UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd LECTURE NOTES 7.5 Disprsi: Th Frqucy-Dpdc th Elctric Prmittivity ad th Elctric Suscptibility Ovr th tir EM rqucy itrval 0 Hz, th spd prpagati v prp mchrmatic (i.. sigl-rqucy) EM wavs i mattr is t t cstat, t idpdt rqucy: vprp cstat; crqucy, vprp, bcaus mattr - at th micrscpic scal - is cmpsit - cmprisd atms/mlculs which hav rsacs i rgy/rgy lvls which ar gvrd by th laws quatum mchaics Th rqucy-dpdc th wavlgth, r wavumbr k, ad liar mmtum p assciatd with macrscpic EM wavs prpagatig i a disprsiv mdium ariss rm th rqucy-dpdc th macrscpic lctric prmittivity (r quivaltly th lctric suscptibility sic:. Th rqucy-dpdc th macrscpic lctric prmittivity is kw as disprsi; a mdium that has c is kw as a disprsiv mdium. Fr -magtic/-cductig liar/hmgus/istrpic mdia, th id rracti. Thus, i: th:. Fr a wav packt (= a grup {= suprpsiti/liar cmbiati} wavs may rqucis as plaid by Mssr. Furir), th vlp th wav packt travls with (i gral, rqucy-dpdt) grup spd = spd at which rgy i th wav lws: A prpagatig wav packt: kz t v g " " d dk dk dk d d v v g k dk d A spac-pit z t th wavrm mvs - with cstat phas kz t with (i gral, rqucy-dpdt) phas spd v k. Hc: zt, k v t. Nt that {i gral}: v. g v Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd I v k is rqucy-dpdt, ad vg dk d v {.g. as i v v } th rlatiship btw v v th cas r surac wavs watr, whr g ad g dpds th dtaild physics th mdium (as w shall s s... ). Nt that i crtai circumstacs, v ca cd c {= spd light i th vacuum} but i ths situatis, rgy (ad/r irmati) is trasmittd at supr-lumial spds rgy/irmati is trasmittd at v g < c always, by causality A physical/mchaical ampl: calculat th phas spd th itrscti pit th scissrs tw halvs a scissrs as th blads th scissrs ar clsd. {Aswr:!!!} Disprsi Phma i Liar Dilctrics I a -cductig, liar, hmgus, istrpic mdium thr ar r lctrs (i.. r r 0 ). Atmic lctrs ar prmatly bud t ucli atms cmprisig th mdium. prrtial dircti / prrtial dirctis i such a {istrpic} mdium. Supps ach atmic lctr (charg ) i a dilctric is displacd by a small distac r rm its quilibrium psiti,.g. by applicati a static lctric ild Er rˆ dircti. Th rsultig macrscpic lctric plarizati (aka lctric dipl mmt pr uit vlum) is: b b 3 r pr whr: = bud atmic lctr umbr dsity # m ad th {iducd} atmic/mlcular lctric dipl mmt is: p rr {hr}, whr r is th {vctr} displacmt th atmic lctr rm its quilibrium { r = 0} psiti. b b r p r r Thus: Th atmic lctrs ar ach lastically bud t thir quilibrium psitis with a rc k N m. Th rc quati r ach atmic lctr is thus: FrErkr. r E r k. b b b b E r r p r r E r k k cstat Hc: Th static plarizati is thrr giv by: Hwvr, i th E -ild.g. varis harmically with tim, i..,; v ikz t E E r t E ˆ du t a mchrmatic EM pla wav icidt a atm, th abv rlati is icrrct! A mr crrct {but smi-classical } apprach t trat this situati is t csidr th bud atmic lctrs as classical, dampd, rcd harmic scillatrs (driv by th icidt lctric ild), as mathmatically dscribd by th llwig dirtial quati: mr t m r t kr t E r t ihmgus d -rdr dirtial q. ; ; ;, ; Th dampig cstat rads sc rprsts th ct EM r-radiati by th atm {hr}. Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd r t; rt; m m kr t E r t t t ma ;, ;.b. w hav glctd th v B E trm hr... Vlcity-dpdt dampig trm dampig cstat Pttial Frc (bidig atmic lctrs t atm) m Drivig Frc lctr mass 9. 