V. Light amplification & Spontaneous emission

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Anis Kathleen Gibson
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1 V. Lgh mplfon & Sponnous msson nrgy Lsrs r bsd on onnous msson nd lgh mplfon, hh r nds of qunum phnomnon. Ths hpr qunum mhnlly dsrbs lgh mplfon. nrgy lvl of n om A mr s omposd of oms, nd n om s omposd of nulus nd lrons s llusrd blo. lron Possbl orbs hr lrons xs r dsr. Th nrgy of n om s dsr. om nulus dsn orb: Th ponl nrgy s hgh. los orb: Th ponl nrgy s lo. In qunum-mhnl rms, gnvlus of h nrgy opror of n om r dsr. nrgy gnvlu dsr shm mg of nrgy ss hgh nrgy s gns orrondng o n lron orb. lo nrgy s Inron bn lgh nd mr bsr om # Whn lghv s nd no mr sysm, undr ondon of h phoon nrgy bng qul o h nrgy dffrn bn o nrgy-lvls n h mr, nron or nrgy xhng bn lgh nd h mr ffnly ours, hh s horlly dsrbd n h follong sons. Trnson from lo nrgy lvl o hgh nrgy lvl hgh nrgy lvl o lo nrgy lvl lgh bsorpon lgh msson

2 Sm-lssl hory nd full-qunum hory Thr r o pprohs o dsrb h lgh-om nron; Sm-lssl hory: n om s qunum, nd lgh s lssl. Full-qunum hory: n om nd lgh r boh qunum. Th sm-lssl hory s onvnn o undrsnd h rsonn nron. Th full-qunum hory xly dsrbs h nron. L us sr h h sm-lssl hory o s n nuv pur of h lgh-om nron. Moon quon of n om n h sm-lssl hory Th om s s hngs hl nrng h lgh. In qunum mhns, h s voluon s dsrbd by h Shrödngr quon. W ll s ho n om volvs hrough h nron, usng h Shrödngr quon. Th Shrödngr quon: d d : nrgy opror of n om mlonn : qunum s of n om Ĥ Frs, xprss h om s s s lnr ombnon of nrgy gnss of h nrgy opror of h om. hr > s n gns of h nrgy opror of h om. Ĥ phs voluon hou h nron Chp. probbly mplud possbly vryng hrough h nron : nrgy gnvlu Th ol nrgy opror hn h om nrs h lgh nluds h nron nrgy opror n ddon. Ĥ n n s h nrgy opror for h om-lgh nron, hh s drvd blo. W suppos h dpol nr, hos lssl nrgy s V n P r D P r D r : lron hrg r : poson of h -h lron lgh fld - + polrzon P

3 Th sm-lssl nron nrgy opror s obnd by rplng D h n opror hl subsung h lssl monohrom lgh for. * n D 3 An quon dsrbng h om s bhvor s obnd from h Shrödngr quon h h bov xprssons s: d d d d n T h nnr produ h d d d d n d n d n n * D m m D D Rsonn rnson W ll solv h bov dffrnl quon, ssumng h s lmos onsn n shor m.., n frs-ordr pproxmon. d d * D r, ssum h h om s nlly on of h gns >, no suprposon s. for Thn, h soluon s obnd s d * D xp[ d * D{ xp[ xp[ } xp[ * xp[ D{ } s lrg hn h dnomnor s zro.

