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# Light Matter Interaction OSE 5312

University of Central Florida

GPA 3.97

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This 24 page Class Notes was uploaded by Rachelle Hilpert Jr. on Thursday October 22, 2015. The Class Notes belongs to OSE 5312 at University of Central Florida taught by Pieter Kik in Fall. Since its upload, it has received 46 views. For similar materials see /class/227450/ose-5312-university-of-central-florida in Optical Studies at University of Central Florida.

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Date Created: 10/22/15

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not the correct answer 39 4 palms palm ShdeW F AV 5 g a scale with a term IJ1 b did not exhibit a zero point energy gt c are evenly spaced 5 gt a any polai crystal b only polar insulators c only polar semiconductois gt a the thermal population ofphom 7 The occurrence ofthe ultraviolet catastrophe was due to 4 paints nniodes h a decreasing optical density of states fol high frequencies co the small aperture size of our ideahzed blackbody radiator palms H Ligm Maller hileraction FaH 2009 7 Class 23 photoniluxThe corresponding absorption coe ide t is given by 7 o With o the 4 palms Slidel OSE5312 Fall 2009 MT2 recap u IIIIIII I iIIequot xltx absorpnon line av 15quot5 cm 1 correspond w n AJO Housman Absorption spectrum of NO moiecules In gas phase I I I I I I I 9 Y 39 X I 39 HtIII II 0 I39qI IIu e39v I II I I1 1870 1580 1590 1900 1910 1920 1930 Energ cm39p A n talruklk he energy ohm NO slr th x39mmlmn I ramm I moanl I em h x 1 IIIum lI39ML39l I blv39blJI39JL1039II 6 39JlJOJ M III 39 IIII I I spnngmnImIIIIIrIIIeNm Emmi III sI IInIIIIN m IapIIIIIIII rquueIIcyIIIIIechImImmmsswan hyw K II I Llln gt1nuI 391391vmI39 rump I 39 l JIII kg we haw 5 llmd 5 RH irl SSElX m Light Matterlnteraction Fall 2003 Class 23 slide 3 c A small fracuon 02 ofoxygen atoms have a mass of18 amII labeled 180 These oxygen atoms NEW 0 molecule woul Show 39 in II IIIII eIIeI 39 You may use energy Imus wavenumbeIs J or eV 6pz7ims New reduced mass II 114 118quot ms amu Ra o in equenmes mumso mumso KI ILImIsoW K Humm quot39ILIINIgoImmo 0974 be 39heavy vasion II III u 39139 III I III Conespondmgm 127 mev 3 52E20 J a wall 6 nn m I F stpeaktothen39ght ofJ0is o a 1 2 sevemh peakis 5 4 7 39 pUImIHVViH m ed on me relaIed absoerion line 6 points andAvgtH hausition 0 J1H39mv nvv From I T gt 11 The specuum showsbmh a AI0 and A F1 and a AJ1 10 to 39 It 39 III he spemum aloranonalu ansitim from0gt 1 Iakes 18817 1876 cm1 5 cmquot meanmg we or 2mm Light Manerlnleraclion Fall 2009 Class 23 r 39 39 The have an T0 slide 4 OSE5312 Fall 2009 MT2 recap an n J Lu 39xlmlrmh gt11 39V 1 n Thesulmm hm remmuymxeletmr consmmomr x 9 m The mom masses nle and r are resyecm39 m amuund Damn caxmmemumm mass y or qu m pmun 10M 1 m a 224 m3 me yaumunm seem reasnmbk39 1111 pomm 39mfu 39 9V Yv39m InI 1w w mng m warAM ml L gm Mauer meracuon Pa 2009 7 CH mxzwu 1mm m Hm Shdei ll 439 nuhmuhl the Well is V0 and mat the well extends from 20 to FZDIEL We m 21 Th Tpmrnn 4 nu dmmm A pomts A consKampmamial Implied Wave mc ons wnh a xed Wavevedor w1zA1 sm 2E9 L gm Matter Hteracuo Pa 2009 r y w Puningthele edge arms well arFo leads m sinusoidalsolunons o hefonn w A 3111 n n z 5 mm W2nm This giV as me rst V0 solutions o he