0 3 kg Supps th drivig/rcig trm varis siusidally/is harmic/pridic with agular it F r,; t E r,; t E i t r bcaus E r,; t E rˆ. rqucy, i.. ˆ.b. Th lctric ild E is cmpl E ad pla-plarizd i th ˆr -dircti. i t Th ihmgus rc quati bcms: mr ˆ m r kr E r with cmpl tim-dmai vctr displacmt amplitud: r t; r t; ˆ r. I th stady stat, w hav: Sic rt; i t mr m r kr E rˆ physically rprsts th cmpl vctr spatial displacmt a atmic i t r t; r t; rˆr rˆ lctr rm its quilibrium { r 0 } psiti, th: Thus: i t mr m r kr E rˆ r t; rt; m m kr t; Er, t; t t m r it imr i t kr i t E i t m k i m r E charactristic quati k Divid this quati thrugh by m : i r E m m k k Di: 0 r: 0 = charactristic/atural rsac {agular} rqucy. Th: m m 0 ir E Bud atmic lctr r: r E cmpl rqucydmai spatial m m 0 i displacmt amplitud Nt that th cmplss rqucy-dmai r is i th dmiatr. W ca mv it t th umratr usig th llwig stadard trick /prcdur: iy iy y I: z whr: z ad: m z iy iy iy y y y Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 3
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Thus: r E E 0 i m m 0 i 0 i 0 i m E 0 0 i 0 m E 0 0 i 0 i i m E 0 i 0 i i 0 * * i E i E Nt that: E E E. Hwvr, w ca always rtat away /absrb th phas E E 0.g. by a glbal rdiiti th zr tim, r a glbal rdiiti/traslati ur crdiat systm. Hc, w ca quivaltly writ, withut ay lss grality: 0 i r E r ir r ral m 0 imag i Th {rqucy-dpdt} ral part th rqucy-dmai displacmt amplitud r it i.. th cmpt r that is i-phas with th drivig rc E r,; t E rˆ ral 0 r r E m 0 r is: Th {rqucy-dpdt} imagiary part th rqucy-dmai displacmt amplitud r i.. th cmpt r that is 90 it -ut--phas with th drivig rc E r,; t E rˆ imag r mr E m 0 is: is: Th {rqucy-dpdt} phas th rqucy-dmai displacmt amplitud r imag mr r r ta ta ta ral r r 0 k Wh: 0, r 0 r lags E. Wh: k 0, r 0 r lads E. m m 4 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd is: Th {rqucy-dpdt} magitud th rqucy-dmai displacmt amplitud r * ral imag 0 i 0 0 m r r r r r i E 0 E m 0 E 0 m Th 8-igur plt blw shws th bhavir a mchaical rsac at 30 KHz, ad FWHM 30 Hz i th viciity th rsac r uit amplitud m E. Th rsac rqucy ad l/hi rqucis that di th FWHM, } ar als idicatd i ach plt: {i.. Nt that th s-calld Q {this} rsac {hr} is: Q 000. Fr small Q ad that dampig, it ca b shw that 0, 0 r 0 ta 90, r ta 35 ad r ta 45. Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 5
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Th 3-D plt blw shws th CW path tak i th cmpl pla i passig thrugh th rsac: FWHM, = Nw:,; b ; b i rt r t r t rˆ whr: r Thus:,; 0 i E m 0 i i rt E r E rˆ b 0 it b 0 ˆ it m m 0 0 it Sic th cmpl tim-dmai rt,; r; rˆ, w s that th cmpl rqucydmai vctr amplitud is: ; I th static limit (i.. 0 ): b 0 i r ˆ E r. m 0 b b b m m r; 0 E ˆ ˆ ˆ r Er Er 0 k k m Static plarizati 0 is i-phas with E 6 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Nt that th phas th rqucy-dmai vctr amplitud r; lags bhid (r lads) E ta by a rqucy-dpdt phas agl : mr, ta r, 0 is rqucy dpdt, k Wh: 0, 0 lags E. Wh: k 0, 0 lads E. m Frm th abv rmula, t that i th dampig cstat 0, th 0, th plarizati, bcaus i 0 m r, t 0, is always i-phas with E i.. th plarizati (FWHM) Hz m, th is purly ral! A dampig cstat 0 als mas that th width th atmic/mlcular rsac is iiitly arrw, ad thus thr ar dissipativ prcsss (i.. rgy lss mchaisms) prst at th micrscpic atmic/mlcular lvl i this macrscpic mdium! Nt als that has physical/si uits radias/scd. Nt urthr that E i th abv prssi is actually E it th itral macrscpic lctric ild th dilctric: E Eit Et EP, th sum th macrscpic tral applid lctric ild ad th macrscpic lctric ild du t th plarizati th dilctric mdium. Th lctric ild du t plarizati th mdium is: Thus: E Eit Et 3 Thrr: E P 3 b 0 i k Et 0 m 3 0 m Nw slv r : Skippig writig ut sm {tdius} cmpl algbra, w btai: b i E m t whr: b 0 0 = 3 m Nt that this rmula is sstially idtical.g. t th {cmpl} displacmt amplitud rmula r a driv harmic scillatr, ad/r that r th {cmpl} AC vltag amplitud i a LCR circuit, ad r may thr physical systms hibitig a {dampd} rsac-typ bhavir. Nw i E t dilctric mdium: E -ild assciatd with a mchrmatic pla EM wav prpagatig i a ikz t Et z, t E, th bcaus th liar rlatiship btw th ikz t E t z, t E, Gauss Law bcms (sic 0 r r ): E 0 plarizati ad.g. ˆ t bud.b. Th dampig cstat has th sam uits as : radias/sc S P435 Lct. Nts 0, p. -6. S als P435 Lct. Nts 9, p. 6. ctiv agular rsac rqucy bud atmic lctrs Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 7
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Th wav quati r a dilctric mdium with r 0 ad J 0 r r bcms: Or: Et b Et Et c t t m t i with: c i b Et Et c m t b with: 0 3 m Th gral sluti t this disprsiv wav quati is th rm: E r,; t E t ikz t with cmpl k ki ad: k c i. b m Thus, w als s that hr {agai} th cmpl wavumbr k ki is plicitly k k i. dpdt th agular rqucy, i.. W urthr s that mchrmatic pla EM wavs prpagatig i a disprsiv dilctric ikz t z ikz t E r,; t E E, mdium ar ptially attuatd, bcaus: i.. th m k t trm crrspds t absrpti/dissipati i th macrscpic dilctric, ad is physically rlatd t/is prprtial t th dampig cstat. Nt that w als hav: rt,, Et rt,,, thus th suscptibility als cmpl, ad rqucy-dpdt: i. Th m {hr} is trm crrspds t absrpti/dissipati i th dilctric, ad is physically rlatd t/is prprtial t th dampig cstat. Th crrspdig dissipativ rgy lsss at th micrscpic, atmic/ mlcular lvl i th dilctric ultimatly wid up as hat! Sic: b i r,; t Et r,; t E,; t r t m b i E, ; t r t m b whr: 0 3 m 8 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd W s that th cmpl suscptibility assciatd with a sigl rsac is: Hc: Ad: i b m i b m b m m Nw br w g much urthr with this, w d t discuss athr aspct ur mdl amly that i mst liar dilctric matrials, th atms cmprisig th matrial ar multilctr atms, ad csqutly thr ar may dirt bidig rgis th utr shll atmic lctrs ar wakly bud, hc hav small k, ad thus small 0 k m, whras th ir-shll lctrs ar much mr tightly bud, hc hav largr k, largr 0 k m. Furthrmr, i cmpl mdia, i.. dilctrics with mr tha kid atm, lctrs ca b shard btw atms i.. thy ar bud t mlculs.g. th π-lctrs i bz rig / armatic hydrcarb-typ cmpuds, which ca b wakly bud i sm mlculs. Thus, thr ca b als b {mlcular} rsacs.g. i th micrwav ad ira-rd rgis th EM spctrum atmic rsacs ar typically i th ptical ad UV rgis {r th utr-mst shll lctrs}, as wll as i th ar UV ad -ray rgis {r th irshll lctrs}! Allwig r all such rsacs, w ca writ th {cmpl} lctric plarizati as a summati vr all th rsacs prst i th liar dilctric as llws: whr: ad whr: Physically: b i sc rt,; E,; t rt m b 3 m 0 sc ad: 0 k m scillatr strgth th rsac, did such that: sc = ractial strgth th th rsac ad = width th th rsac. sc Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 9
