih is due to the soureD s shown in pigF fFQ@AD is e'etively removed in this wyF xextD the eletri (eld orresponding to n emitter in multilyer system needs to e ddedF por thisD nother numeril method is used tht is more pproprite to suh multilyer system nd onverges muh fsterF his method onsists of expnding the eletri (eld in plne wvesD nturl sis of the systemF he solutionD sed on plne wve expnsion @iAD is dded to the (eld lulted y mens of the periodi pww nd we end up with n eletri (eld tht does not hve the prolem of soure tht is spred out over irleF he improvement of pigF fFQ@A is depited in pigF fFQ@AF es n e visully oservedD the solution of pigF fFQ@A is indeed muh etter onverged thn the solution of pigF fFQ@AF imilrlyD we see n improvement of the imginry prt of E z when ompring pigF fFQ@dA with pigF fFQ@AF

B.2.4 Convergence

xextD we study the onvergene quntittivelyF foth methods @with nd without iA hve een used to lulte the totl emitted power P for n xEpolrized point soure loted t (x, y, z) = (150 nm, 0, z QD )F por the moment we trunte the sum over L to L max = 3F pigure fFS@A shows the totl emitted power s funtion of the trunture of the sum over the order of the ploquet modesF st is seen tht the method whih mkes use of the i onverges with PFS times less modesF pigure fFS@A shows the omputtionl time for single vlue of L s funtion of the trunture m max for oth methodsF st is seen tht onvergene is rehed VFQ times fster with i thn withoutD eventhough we perform two periodi pww lultionsD s n e seen in pigF fFR1 F eprt from the fster onvergene in terms of mD there is lso fster onvergene for L when the i is ppliedF le fFI shows the emitted power per zimuthl numer L reltive to the totl emitted powerD where we de(ne totl emitted power in this tle s the totl emitted power for L = {-9 • • • 9}F his lultion hs een done for the sme sitution s desried for the results of pigF fFSF e note tht with only L = {-2 • • • 2}D the solution tht uses the i is su0iently well onvergedD s opposed to the solution where the i hs not een usedF he reson for this di'erene is simpleF hen i is usedD the emission whih is due to soure in the multilyer is lulted very urtely without the use of n expnsion sed on LF he periodi pww is only used for the prt whih is due to the ft tht the gold disk hs (nite rdiusF e hve therefore inluded this ontriution to L = 0 in the tle nd it shows tht WRFU% of the totl emitted power is lredy ounted forF por the method where the i is not ppliedD the soure needs to e pproximted y dding up ring souresF hepending on the position of the soureD mny zimuthl numers L re needed to pproximte the ring soure orretlyF prom the tle we (nd tht t lest L = {-5 • • • 5} re needed for this methodF xote tht the omputtionl time grows linerly with L max F e see tht the trik s illustrted y pigF fFR drmtilly improves the lultionF hue to this improvementD we hve now ess to the eletri (eld t the height of the quntum dotsY prmeter tht is importnt to evlute the dipoleEdipole intertions etween two quntum dotsF

  WR TFQFQ hesription of dense ensemle of olloidl quntum dotsF F F F F F WS TFR iletroluminesene model for vih F F F F F F F F F F F F F F F F F F F F F F WW Q gyxix TFS iletroluminesene model in the presene of nnontenns F F F F F F F F F F IHHTFSFI qenerlized model for hEwvihEsed eletroluminesene experiE ments

Figure 2

 2 Figure 2.2: (a) Schematic of reciprocal space (k x , k y , k z ). The states on shell, i.e.

  2 y + k 2 z = ω 2 /c 2 , are indicated by a blue dashed circle. The red line corresponds to the collection of o-shell states that form together the evanescent wave which is indicated by a red dot in (b). (b) Schematic view of (k x , k y ; z) space. Propagating waves lie in the band within k 2 x + k 2 y < ω 2 /c 2 . Evanescent waves lie outside this band. One evanescent wave is indicated by a red dot.