form and way A sm27rzlE9 Shde6 OSE5312 Fall 2009 MT2 recap bAn139ucxdenr elecn39omaau u 439 canexcue 39 mmm ans transinon ripomrxj v rmwvr Schr dinger equation we know that With here 111 me elemon mass From H w E w we nd n z W 1511010 943 meV 5391 H N S A 211 2 lt 5041049 377 meV E2 2 109 39 quot79432253 meV Thxsconespondsoawavelengthof438 pm 012 431514 lads Lmht Matter nteracuon Pa 2009 r 0855 23 Shde 7 cl uf hefwm 7 39 Zl wmlswvgtlquot 39 39 th H mm in mm m m u 40K 39 I mun u aquu 3910 th16 nnmeav me mm m2 and 1011hehrmsmmlltl mung m we numberc nndmkl 391 Ms mu lnll m lblu quot 585 1rt1055 10 m v gt1qu r 30 m 7L 10 1m 10 w I 4 How A nu wluclm I mm um am39wm 39 V 7 I gt dmded by m Dulmumphrdb Umgnmgwughh 3 10 U9 Mauer meracuon Pa 2009 r MES 23 SMQX OSE5312 Fall 2009 MT2 recap PreVIous lectures 13 Found description of propagating plane waves J ill 14 H7 e ee t where the dispersion relation in a medium was iven b electric res onse onlh 1 LI2 4 Tc IFX LLI Taking into account the magnetic response and remembering c E my 1 k int u39 1 X Lu L l JXmlh 1 Or simply k pm with permittivity c and permeability given by HXQ and y y LHXM where xm ltlt 1 or pie pg in most materials and s1XEm the dielectric function Light Matter lritelattlml Fall znnar Class slide i PreVIous lectures 23 ln vacuum plane waves propagate with a phase velocity Vre f VI 3 c r while inside a lowloss medium the phase velocity is Vr k e y i K with n the real re 39active index In isotropic nonmagnetic lossy materials wave propagation is governed by the complex electrical susceptibility X with XUJ x39 u t ixl39lu In this case we can write the dispersion relation k2 pe a in terms ofa complex wave vector k k H k and a complex refractive index n n i K IU The complex wave vector Is now given by k 7 or kyk with k1 2m the 39ee space wave vector and A cf the 39ee space wavelength Note ifwe can model the complex susceptibility or index we can predict absorption Light Meter inieiaciinn Fall zuuar Class slide 2 OSE5312 Fall 2009 Class 05 Lorentz model PreVIous lectures 33 The relation between the complex re 39active index n and the susceptibility X is quiz no it Kai ll l 39u e Xvw Taking the square ofthis and equating imaginary and real parts we found i n u k lu a H39X39lu t o 5V lulu Klu X lu with 2 and cquot the real and imaginary parts ofthe complex permittivity U and a complex wave vector kzy39kv resulting in absorption a z 139 id ZK Z In the case ofdilute media small X amp X the relations between n and X became i u Mu I XT and KUI 3 Note these relations are WRONG in the case of large n even ifyris small Light Mattel llltelattlull Fall znnar classa slide 3 Today s lecture EM waves move charges and accelerating charges emit EM waves lfwe can model howmuch the charges move we should be able to predict the complex refractive index Today 1 Simple model ofa polarizable atom the Lorentz model single resonance frequency one absorption only expressions for Xw and consequently nw 1a and Rw 2 Extended model multiple absorptions 3 Modi ed Lorentz model absorbers in a dielectric host material xiii n39 Light Matt lnieiaciian Fall zuuar Class slide A OSE5312 Fall 2009 Class 05 Lorentz model Equation of motion iv Illuminate atom f Charge displacement E 1 E E Equation ofmotion ma Frat Average electron position 7t Q quot5511011 mm 7 Allforces acting on the electrons Ft