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd is: Thus, w s that th multi-rsac cmpl lctric suscptibility i i b sc m i Hc: b sc m Ad: b sc m m Th cmpl lctric prmittivity i dilctric mdium is: a disprsiv, liar b i sc m with th rlatis: ad: b sc i m m m b sc. m i 0 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Mchrmatic pla EM wav slutis t th disprsiv wav quati ar th rm: E r,, t E Thus:,, ikz t with cmpl wavumbr k k i z E r t E E ikz t ikz t t ptial dampig EM wav r a -magtic mdium, i... is quivalt Itrducig a {rqucy-dpdt} cmpl wavumbr kki t itrducig a {rqucy-dpdt} cmpl id rracti i. Fr a disprsiv dilctric, th cmpl id rracti ad cmpl wavumbr ar {simply} rlatd t ach thr by: k c k i i i c k c c c ad c Fr a -magtic mdium ( ), th cmpl id rracti is rlatd t th cmpl lctric prmittivity suscptibility via th rlati: Squarig bth sids: But: ad thus th cmpl lctric, k b i sc m b i sc c m k i k ik Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd c th: Sic: k b c i sc k m i i Equatig th ral ad imagiary parts th LHS & RHS th abv quati, w btai: sc b m b sc m quatis ad ukws: & slv r & First di: sc b m sc b m (.b. 0, is always psitiv {hr}) Th: ad Thus: 4 0 4 Or: Di:. W tmprarily supprss th Th: 0 multiply quati thrugh by a b c -dpdc i th llwig: 0 with: Th slutis / rts this quadratic quati ar th gral rm:.b. This may lk lik a quartic quati, but it is actually a quadratic quati!!! a, b, c b b 4ac a Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd 4 i...b. th trm: 0 Must slct +v rt physical gruds, sic 0. Fially, w btai: Cmpl id rracti: i m Whr: sc b m sc b m Obviusly, plicitly writig ut th ull mathmatical rmula r ad is quit tdius but ths ca b rasably-asily cdd up {i.. a cmputr prgram} ad plts vs. ca b btaid. W ca als th btai th llwig: vs. ad c Th cmpl rlatis: i ad k k i c c ad thus: k ad Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.. 3
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Th {rqucy-dpdt} itsity/irradiac Ir, Sr, t; assciatd with a mchrmatic pla EM wav prpagatig i a liar, disprsiv dilctric is als ptially dcrasd by a actr its rigial valu i gig a charactristic distac : z i.. diig: att att = itsity attuati lgth which is ~ aalgus t th ski dpth, r mtals / cductrs. Hwvr, t that is assciatd with th attuati th E ad B - sc ilds, whras attuati cts i itsity/irradiac, I varis as th squar th E -ild: I r S r t E r t,, ;, ; t hc: Ir (, ) E E sc z z I th ptial z-dpdt trm, sic th rgy dsit(is) uem, r,; t I r, S r, t; ar bth prprtial t E i.. bth prprtial t itsity z z, w di th {rqucy-dpdt} absrpti cicit: ad att. Similarly, r th {rqucy-dpdt} cmpl id rracti i w ca als di th {rqucy-dpdt} ticti cicit:. c Sic: c c c thus:. att c c ad. Th absrpti cicit: Th ticti cicit: Typical valus th (ral) id rracti r slids ad liquids ar.3.7 i th visibl light rgi EM spctrum,.g. glass.5, HO.3, plastic.7. Th i: Th: id rracti.5 glass i th visibl light rgi.5.5 O quati & tw ukws: 4.50 ad Thus: Nd athr rlati / idpdt cstrait!! 4 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Nt that glass ds t hav sigiicat absrpti i th visibl light rgi, typical such slid/liquid matrials hav (masurd) absrpti cicits r visibl light i th rag : c 0 0 m Itsity I(z) dcrass t / = = 0.368 iitial I(z = 0) valu atr light travls a distac z = / () ~ 0 00 m i glass. 