4 Th frs rm s lrg hn - = Th om rnss from h nl s o lor nrgy s h n nrgy dffrn of, hh quls o h phoon nrgy of h ndn lgh. W r no onsdrng losd sysm. Thus, h nrgy h h om loss should go somhr.., nrgy onsrvon. Th ndd s h lgh, hh s rsonbl sn h nrgy los by h om quls o h phoon nrgy. Ths onsdron suggss h h om rnss from h nl s o lor nrgy s hl h los nrgy urns o b phoon myb. > Th sond rm s lrg hn = Th om rnss from h nl s o hghr nrgy s h n nrgy dffrn of, hh quls o h phoon nrgy of h ndn lgh. Consdron smlr o s suggss h h om rnss from h nl s o hghr nrgy s, obnng h nrgy from phoon myb. > > > Rsonn pross: Inron ours mos ffnly hn h phoon nrgy s ondn h n nrgy dffrn of n om. Th probbly of h s rnson s smd s follos. Frs, ssum ~. Thn, frs rm >> sond rm D * xp[.., msson Th probbly of h s bng h -h nrgy s m s D sn [ hh s quvln o h rnson probbly from h -h s o h -h s sn h om s nlly h -h s. Ths onsdron suggss h h rnson probbly s proporonl o h lgh nnsy.

5 In summry, - Trnson ours ffnly bn nrgy ss hos nrgy dffrn quls o h phoon nrgy. An om rnss from n uppr nrgy s o lor on lor nrgy s o n uppr on - Th rnson probbly s proporonl o h lgh nnsy no rnson hn no lgh no onnous rnson lgh msson myb lgh bsorpon myb 5 - Th rnson probbly from h uppr o h lor nd h from h lor o h uppr r qul. Full qunum hory Th prvous son suggss h rsonn nron,.., n om nd lgh ffnly nr h h ohr hn h phoon nrgy quls o n nrgy dffrn of h om sysm. ovr, hr r som ssus h r no lrfd: - Phoon msson nd bsorpon r suggsd by rumsnl vdn, no by dr drvon. - Sponnous msson s no drvd. - Ohr nformon rld o lgh mplfon r mssd suh s lgh mplud, mplud fluuon, phoon numbr fluuon In ordr o fully undrsnd lgh mplfon, nd h full-qunum hory, hh s dsrbd n hs son. r, h lgh s qunzd s ll s oms. Th nron nrgy opror s xprssd s: hr n g Th subsrp rprsns h -h om. : h om s rnson opror from h uppr s > o h lor s > : h om s rnson opror from h lor s > o h uppr s > A phoon s bsorbd. An om s xd. > A phoon s md. An om s rlxd. > > > Th nron nrgy opror n h full qunum hory s gvn by n g Th subsrp rprsns h -h om. hr : h om s rnson opror from h uppr s > o h lor s > : h om s rnson opror from h lor s > o h uppr s >

6 A phoon s bsorbd. An om s xd. > A phoon s md. An om s rlxd. > 6 > > In h follong, h snbrg pur s mployd. Th snbrg pur: m-dpndn opror s drvd frs, nd hn s xpon vlu h r o n nl s s luld. A xp A xp A : m-dpndn opror A follos h snbrg quon of moon: d d A [, A { A A } Ĥ : mlonn nrgy opror Th nrgy opror n h prsn sysm onsss of h orgnl nrgs nd h nron nrgy: lgh oms nron : proporonl onsn Abou An om s s suprposon of > nd >. * * * * A qunum s s normlzd.

7 7 Applyng h dny oprors o h nrgy opror.,, ' ' ',, ' ' ' W rgrd s rfrn of h nrgy vlu, nd ssum =. = - An nsmbl of oms: hr for h om dny opror d d, [ snbrg quon for lgh:, [, [ } {, [ } { } { snbrg quon for n om: d d, [, [ } {, [ } {

8 8 d d xp[ xp[ Th bov dffrnl quons r smplfd by vrbl rnsformon s: d d W ll solv hs quons by h mhod of sussv pproxmon. d d d d : nl ss : frs-ordr soluons,, g d d d g d d d d d d d : sond-ordr soluons, d d d d

9 os[ 9 P P os[ W rgrd hs sond-ordr soluon s h m voluon durng shor m: Th vrgd vlu of physl quny s gvn by h nnr produ h n nl s. W ll s h vrgs of mplud: mplud fluuon: phoon numbr: n x x x In ordr o vlu h vrgd vlu, nd h nl s >. rl pr r, ssum h h nl om sysm nluds N oms h lor nrgy s nd N oms h uppr nrgy s. Thn, h nl s s xprssd s N r { } r oms lgh [Amplud { P} r r P r r