Figure 3

 3 Figure 3.1: (a) The normalized scattering cross section for a system of 5 homogenously distributed atoms inside a rectangular box with dimensions 10×10×10 (µm) 3 is Lorentzian. The double arrow indicates the FWHM of the spectrum which is equal to the single atom spontaneous emission rate Γ 0 . (b) The scattering pattern ( |E sc | 2 ) after an ensemble average over 10 000 realizations for a resonant linearly polarized incident plane wave. Light gets scattered in many directions. (c) The normalized scattering cross section for a system of 100 homogeneously distributed atoms inside a rectangular box with dimensions 1 × 1 × 1 µm 3 is not Lorentzian. (d) The scattering pattern ( |E sc | 2 ) after an ensemble average over 1 000 realizations for a linearly polarized incident plane wave with δω L = 0. The scattering pattern has changed into a diraction pattern.

3. 3 Figure 3

 33 Figure 3.2: A wavelength-size cloud of cold Rb-87 atoms is loaded in a dipole trap. (a) Experimental setup of Ref. [51]. A wide laser beam gets scattered by the atomic cloud. The light scattered at a 90 • angle is collected by means of an NA = 0.5 aspheric lens, after which it passes through a polarizer and a focusing lens. Courtesy of Joseph Pellegrino. (b) Experimental setup of Ref. [7]. A Gaussian probe beam gets focussed by an aspheric lens on the atomic cloud. The light scattered in the forward direction is collected by passing through a NA = 0.5 aspheric lens, polarizer and focusing lens. Courtesy of Stephan Jennewein.

  Figure 3.4: Numerically obtained on-axis spectrum from a single atom being positioned at three dierent places: before, in and after the beam focus. Input eld is a Gaussian beam with waist w 0 = 1.2 µm.

  Figure 3.5: (a) Experimental results of Pellegrino, published in Ref. [51]. The normalized scattered power as a function of the number of atoms inside the cloud (horizontal axis) and for dierent detunings of the laser beam (markers): ∆ω L = 0Γ 0 (red circles), ∆ω L = ±Γ 0 (up or down open triangles), and ∆ω L = ±2.5Γ 0 (up or down lled triangles). The power has been normalized to the power that would be detected for a single atom in the cloud being illuminated with the same detuning. Dashed line: case of non-interacting atoms. (b) Theoretical results. The red dots represent the total scattered power at 90 • normalized by the power that a single atom would scatter in the same solid angle as a function of the number of atoms in the cloud. The dashed line corresponds to the line y = x, which would be the result in the single-scattering regime. Scattering suppression can be observed when the number of atoms increases. (c) On-resonance ensemble-averaged scattering pattern for a cloud containing N = 300 atoms. The big lobe in the forward direction is reminescent of diraction from the atomic cloud. Incoherent light scattering occurs quasi isotropically, as can be seen at the height of the arrow indicating 90 • .

Figure 3

 3 Figure 3.6: (a) Experimental data, obtained by Pellegrino et al.. Normalized scattered power at a 90 • angle with respect to the optical axis as a function of the laser detuning ∆ for dierent number of atoms inside the cloud, the latter being the color that makes the distinction between dierent number of atoms. From bottom to top we have N = {1, 5, 20, 50, 200, 325, 450}. (b) For the same values of N , we have calculated the normalized scattered light at 90 • for a circularly polarized incident plane wave.

Figure 3

 3 Figure 3.7: Numerical results of coherent light scattering with a Gaussian beam (red) incident on a cloud of N = 180 atoms (blue) with zero detuning. The right part of the gure shows the numerically obtained diraction pattern, where the detection system can measure the light over a full angle of 60 • (NA = 0.5). The schematic view of the incident light does not have the same scale as the numerical results at the right.