6 0 m i glass r visibl light, vis 0 radias / sc c S supps: c 0 8 3 0 9 6 0 3 0 Nw: ad.5 r glass i visibl light rag EM spctrum. 9 r: 3 90 3 i th visibl light rag r glass Th: Ca w slv r : 4.50 90. 9 ad: 4.50 4.50 4.50 4.50 9 This has a sluti wh:.5 r: 9 0 Obtaid via umrical mthds usig a cmputr Thus, r.5 r glass i th visibl light rgi th EM spctrum, with.5 9 ad 9 0 (i.. ), as a plicit chck, w s that: 9.5 90.5.5.5.5.5 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 5
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Thus w als s that: sc b m i.. r typical matrials glass, watr, plastic i th visibl light rgi th EM spctrum, 6 0 radias / sc. Whras: b sc m r ths sam matrials glass, watr, plastic i th visibl light rgi th EM spctrum, 6 0 radias / sc. Our rigial quatis wr: ad 9 with:.5 ad 90 r.5 9 with visibl light ad: 30. W w s mr clarly that: r: (r glass) i th visibl light rgi th EM spctrum i i r glass is 9 r glass, i.. th cmpl id rracti.5 90 prdmiatly ral i th visibl light rgi th EM spctrum. Thus, r glass i th visibl light rgi th EM spctrum: sc b.5.5 m ad: b sc 9 0 0m sc b.5 m 9 Nt that ths rsults that w ust btaid r glass i th visibl light rgi th EM spctrum d t hld tru r all rqucis EM wavs {visibl light rgi is i act ly a arrw prti th EM spctrum}!!! I particular, ths rsults d t hld at {r ar} a atmic (r mlcular) rsac! 6 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Lt us csidr a simpliid atmic/mlcular systm, that havig ly a sigl rsac rqucy (i.. a sigl bud-stat quatum rgy lvl), th: r: b 0 3 m b 3m 0 with: 0 k m sc.b. Oscillatr strgth {hr} bcaus hav ly a sigl rsac! Th: sc b m b sc m Th igur th lt (immdiatly blw) shws th bhavir th ral ad imagiary parts vs. } ad th cmpl id rracti a disprsiv, liar mdium, i.. { { vs. } r a sigl atmic rsac. Th igur th right (immdiatly blw) shws th bhavir { vs. } ad th absrpti cicit { vs. } r a sigl atmic rsac. = Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 7
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd.b. Th abv curvs ar classic aturs a cmpl rsac with ctr / rsac rqucy ad = dampig cstat Γ = width (= FWHM) th th rsac. Nt that th width Γ Hz. I th - cmpl pla: (FWHM) 0 = = 0 I th visibl light rgi th EM spctrum, th graph blw shws bth th rqucy ad vs. {dttd li} ad wavlgth bhavir th {ral} id rracti glass, i.. v v vs. {slid li}. Nt that sic prp prp r: vprp { Ågstrm = 0 0 m = 0. m}.b. Mdia which ar vry traspart.g. i th visibl light rgi ar t almst (r ar) paqu i th s-calld amalus disprsi rgi a rsac, R i.. i th FWHM rgi th atmic rsac, whr th ticti cicit bcms vry larg EM wavs ar th rsac rqucy R ar vry rapidly ptially attuatd! Th Gral Bhavir Classic Cmpl Rsac: z iy.b. i sm cmpl systms.g. th rsac a LCR circuit, z & mz ar itrchagd rm what is draw hr! i.. z mz 8 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Nt that th shap th curv r th magitud z, z z z * y is vry similar t shap th mz curv {as shw hr}. Th tractry z i th cmpl pla:, A mr ralistic micrscpic pictur a atmic systm with may lctrs with may quatum bud stats may rsacs i a disprsiv, liar macrscpic dilctric!!! ad Ercis(s): Draw ut th crrspdig tractris cmpl i i r th abv tripl-rsac cass i th cmpl pla! Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 9