10 os[ P os[ N N os[ [ sn os[ d [ sn nrgy lvl s dnsly dsrbud so h rgrd s onnuous vrbl. [ sn d : dsrbuon dnsy of s rgrdd s onsn, omprd h h vron of sn[[. s lso rgrdd dnl for h. N N N N d [ sn N : Th numbr of oms h lor s round = N : Th numbr of oms h lor s round =..,

11 P { N N } { N N } xp[{ N N } << g g N N [Phoon numbr n { P }{ P} { P P r r P r r P r r P P P P} { } os[ N N { ghr-ordr rms r ngld. } { } P P P P

12 N os[ { os[ } n n n N N N n n N N N h rm n b nrprd s follos: n : Indn phoon numbr n N N n : Phoon-numbr nrs proporonl o n nd N : Phoon-numbr drs proporonl o n nd N smuld msson bsorpon N : Phoon-numbr nrs proporonl o N onnous msson > > > > smuld msson bsorpon > > onnous msson W furhr smplfy h oupu s: n n g n g g g n n npu sgnl g n n g gn onnous msson g N N N N N N N N N g n N N N N n N N N g populon nvrson prmr nos for

13 [Amplud fluuon Fluuon of h lgh mplud n b vlud by h vrns of h rl nd mgnry prs of h nnhlon opror Chp. IV. x x x x x x x x x x x x rl pr mgnry pr r, nrodu modfd mplud opror nd rdfn h rl nd mgnry pr oprors for smplfyng h drvon: b x b b x x x b x b b x x x g g x b b { } { } { } x x { b b }{ b b } { b b b b b b b b } 3 g b b b { [ r r r r { P P } {} P}{ PP { g } P} g {} {} { N N } g { P P } P P b { r r P r r P }{ r r P P P g g N N g P P P P N n g g P}

14 b b x { }{ P P } r r r r P r r P PP g g N N b b { {} { {} { {} { { x } P }{ { P } [ r { } r P r r P P P { r r g } g } g g } n { n g g { x x x g { } N g { } g g { x } n x g g { { x } x } n g g x n } g g } } g g { N N N } N { g } N N N N N N N N N N n N N g g g g } nl vrn ddonl vrn Opl mplfr Th prvous sons dsuss h lgh-oms nron n shor m <<. In n opl mplfr, hovr, lghv nrs squnlly h oms long h mplfr lngh, nd h shor-m pproxmon nno b ppld. Th mplfon proprs n n opl mplfr n b onsdrd s n hs son.

15 For h dsusson, dvd h mplfr lngh o shor sgmns hn hh h shor-m pproxmon s pplbl. 5 IN OUT Lgh psss hrough h sgmn hl hngng s s. Th om ondon s ssumd o b unform ovr h mplfr lngh. Thn, h s hng durng shor m s rnsld o h s hng from h h o + h sgmns s: g n n n g g G n n G n G g G g g x x n x x G n G : mn mplud h -h sgmn n : mn phoon numbr h -h sgmn : h rns m hrough on sgmn Th oupu s s obnd by squnlly pplyng h bov hng from h frs o ls sgmns. 3 3 M M G M G M G M 3 G n M n M G n M G { nm G n G } G n G nm G n G G 3 { G nm 3 G n} G G n G nm 3G G n G G M M n G G n G G G M M G M M G n ng G G n G n x M x M G n G G G G { G x M n } n G G x M G n G G G { G x M 3 n } G n 3 G G x M 3 G G n M M G G G x n G G G x n