  . The number of atoms inside the cloud are N = {10, 83, 180} (from top to bottom). (b) The experimentally obtained minimal value of the spectrum as a function of the number of atoms in the cloud. This value seems to saturate to the value indicated by the dashed line. The interpolation is done with the function: c 2 (N ) = 1 -a + (a -a)e -bN . (c) The shift of the resonance as a function of the number of atoms deduced from the experimental data. It has been linearly tted (solid line). (d) The linewidth derived from the experiment plotted as a function of the number of atoms. It has been tted with a linear function. theoryD euse resonnt dipoleEdipole intertions eome weker SVF sndeedD for the hot tomi vpor of ueveney et al.D oopertive vm shift ws oserved nd greed grei QF vsqr geisxq pyw e hixi gvyh yp gyvh eyw SP

  Figure 3.9: The group delay for N = 170 atoms inside the cloud, measured by Jennewein.

ee

  Figure 3.10: (a) The phase of the total electric eld in the far eld as a function of the laser detuning for N = 12 atoms (black circles) and N = 170 atoms (red dots). To interpolate, for N = 12 the t function: y = -|c 1 -c 2 1-2i x-c 3 c 4 e ic 5 | 2 has been used. For

  1: (a) System under study: N = 450 atoms uniformly distributed in a rectangular box. The atomic density is ρ/k 3 = 1. The incoming light excites the collective eigenmodes of the system (visualized by the blue sinusoid), which gives rise to a scattering pattern in, essentially, the forward direction. Here, we have plotted the modulus square of the electric eld radiated in the far eld at resonance, averaged over 1 500 realizations, |E| 2 . (b) Homogeneous medium described by an eective dielectric constant ε e exhibiting optical resonances. The scattering pattern, |E| 2 , is found to be similar to (a) except for the diuse light, which is by denition null in (b).

  Figure 4.2: The three types of eigenmodes are indicated by the numbers (1) through (3). The inset shows a closeup of the boxed part of the main gure, where the polaritonic modes (a), (b), (c) and (d) for both (A) ρ/k 3 = 1 and (B) ρ/k 3 = 5 are visualized in Fig. 4.7.

  Figure 4.9: Scattering pattern of an x-polarized plane wave incident on the atomic cloud (schematically indicated by the purple box) with density ρ/k 3 = 1 and with ω = ω 0 . The arrow indicates the incident wavevector, and the double-arrow its polarization. The red curve corresponds to the coherent scattered intensity from all modes. The blue curve comes from coherent light scattering from the polaritonic modes as dened by the rectangles in Fig. 4.2(A). Only 0.54% of all collective eigenmodes are within the black rectangles. A signicant part of the scattering pattern can be attributed to only a very few modes, which are called polaritonic modes.

  whih nnot e negleted in this dense regime IPD PWD WUF he sttered eletri (eld E sc n e lulted from the set of oupledEdipole equtions @see iqF PFIS nd iqF PFITAF he frE(eld sttering pttern is proportionl to |E sc | 2 F efter severl reliztionsD the ensemleE verged sttering pttern is otined nd deomposed in oherent nd inoherent prt respetivelyX |E sc | 2 = | E sc | 2 + |δE sc | 2 F he oherent sttering pttern orresponds to the di'rtion pttern of homogeneous ojet desried y dieletri onstnt nd the inoherent sttering pttern is qusi isotropiF pigure SFI shows the sttering pttern for single realizationF st is seen tht there is one ig loe in the forwrd diretion nd in4

Figure 5 . 1 :

 51 Figure 5.1: Angular scattering pattern for a single realization of a cloud illuminated by a plane wave. The volume has dimensions 4.8λ 0 × 0.6λ 0 × 0.6λ 0 . Several peaks are visible. A big lobe in the forward direction indicates that the optical response is dominated by a coherent contribution of all scatterers. |δE sc | 2 and | E sc | 2 indicate what are the main contributions to the scattering pattern.

eFigure 5

 5 Figure 5.2: Scattered power calculated for a linearly polarized incident plane wave propagating along the long axis of the cloud with a frequency detuning δω L = ω L -ω 0 = -10 4 Γ 0 .All powers are normalized by the scattered power for a single atom at the same detuning and plotted on a loglog scale. The fraction of coherently scattered light increases as a function of the number of atoms. The red dashed line is a quadratic t and the black solid line a linear t.

dFigure 5

 5 Figure 5.3: An example of the pair correlation function for the case of the hard sphere approximation. The atoms cannot approach each other within a distance d.