Felecmml Ffmmm Frame ngm Mattel liltevattlml Fall znnar Class slldz 5 IndIVI ual forces on Single atom 1 Electrical force E 76E Nomenclature 2 Restoring force 7K imwgi with 00 Assumes force linear in displacement always true for suf ciently small amplitude This linear dependence ofrestoring force on position is Hooke s Law 3 Friction loss EmFar t Nuts 1quot eva mm 1quot1 at r r Damping assumed linearly related to velocity with F units Is the damping rate Contributions to 1quot Radiation Energy loss due to emission fast relaxation high F Scattenngcollisions Eg excited ions colliding with other ions in gas phase oscillating electrons exciting vibrations in a material ngm Mann lnlmellun Fall zuuar Class slldz a OSE5312 Fall 2009 Class 05 Lorentz model FourIe ransform of the equation of motion drivingterm Equation ofmotion m 1 Ff F39 Fe 2 or mar 9 mr ar mwzee3z at at To get a relation for roo write rt and Et in terms of their Fourier components 39 i t we at 4 Flu note quot w and gut Ewe Aw w Write out all the derivatives For example A l m rUam do mu10cuvuua 39oo Light Matter lntevactimi Fall znnar ciassa siide 7 Complex amp I ude vs frequency Then FT this to obtain the Fourier component at a single 39equency to 7mm2 iimmlquot mco 2 Fco 7250 giving us the 39equency dependent charge oscillation and thus polarization Note that again the loss temw is responsible fora phase shilt rw shows the features ofthe driven damped harmonic oscillator i Maximum amplitude when denorninator rninirndrni when at memu resonance w39 A 2 Note ror excitation at resonance own ro is entirey imaginary at resonance 90 degree onase dirrerence between E and r 3 At nign frequencies a gtgt mu tne amplitude rm vanishes 4 At low rreodencies 0 ltlt mm a fmIe ampNude is obtained t Light Matt lntevactiun Fall zuuar ciasss siide 5 OSE5312 Fall 2009 Class 05 Lorentz model Amplitude and phase of a driven osCIllator Behavior ofa driven and damped harmonic oscillator can be summarized as follows law 5 frequency 2 f W9 re HEN hm 2quot mll y s lt e a 39l 3 w i L s 1 5 E g i r 4e a i ME QutuL Kma Em m ageazemr This type of response of bound charges l is typical for many materials Next ifwe convert our single atom response to polarization per volume we can link roo to Xm giving us a microscopic model for Xm sm and 15 Light Mattel lilteiattlml Fall znnaa classa slide a From microscopic polariza i i y to Pa First relate single atom roo to single atom dipole moment 1a seam E a lid Using 7m thISQIVes w waf We now need to convert this single electron response to P dipole moment per unit volume 2 sum dipole moment over all electrons in a volume V then divide by volume lfwe have Nvalence electrons lm3 this gives N V electrons in volume V average dipole moment per atom vectort orm ami nwkNw V Z Z NeZ m anew atFw Light Mailer interaction Fall 2nnae Class slide in OSE5312 Fall 2009 Class 05 Lorentz model Lorentz mode complex susceptibility Remember that we previously de ned FUJ Xlu EllI which we can now relate to the microscopic model Comparing NeZ m tonZ 750571130 gives L AL a is de ned in L w Assumptions restoan force linear in r damping linearin drdt millions ofclosely spaced atoms can be described as independent oscillators dipoles all point along the applied eld isotropic Light Matter liiteiactimi Fall znnar Class slide ii Lorentz mode real and imaginary suscepti ii