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd I th high-rqucy rgi, abv th highst rsat rqucy (typically i UV r -ray rgi), th id rracti is prdictd t b.0 (i.. actually lss tha that th vacuum). Idd, this phm has plicitly b bsrvd/masurd.g. i quartz (SiO) usig -rays: Nt Supprssd Zr! Nt that physically th dampig cstat = width th th rsac is ivrsly rlatd t th litim assciatd with th crrspdig citd stat th cstitut atms/mlculs th disprsiv, liar dilctric, sic at th micrscpic lvl, th {ral} phts assciatd with th mchrmatic pla EM wav hav rgy E h ad {assumig th atms/mlculs th disprsiv, liar dilctric t all b i thir grud stat, with grud stat rgy E }, th i th mchrmatic pla EM wav has {agular} rqucy R R = th rsac rqucy th bud atmic lctrs, th w s that E E E E hr R at rsac! At a rsac,.g. wh, th {ral} phts i th mchrmatic pla EM wav asily stimulat th atmic lctrs, causig thm t rsat th {ral} phts ar absrbd, th ablig th atmic lctr t mak a trasiti rm th grud stat {with rgy E } t th citd stat {with rgy E } via a lctric dipl trasiti, i s allwd by quatummchaical slcti ruls. Th citd atmic stat has {ma} litim th assciatd with it, thus th atmic lctr d-cits back t th grud stat by mittig a {ral} pht this sam rqucy. Th miracl all this is that {ral} phts assciatd with th EM ild ar actually itractig simultausly with all th atms i th disprsiv liar dilctric (withi th chrc lgth th pht c ) at ay giv istat i tim, thus th rsultat scattrd pht that is {ultimatly} mittd, actually must b summd vr th rsps th smbl may atms th miraculus rsult which is rward scattrig th phts assciatd with th macrscpic EM wav, but with a {rqucy-dpdt} phas shit, which is rlatd t th rsac lishap {ad th iit litim } th citd stat th atm! 0 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd At a rsac,.g. wh k, a larg, trasitry/trasit {cmpl ad rqucy- p r, r r r is iducd i th atm, whr: dpdt} lctric dipl mmt ˆ b i sc r r r i m Nt hr w ca als mak a dirct ccti with quatum mchaics th lctric dipl mmt pratr p r, r ad psiti pratr r pratig.g. th grud stat wav ucti th atm/mlcul p r, r r r. r, i.. ad W ca.g. cmput th pctati valu th mdulus squard th lctric dipl r p r, r th atm/mlcul. Isrtig a cmplt st stats mmt r r r r mchaical prdictis r th {squars} th scillatr strgths it this prssi, w ca th btai th quatum sc : * r p r, r r pr, r r p r, r Th trasiti rat (= # atms/mlculs pr scd) rm th grud stat t th citd stat {via a lctric dipl trasiti, as allwd by quatum mchaical slcti ruls} is prprtial t r p r, r, whras th trasiti rat (= # atms/ mlculs th pr scd) rm th citd stat t th grud stat {via a lctric dipl trasiti, as * r p r r. allwd by quatum mchaical slcti ruls} is prprtial t, Nt that by th {micrscpic} maist tim-rvrsal ivariac th lctrmagtic itracti, th trasiti rats ar idtical, i.. = dampig cstat i ur smi-classical mdl! Nt urthr that th litims th citd stats atms ar {ivrsly} rlatd t th widths th pricipl: E t miimum, i.. th rsacs/widths th th citd stats by th Hisbrg ucrtaity, whr h ad h = Plack s cstat. I w st this rlati t its Et th: r: th Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd I stays wll away/ar rm {all} th rsac rqucis bud-stat atmic lctrs, th rsac actr bcms: i i.. ar rm a rsac: Thus, ar rm a rsac / all rsacs, rlativly littl absrpti/dissipati ccurs is