Figure 5

 5 Figure 5.5: (a) Scattered power has been calculated for a resonant (δω L = 0) linearly polarized incident plane wave propagating along the long axis of the cloud. All powers are normalized by the scattered power for a single atom on resonance. The black arrow indicates for which number of atoms the spectrum has been plotted in Fig. 5.5(b). (b) For N = 450 atoms, the scattered power for a single realization has been plotted, together with the ensemble averaged coherent, incoherent and total scattered powers. All quantities are normalized by the single atom scattered power on resonance.

Figure 5

 5 Figure5.8: Typical spacing δω between two modes is given as a function of the volume of the box for three dierent numbers of atoms inside. The numerical results are tted with a power function. (a) The entire Green's tensor has been taken, i.e., also the 1/r 3 term. It is observed that the typical spacing is independent of the number of atoms and δω ∝ V -1 . (b) Only the 1/r term of the Green's tensor has been used in the calculation. It can be seen here that the typical spacing depends on the number of atoms and that δω ∝ V -1/3 .

Figure 5

 5 Figure 5.9: The t parameter used in Fig. 5.8(b) is plotted as a function of the number of atoms inside the cloud. The t function of c 1 (N ) scales as N -α , where α = 0.68 ± 0.07 with 95% condence bounds.

Figure 6 . 1 :

 61 Figure 6.1: Simplied vision of a quantum-dot based light-emitting device. Electrons leaving the cathode are transported via an electron transport layer (ETL) towards the lightemitting layer that is composed of colloidal quantum dots. From the anode there are holes which are transported via a hole transport layer (HTL) towards the colloidal quantum dots.

Figure 6

 6 Figure 6.2: Schematic view of the device used by Le-Van et al. in Ref. [11]. cQDs are densely packed and put close to metallic nanorings which form a periodic lattice. Courtesy of Quynh Le-Van.

Figure 6

 6 Figure 6.3: (a) The EL intensity in arbitrary units is given as a function of the applied voltage for dierent devices. From left to right: small rings, large rings, and no gold nanoantennas. The small and large rings have been visualized in the graph. The turn-on voltage goes down from 6 V to 1.3 V when gold nanorings are introduced. (b,c) A schematic view of experiments without (b) and with (c) a 5-nm-thick TiO x spacer between the PbS cQDs and the nanorings are presented in the upper parts of Fig. 6.3(b) and Fig. 6.3(c). Below the electroluminescence signals are shown. The applied voltage was (b) 5 V and (c) 10.5 V. The yellow rectangles indicate the regions in which gold nanorings are present.

Figure 6 f

 6 Figure 6.4: (a) An SEM image of the PbS QD lm that is used for the electroluminescence experiments of Le-Van. Courtesy of Quynh Le-Van. It is seen that the quantum dots are touching each other. (b) An exciton in cQD 1 can either decay radiatively within a time 1/Γ r , or nonradiatively be exchanged with cQD 2 by means of Förster resonance energy transfer (FRET) within a typical timescale 1/Γ D→A , where D stands for donor and A for acceptor. The diameter of each quantum dot is 9 nm.

  Figure 6.6: (a) Schematic of a single gold nanoring embedded in a QD-MLED system. The stripes indicate where the absorption has been calculated. The dashed line corresponds to the symmetry axis of the cylindrically symmetric system. (b) Map of the real part of E z for a linearly polarized plane wave incident along the z axis from above (λ = 1.2 µm).

  Color scale from blue to red: negative to positive values. The contours are indicated by the dashed lines. Compare the structure with the structure given in (a). It is observed that the QDs above gold absorb most of the incident light.