y We can now revwite Xm as 2 a l a 7502 firm co Logicozirco 7a firm mgimZJrirco Complex conjugate mild acoz rm a 7502 aria which gives us the real and imaginary components of complex susceptibility Xoo 2 2 ll 1 l39 1 12 D P rm2z1quot2m2 and 1quotm 02 P a 7422 F2m2 Liam Mann lnleiacliun Fall 2nnar classs slid l2 OSE5312 Fall 2009 Class 05 Lorentz model Lorentz model ne shape dielectric functions mm mm mama 1X39lu Q a r2412 1quotw 0 m2 7 u 11quot X Li a Light Matter irvtevattiun Faii zanar Ciass 5m 3 Lorentz model esonance approximation I39m 7 m2 2 M d an i gem r2m2 The rst term in the denominator can be written as We had 1W Z 4 Fa 2 WW m3 7 m2 mtgr meu rm Near the resonance 39equency we have mu m a In a all ltlt tun so we can write 2mm tan 7 0 which gives 2 a 7 and 1quot10 a r W 4 L 4 l 2mD maimf rzf 2mum rmfF22 1 1 On resonance 1quot10 C quotJG quot02 r NZ Halfthe height at AuFFZ Near resonance 1 1 1quot FWHM of lquot 1quot at In 744 maim2lquotZz 21quot22 X has a Lorentzian line shape with a Full V dth at Half Maximum FWHM oflquot Liam Mann interactiun Faii zuuar Ciass sim in OSE5312 Fall 2009 Class 05 Lorentz model entz odel example hig requency resonance N2 1 A 300nm N1028m3 r10Ws 2aeem n nltwgtlslswl xltwgtJlsleswl um no Ex to 1x10 5 i i i i mi 4x10 6x1015 u 7 0 2x10quot 4x10quot 6x10quot 3x10quot 1x10 9 2X10 Note realistic concentration resonance in UV 300nm index 1416 slide 15 Light Matter Interaction Fall 2009 Class 5 entz odel example sharp hig quency resonance A 300nm N1028m3 F10l3ls 7160 N ez 39 enm 605760271130 EN 1w t x l 6x10 8x10 um l l l x l o 1 toquot ma ma 3x10 me n Ma Ma Note damping reduced larger absorption on resonance same lowfrequency index slide 16 Light Matter Interaction Fall 2009 Class 5 OSE5312 Fall 2009 Class 05 Lorentz model Lorentz model example low freqeuncy resonance l 900nm N 1028 m3 P 10l4s xix m Wm 07 39 0 2x1017 mo15 6x1015 3x10 1x10 0 2X10 4x10 5x10 mo 1x10 Note lower resonance frequency NIR lowfrequency index increases 3 slide 1 7 Light Matter Interaction Fall 2009 Class 5 Lorentz model example effect on reflection spectrum fii39ii 701702 18 n1Z K2 75 300nm N 1028 m3 r 1014s i 7 l i gt A 300nm N 1028 m 3 F 10 4 Is Au 900nm N 1028 m3 L F lOM ls Light Matter Interaction Fall 2009 Class 5 slide 18 OSE5312 Fall 2009 Class 05 Lorentz model 2a 541 2 11101 sf ma 1a 2762 5 Mil2 I n12 K2 R co ve edance mm aw at nuvmal momenta mm us Note prupnmunaltn absuerun augmenmmutscaleu by Empfurplut ngm Manev lmevamun Fall znnar Class 5m m Lorentz model examples shape of absorption spectrum a 4 mp 8 139o3 a m nll nm1 Y Y39 7lzl 5 7 39 1a 2762 c 2 a Z a m 2 nily g n12 K2 ve emancelmmawalnuvmalmmdence mum Mann lmevacuun Fall zuuar Class 5m 2m OSE5312 Fall 2009 Class 05 Lorentz model Lorentz model typ alfrequencydependence Four distinct spectral regions 0 Transmissive ener Absorplve murF2ltmltmu Reflective mm m lt a lt 0 2 Transmissive T A R T The regions are more distinct for smiledquot and iargemp i i i i i we us i i i i i 2 i 5 a in Light Manei iiiteiactiuii Fall ZEIUW classa slide 2i alatoms qua mmechanicalpicture Real atoms up to tens of electrons many 39 39 39 39 2mg quot X f2 f5 A Mi Ground State on Ne2 From perturbation theory 1a Z ED m J 0 7a 711350 with or the resonance 39equency ofa transition with AEJ hm is the oscillator strength and Pi is the decay rate of state aiunii iiUiiibEi 