prdmiatly ral} ad hc: {i.., such that b sc m Nw: 4 b sc sc 4 m Th: I ad Thus, ar rm a rsac/rsacs: b sc sc 4 m But: c c = vacuum wavlgth, hc:, thus: k w btai Cauchy s Frmula: sc sc b c 4 m B A Whr: A = Cicit Rracti ad: B = Cicit Disprsi. Cmparig th quatis, w s that: b sc A m sc sc ad: B c 4 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd c Sic: i ad/r: k kik ad: k c Th: k c ad:, thus: Th phas spd: v c i k Th grup spd: v g c " " d dk dk d < Nt that at th turig pits ithr th { vs. } r: { k vs. } graphs, i.. at whr th slp dk d 0 v!!! {agular} rqucis ad/r g Nt urthr that i th {agular} rqucy rgi {th amalus disprsi dk d v dk d 0!!! rgi}, sic th slp 0 th th grup spd g {Hc th am amalus disprsi } This phm has b primtally vriid (s.g. C.G.B. Garrtt & D.E. McCumbr, Phys. Rv. A,, p. 305 (970). I th disprsiv mdium is t t thick, a Gaussia puls with a ctral rqucy ar a absrpti li (i.. ar a rsac, ) ad with puls width t prpagats with apprciabl absrpti, but (mr r lss) rtais its shap. R R Th pak th Gaussia puls prpagats at vg v wh th grup spd is gativ!!! Usul r puls r-shapig applicatis - ladig dg is lss attuatd tha trailig dg. R Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 3
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd Ca actually hav th pak a gratly attuatd puls mrg rm th absrbr br th pak th icidt puls trs th absrbr ( diiti gativ grup spd)!!! {i.. micrscpically, i th absrbr is t t thick, th sm phts ca mak it all th way thrugh th absrbr w/ itractig at all this prbability is ptially supprssd. Has applicatis/uss.g. i ptical mammgraphy/brast cacr scrig r wm...} S.g. J.D. Jacks s Elctrdyamics, 3 rd Editi, pags 35-6 r mr dtails! Fially, i w st 0, th w btai th static (i.. zr-rqucy) limit {all } ths quatitis. Nt that thy als {all} bcm purly ral i this limit: Static Plarizati: b (0) m sc whr b 0 3 m ad sic Et Static Elctricity Suscptibility: b sc (0) ad 0 m k m Static Id Rracti: 0 0 (0) K (0) But: K 0 0 Static Dilctric Cstat: K b sc m But: 0 0 b sc ad thus: 0 0 0 0 0 K 0 0 ad: K m 0 0 Nt that th static dilctric cstat {as masurd at = 0 Hz/DC} is b sc m K 0.0 bcaus it ctais irmati abut all th {quatum mchaical} rsacs/citd stats 0 89 prst i th disprsiv, liar mdium, v it th -ray rgi at 0 Hz ad byd!!! Equivaltly, armd w with this kwldg th micrscpic bhavir a disprsiv, liar mdium, a lctric suscptibility 0 > 0 {r quivaltly, a dilctric cstat K 0 >} istatly tlls us that thr ar idd {quatum mchaical} rsacs/citd stats prst i th {cmpsit} atms/mlculs that mak up th macrscpic matrial th disprsiv, liar mdium!!! 4 Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd.
UIUC Physics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 7.5 Pr. Stv Errd A wdrul macrscpic ampl disprsi i atur is th raibw. At th micrscpic lvl, th rqucy-dpdc th id rracti light () ariss as a csquc th rsat bhavir quatum mchaical bud stats lctrs i th atms th watr mlcul (HO) rspdig t EM light wavs{= visibl light phts} cmig rm ur su. I such cmpsit bhavir istd at th micrscpic lvl, thr wuld b raibws t y i th macrscpic vryday wrld! Eprimtal masurmts th absrpti cicit pur watr: Visibl Light Vilt Rd UV Rgi IR Rgi Absrpti cicit pur watr - HO is larg i UV ad IR rgis du t rsacs! Prssr Stv Errd, Dpartmt Physics, Uivrsity Illiis at Urbaa-Champaig, Illiis 005-05. All Rights Rsrvd. 5