  6.5.1 Generalized model for QD-MLED-based electroluminescence experimentsefter hving disussed eletroluminesene from onventionl vihsD we propose model for eletroluminesene from hEwvihsF ery reentlyD it hs een shown y qre'et et al. tht uirhho'9s lw n e generlized to system tht is t lol thermodynmi equilirium IPW!IQIF e hve seen in etion TFQFQ tht due to the nerE(eld intertions etween the hsD our system is t thermodynmi equiliriumX we n de(ne lolly the sorptivity nd the eletrohemil potentilF reneD we n use the extension of uirhho'9s lwF sn efF IPWD the uthors hve shown tht in order to lulte the totl grei TF riy yp ivigyvwsxigixgi f exwEhy feih wiepegi vsqrEiwssxq hisgi IHP emitted power P e under the pplition of voltgeD the sored power needs to e lulted lolly t the position r for plne wve inident long the diretion u

  ❈❍❆ ❚ ❊|✻ ✳ ❚❍ ❊❖| (❖ ❋❊ ▲ ❊❈❚|❖ ▲❯▼ ■◆ ❊ ❙❈ ❊◆❈ ❊❇(◗❯❆◆❚❯▼ ✲❉❖❚ ❙ ❇❆ ❙ ❊❉ ▼ ❊❚ ❆ ❙❯| ❋ ❆ ❈ ❊▲ ■•❍❚ ✲ ❊▼ ■❚❚ ■◆•❉ ❊❱ ■❈ ❊

Figure 6

 6 Figure 6.9: (a) The experimental electroluminescence results are presented for both light polarizations as a function of the wavelength for a setup with nanorods. Light polarized along the long axis of the nanorods gives more electroluminescence signal than light polarized along the short axis. Courtesy of Quynh Le-Van. (b) Numerical results of the calculation of the absorption cross section for quantum dots on top of a single nanorod with dimensions 170 × 50 × 50 nm 3 .

Figure B. 1 :

 1 Figure B.1: Schematic of the structure. All dimensions, apart the dimensions of the quantum dots and the substrate, are on scale. The diameter of the gold disk is 2.6 µm. The materials, in order and per layer are: Au, Au, GaAs, Al 0.1 Ga 0.9 As, GaAs,QDs,GaAs, Ga 0.05 Al 0.95 As, {GaAs, Ga 0.05 Al 0.95 As} nm (between curly brackets: repeated 40 times), GaAs. The heights are from top to bottom: 45, 5, 10, 10, 24, QDs, 19, 74, {60, 72} (between curly brackets: repeated 40 times). The z position where the QDs are found is indicated by z QD and is 44 nm underneath gold.

F

  Figure B.2: (a) Example of a rotationally invariant nanostructure with the presence of a ring source. (b) The aperiodic nanostructure is periodized with period d along the symmetry axis. The cells are separated by either a PML or, in our calculations, by complex coordinate transforms. (c) In the radial direction, the structure is divided in slabs for which the optical properties are invariant along the radial direction. Figure has been extracted from Ref. [140].

  Figure B.3: A map of the electric eld in the plane of an x-polarized quantum dot positioned at (x, y) = (50 nm, 0). m = {-100 • • • 100} and L = {-3 • • • 3}. (a) log 10 |E x | 2 obtained without a plane wave expansion (PWE). The source is indicated by QD 1. QD 2 indicates the place where the other quantum dot could be positioned. (b) log 10 |E x | 2 obtained with a PWE. (c) Im(E z ) obtained without a PWE. (d) Im(E z ) obtained with a PWE.

Figure B. 4 :

 4 Figure B.4: A schematic view of the method used to improve the convergence of the electric eld calculation in the vicinity of the quantum dot. From the electric eld calculated by means of the aperiodic FMM as explained in the main text, we subtract the electric eld calculated by the same aperiodic FMM, but with the gold disk extended in two dimensions, eectively making it a multilayer system. The electric eld created by a point source in this multilayer system is calculated by a plane wave expansion and added.