7 nLithiiJlElE rni iieutiai aici The ThomasReichKuhn sum rule states that slide 22 nghl Matt inieiaeiinn Fall 2mm 555 OSE5312 Fall 2009 Class 05 Lorentz model Real atoms quantum mechanical picture identifying number of the level Idealized case purely radiative relaxation decay rate F units s4 is related to the FWHM of an emission line fast decay high damping gt broad resonance broad emission spectrum In molecules solutions solids broadened absorption bands Energy levels broaden due to vibrational and rotational states interactions between electrons on neighboring atoms band structure Doppler shifts gt broad emission absorption line is not always due to fast decay Light Matter Interaction Fall 2009 Class 5 slide 23 Real world example a KCI crystal R03 05 Reflectance o JlJl Fig 36 The spectral dependence of the re ectance of KCI The region of transparency extends to about 7 eV Above 7 eV there are a number of sharp peaks related to narrow energy Reflectance for single transition one broad reflection band Reflectance for polar solid KCI Below 7 eV transmissive and normal dispersion Higher energy several transitions High resonance frequencies KCI looks transparent at visible frequencies bands and excitons From H R Philipp and H Ehrenreich Phys Rev 131 2016 1963 4 Light Matter Interaction Fall 2009 Class 5 slide 24 OSE5312 Fall 2009 Class 05 Lorentz model Isolated absorbers in a transparent host material visible H333 Absorption Vmus wxvcbcngll 2 P S a 21 r Zhoxt a2i1 a 2 mo Light Matter Interaction Fall 2009 Class 5 glmu Mammal EDrl mimic 63 mm Conse uentll near a single resonance the dielectric function is a I roximatel or 84w 800 where a is known as the high frequency dielectric constant high well above 030 At zero frequency this gives a modified static dielectric constant SS 2 800 Example Nd3 ions in glass 2 Distinct absorption lines within host glass transmissive region Near a single resonance we can describe P as a superposition Bot 131103 Pdopan a E 8 in 0 0 002 02 sz with Xhost assumed constant 2 0 19 mg a2 iFa 2 a 0 slide 25 Summary 1 of2 Lorentz model predicts electron amplitude And the corresponding dipole moment Resulting in Lorentz susceptibility Calculate oc Calculate R Approximate transmission T1R2 equotXZ With only 030 and N you can predict nO Light Matter Interaction Fall 2009 Class 5 With the resulting dielectric function sr1x you can 2 e r a m 002 02 in 2Ea ma02 a2 1Ta Zltwgt Nej 1 2 2 60 m mo 0 1Fa mm quot39l slide 26 OSE5312 Fall 2009 Class 05 Lorentz model mmary 2 of2 Considering dopants in a host material with constant refractive index nm we can estimate the optical properties by considering 1m 1m mm 5mm 39 1w 1mm CAREFUL 5mm a 9m Sdupanl am mm M5 Imam dopnvt e N With new 66 2 an m 150 de allwa See homework for example questions Lighl Matter lrllelatllml Fall zuuar Class slide 27 Next lecture Extension of Lorentz model consider case in which spring constant depends on amplitude Dependence of restoring force on displacement no longer linear 2 Nonlinear response Gives rise to second harmonic generation optical recti cation sum fre uenc eneration difference 39equency generation nonlinear optical re 39action nonlinear optical absorption eg two photon absorption 5MB 23 nghl Mann lnlevacliun Fall 2mm OSE5312 Fall 2009 Class 05 Lorentz model

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