16 In summry, 6 mplud phoon numbr ou G n M nou Gnn G n M G G : h mplfr gn mplfd phoon onnous msson mplud fluuon G x ou G x n n mplfd nos ddonl nos Th bov mplfon pross n b llusrd n h omplx mplud onsllon s Im[ npu s R[ r, h npu lgh s supposd o b pur ohrn s. In hs s, h mplud vrns of h npu lgh r: Thn, h mplud vrns of h oupu lgh r: x n x n G x ou x ou G n G n vr. of ohrn s ddonl vrn Ths xprssons n b nrprd s: h mplfd lgh s omposd of ohrn lgh nd onnously md lgh. Im[ ohrn lgh Im[ G + onnous lgh R[ R[

17 A ohrn s orronds o nos-fr lgh n h lssl orld. Thus, h bov onsdron suggss h h rnsfr funon of n opl mplfr n b lsslly xprssd s: 7 ou G n mplfd sgnl lgh < > = onnously md lgh or mplfd onnous msson: AS hr h mn vlu nd h vrn of AS lgh r gvn by {R[ } {Im[ } G n hf f hf s mulpld bus h nrgy un n h bov dsusson s on phoon. f s mulpld bus h bov dsusson s for on rsonn mod. hf : on-phoon nrgy f : frquny bnddh of AS r, h ol vrn of h AS lgh xly qul o h mn por of h AS lgh. Rllng h h vrn quls o h por for n ddv Gussn nos, hs propry of h AS suggss h h dsrbuon profl of h AS lgh mplud s Gussn. In h bov, mplud fluuon hrough n opl mplfr s dsussd. Phoon numbr fluuon.., nnsy fluuon n b lso dsussd h smlr lulons. ovr, h drvon s rhr ompld, nd us sho rsuls blo. n ou nou nou ou ou Gn n G n G G n n G n G { n n n nn } n n sgnl sho nos onnous sho nos sgnl-onnous b nos mplfd xss nos onnous-onnous b nos Nos Fgur NF Th nos prformn of n mplfr s usully ndd by nos fgur NF. W dsrb NF of n opl mplfr n rms of qunum mhns n hs son. Th nos fgur s gnrlly dfnd s: SNR NF SNR n ou S N S N n ou SNR n hr SNR: Sgnl-o-Nos Ro, S: Sgnl por, N: Nos por SNR ou Trdonlly, SNR s vlud n h phoo-urrn gnrng from dr lghv don. = lgh nnsy = phoon numbr

18 In our suon, S = mn phoon numbr = n N = h vrn of h phoon numbr = n = n n 8 From h prvous rsuls, S ou n ou { Gnn G n} N ou n Gn n G n G G nnn G n G { n n nn nn} Usully, {n >> n, G >> }, nd hn h bov xprssons r pproxmd s: S G N ou nn G G n n G n n ou n n.., sgnl-onnous b nos s domnn Thn, SNR ou G G n nn n n On h ohr hnd, SNR n nno b xplly xprssd, bus dpnds on h npu lgh s. In ordr o nd h nrns nos propry of mplfrs, h npu lgh s ssumd o b nos fr,.., ohrn s. Thn,. Sn n n n N n n n n n SNR n n n n n From h bov quons, SNR NF SNR n n n ou nn n n n n n n n n n n n n n Usully,. db-rprsnon s usd for h nos fgur; NF db log NF logn By h y, n n h bov xprsson s: Thus, N n N N N N NF 3 db n n N : h numbr of oms h uppr nrgy s N : h numbr of oms h lor nrgy s NF = 3dB s h qunum-lmd nos fgur of mplfrs. Any mplfr bsd on lgh-om nron nno sho br nos prformn hn NF = 3dB.

Chapter 1: Review of Quantum Mechanics. Postulates of Quantum Mechanics: 1-3

Chapter 1: Review of Quantum Mechanics. Postulates of Quantum Mechanics: 1-3 Chr : Rw of Qunum Mhns In hs lur you wll lrn..ll h you mgh h forgon: Posuls of qunum mhns Commuon rlons Shrongr n snrg urs Tm lomn Dnsy orors n nsy mrs Dohrn n qunum mhns C 47 Srng 9 Frhn Rn Cornll nrsy