Figure

  Figure B.5: (a) The total emitted power has been calculated for an x polarized point source at (x, y, z) = (150 nm, 0, z QD ) and L max = 3. The dashed lines indicate the regions for which the calculated power deviates by 1% of the value obtained when m max = 500. PWE stands for plane wave expansion and is used to improve the convergence. (b) The computation time as a function of the truncature on m. This calculation is done for L = 0.

  -0.001 -3 • 10 -4 1 • 10 -4 -1 • 10 -5 IPS fFPF xwisgev wiryh

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light-emitting diodes by incorporating gold nanoparticlesD yptF ixpress 24D eQQEeRQ @PHITAF IIQ F figioniD tFEF rungD nd fF rehtD Nanoantennas for visible and infrared radia-tionD epF rogF hysF 75D HPRRHP @PHIPAF IIR FF eperik nd eF hegironD Numerical analysis of an optical toroidal antenna coupled to a dipolar emitterD hysF evF f 83D PRSRHV @PHIIAF IIS tF ezoulyD eF hérreD eF ihrdD nd F hénioD Quenching and enhancement of single-molecule uorescence under metallic and dielectric tipsD iurophysF vettF 51D QUR @PHHHAF IIT wF homsD tFEtF qre'etD F grmintiD nd tFF erisEqonzlezD Single-molecule spontaneous emission close to absorbing nanostructuresD epplF hysF vettF 85D QVTQ @PHHRAF IIU FEF rsiehD gFEF vingD FEpF ghenD gFEF viD nd FEF ghouD Mechanism of giant enhancement of light emission from Au/CdSe nanocompositesD xnotehnology 18D RISUHU @PHHUAF IIV uF rosokiD F ygkiD F mmotoD uF wtsudD nd F unemitsuD Direct and stepwise energy transfer from excitons to plasmons in close-packed metal and semiconductor nanoparticle monolayer lmsD hysF evF vettF 100D PHURHR @PHHVAF IIW wF rridsD vFxF ripthiD nd tFuF fsuD Photoluminescence enhancement and quenching in metal-semiconductor quantum dot hybrid arraysD eppF hysF vettF 98D HTQQHS @PHIIAF IPH eFiF gD eFEF qdllhD F h osD wFfF wohmedD nd sFwF ezzouzD Ag surface plasmon enhances luminescence of CdTe QDsD yptF gommF 314D VTEVW @PHIRAF IPI vFxF ripthiD F ungD FEwF fhkD F rnD qF ghoiD tF hieD tF teongD nd hFE F uimD Quantum dots-nanogap metamaterials fabrication by self-assembly lithography and photoluminescence studiesD yptF ixpress 23D IRWQUEIRWRS @PHISAF IPP F veEnD F ve ouxD FF eperikD fF rertD pF wrquierD tFEtF qre'etD nd eF hegironD Temperature dependence of quantum dot uorescence assisted by plasmonic nanoantennasD hysF evF f 91D HVSRIP @PHISAF IPQ eFtF workD wFgF eidmnD pF rinsD nd FeF isdleD Magnitude of the Förster radius in colloidal quantum dot solidsD tF hysF ghemF g 118D IQWPHEIQWPV @PHIRAF IPR F foseD tFpF wwillnD tF qoD uFwF ikeyD gFtF ghenD hFF lpinD gFfF wurryD nd gFF ongD Temperature-tuning of near-infrared monodisperse quantum dot solids at 1.5 µm for controllable Förster energy transferD xno vettF 8D PHHTEPHII @PHHVAF IQS fsfvsyqer IPS qF uirhho'D Ueber das Verhältniÿ zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und LichtD ennF hysF 185D PUSEQHI @IVTHAF IPT F hokley nd rFtF ueisserD Detailed balance limit of eciency of p -n junction solar cellsD tF epplF hysF 32D SIH @IWTIAF IPU F ürfelD The chemical potential of radiationD tF hysF gX olid tte hysFD 15D QWTUEQWVS @IWVPAF IPV xF fonodD iF opovD nd wF xevièreD Dierential theory of diraction by nite cylindrical objectsD tF yptF oF emF e 22D RVIERWH @PHHSAF IPW tFEtF qre'etD F fouhonD qF fruoliD iF ktD nd pF wrquierD Generalized Kirchho lawD rivXITHIFHHQIPvI @unpulishedAF IQH gFrF renry nd FpF uzrinovD Quantum noise in photonicsD evF wodF hysF 68D VHIEVSQ @IWWTAF IQI gF ytey nd F pnD Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plateD hysF evF f 84D PRSRQI @PHIIAF IQP FuF ghettir nd xF inghetD Internal homogenization: Eective permittivity of a coated sphereD yptF ixpress 20D PPWUTEPPWVT @PHIPAF IQQ F irnst nd F tehleD Emission of radiation from a system of many excited atomsD hysF evF 176D IRST @IWTVAF IQR xFiF ehler nd tFrF ierlyD SuperradianceD hysF evF e 3D IUQS @IWUIAF IQS rF vuhnD hF frredoD F vetsD F de véséleuD F wrìD F vhyeD nd eF froweysD Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising modelsD xture 534D TTUETUH @PHITAF IQT F voudonD The Quantum Theory of LightD @glrendon ressD yxfordD uD IWVTAF IQU fF ruttner nd FwF frnettD Quantization of the electromagnetic eld in dielectricsD hysF evF e 46D U @IWWPAF IQV F qruner nd hFEqF elshD Green-function approach to the radiation-eld quantization for homogeneous and inhomogeneous Kramers-Kronig dielectricsD hysF evF e 53D Q @IWWTAF IQW F qruner nd hFEqF elshD Correlation of radiation-eld ground-state uctuations in a dispersive and lossy dielectric D hysF evF e 51D R @IWWSAF IRH pF figourdnD tFEF rugoninD nd F vlnneD Aperiodic-Fourier modal method for analysis of body-of-revolution photonic structuresD tF yptF oF emF e 31D IQHQEIQII @PHIRAF

  , • • • , e j , • • • g N |0 D the seond with |g 1 , • • • , g N |1 k ,λ D nd the third projetion with the wve funtion |g 1 , • • • , e m , • • • , e n , • • • , g N |1 k ,λ Ffy ompring the oe0ients of the sme sttes on oth the leftE nd rightEhnd side of iqF PFQID the following three di'erentil equtions n

	Ĥ|Ψ(t) = i he following step is to projet the hrödinger eqution three times y di'erent wve d|Ψ(t) dt . @PFQIA geisxq PR funtionsF he (rst projetion is done with |g 1 grei PF gvesgev exh exw iewix yp vsqr e otined for the oe0ientsX

  ). All possible second-order processes are shown between two two-level systems.The intermediate states |m 1 correspond to the wave functions having γ as coecient, and |m 2 to the wave functions having α as coecient. Therefore, |m 1 = |g 1 , g 2 |1 k,λ , and |m 2 = |e 1 , e 2 |1 k,λ .

	grei PF gvesgev exh exw iewix yp vsqr
	geisxq	PT
	Figure 2.1:
		@PFRPA

  As an interesting side note to put Kirchho 's discovery in historical context, let us mention that when Kirchho published his seminal work on the relationship between emission and absorption in 1860, Planck's function was not known yet, but Kirchho knew it should be independent of the properties of the object and only depend on the wavelength and temperature[125]. He wrote: Es ist eine Aufgabe von hoher Wichtigkeit, diese Function zu nden, which means It is of very high importance to nd this function. He continued: Trotzdem scheint die Honung gegründet, sie durch Versuche ermitteln zu können, da sie unzweifelhaft von einfacher Form ist, wie alle Functionen es sind, die nicht von den Eigenschaften einzelner Körper abhängen, und die man bisher kennen gelernt hat, which can be translated by Nevertheless, it appears that there is foundation for the hope, that it can be determined by research, as it is undoubtedly of a simple form, like all functions, that are not dependent of the characteristics of a single body, and which are known sofar. Kirchho was right and it was Max Planck who correctly described black-body radiation in 1901 by quantizing radiation. Planck's law was the birth of quantum mechanics.

  k y , -γ , present n improvement of n periodiEpourier modl method @periodi pwwA tht hs een developed y figourdn et al. IRHF his ode enles to lulte the eletri (eld produed y point soure inside ny ylindrilly symmetri systemF hen the point soure is positioned wy from the symmetry xisD the lultion eomes numerilly more demndingD sine the symmetry of the prolem is rokenF rereD we present numeril trik to improve oth onvergene nd onvergene time of this numeril method for soure tht is positioned o' xisF B.1 System description he system tht ws studied with the periodi pwwD nd ws used to improve the onE vergene of the numeril lultionD hs two snesGqes quntum dots tht re sptily seprted y roughly QHH nmF hey re emedded in multilyer systemF his multilyer system ontins distriuted frgg re)etor @hfA nd on top there is gold diskD see pigF fFIF his gold disk hs dimeter of PFT µm nd is positioned suh tht the quntum dots re symmetrilly positioned round the enter of the gold diskF he quntum dots re RR nm underneth the goldF por the other system prmetersD refer to the ption of pigF fFIF sn order to otin the qreen9s tensor of the prolemD we need to lulte the eletri (eld for pointlike eletril dipoleF sn the next setion we rie)y disuss the ide ehind the numeril method tht is usedF

	Appendix B
	Numerical calculations on o-axis source in a cylindrically symmetric nanostructure
	@eFPIA
	with γ ∈ RF his ommuttor is not stritly zeroF sndeedD propgting modes re physilly llowed in uniformD nonsoring medium without n eletri soureF B.2 Numerical method
	A.4 Conclusion B.2.1 Electric eld decomposition

e presented the quntiztion of evnesent wves in n soring mediumF e hve formlly estlished the reltionship etween evnesent (elds nd virtul photonsF he retion nd nnihiltion opertors for evnesent wves n e expressed s liner suE perposition of retion nd nnihiltion opertors of photoni modes tht do virtulF e here re severl numeril methods possile to model ylindrilly symmetri systemF sn this eppendixD we study n periodiEpourier modl method @periodi pwwA tht is developed y figourdn et al. IRHF sn the urrent setionD we rie)y disuss their methodF por more thorough disussionD refer to efF IRHF por this periodi pwwD the periodi struture n exmple is given in pigF fFP@A is periodilly repeted in the z diretion s eixhs fF xwisgev gevgvesyx yx yppEes ygi sx e gvsxhsgevv wwisg xexygi IPH

Table B .

 B 1: The fraction of power emitted by the mode with azimuthal number L is given for both methods where either the PWE is applied or not. It is seen that the method that uses a PWE converges faster than the method without.

, @QFSA

In atomic physics, this notation for the atomic energy levels is very common. The dierent symbols in the notation (nL J , F ) stand for the principal quantum number (n); the orbital angular momentum (L), where L is given as a letter (S corresponds to L = 0, P to L = 1); the total angular momentum, taking into account the spin angular momentum of the electron (J ); and the total atomic angular momentum, where the total angular momentum of the nucleus has been taken into account (F ).

For the experiment of Jennewein, there are 5 degenerate states for (5S 1/2 , F = 2), and 7 degenerate states for (5P

3/2 , F = 3). They are degenerate, since no external magnetic eld is applied that lifts the degeneracy.

This is not true when one looks towards the sun, since in that direction we observe coherent light scattering, where coherent should be interpreted in our way as an ensemble average.

If a detector is placed in the forward direction, with a polarizer and a small collecting aperture in front of the detector, one can reduce signicantly the amount of detected incoherent light. This situation corresponds to the experiments of Jennewein.

In principle it should be possible to combine both calculations in only a single calculation, because of the linearity of the electric elds. This improvement of the calculation has not been implemented yet.

Acknowledgements

st is with plesure tht now s n thnk those with whom s hd the opportunity to intert with nd to interfere with onstrutively over the lst ouple of yersF