2.2.C Analogy between electronic excitations in an atom and the mechanical motion of a forced harmonic oscillator"

Transcription

1 ..C Analgy btwn lctrnic xcitatins in an atm and th mchanical mtin f a frcd harmnic scillatr" Hw t chs th valu f th crrspnding spring cnstant k? Rsnant Absrptin Mchanical rsnanc W idntify th mchanical rsnanc frquncy with th ptical rsnanc frquncy ω. Lt's assum w knw th particular transitin nrgy ΔE. Thrfr w knw ω, als calld ω rs ptical mchanical

2 Rsnant Absrptin Hw t chs th valu f th damping cnstant "b"? Light Chargd mchanical scattring scillatr Input nrgy Atm drivn by xtrnal radiatin f frquncy ω f Input nrgy ω f ω, k R-mittd light Lss f nrgy ω f

3 Th missin f lctrmagntic wavs by th acclratd charg cnstituts th channl f dissipativ nrgy ω f Extrnal xcitatin f driving frquncy ω f ω, k Th spcific valu f th lctrmagntic damping paramtr "b" (in trms f th atmic physical cnstants) is prvidd in vry much ail in sctin IV f ths Applid Optics nts. Hr, in th nxt sctin, w prvid a summarizd vrsin.

4 A. La Rsa, Applid Optics Lctur Nts Prtland Stat Univrsity Elctrmagntic radiatin damping Mdling ptical xcitatin with a simpl harmnic scillatr 1. Emissin f radiatin by an acclratd charg. Emissin f radiatin by charg undrging harmnic scillatins 3. Mdling ptical xcitatin with a simpl harmnic scillatr Rf: Fynman Lcturs Vl-1, Chaptr 3 1. Emissin f radiatin by an acclratd charg Elctrmagntic radiatin is mittd by chargs in acclratd mtin. Acclratin a q r E Th lctric fild E, at a distanc r frm th charg, and at an angl frm th axis f th charg mtin, is prpndicular t th lin f sight. Fig. 1 A charg q that mvs with acclratin a mits lctrmagntic radiatin. q sin( ) 1 E( r, t) a( t r / c ) 4 c r Rtardatin tim Elctric fild prducd by an acclratd charg whr a is th acclratin f th particl, th angl btwn th vctr acclratin a and th vctr psitin r. c is th spd f light () 1

5 Travling lctrmagntic nrgy: Th Pynting vctr E Its magnitud S givs th amunt f nrgy pr squar mtr pr unit tim that passs thrugh a surfac that is nrmal t th dirctin f prpagatin. (3) Radiatin pwr P mittd by an acclratd charg a q r E P S [ da Intgral vr a sphr f radius r q sin ( ) c (4 c ) 1 r a' ] da Fig. Elctrmagntic pwr radiatd by an acclratd charg. P q 6 c 3 a' whr a a( t )= a( t - r/c ) q 6ππ c P 3 a' Elctrmagntic pwr radiatd by an acclratd charg (4). Emissin f radiatin by charg undrging harmnic scillatins Cnsidr a charg q attachd t a mchanical spring that is scillating with angular frquncy ω jt x( t) al { x } x Cs( ωt) Cmplx variabl Hr, is nt ncssarily th natural frquncy ω 0 f th (5)

6 mchanical scillatr (s Fig ). Fr xampl, th scillatr may b bing drivn by an xtrnal surc f arbitrary frquncy. q r E Fig. 3 Charg attachd t a spring f apprpriat natural frquncy ω. In what fllws w will b valuating tim avrags f physical quantitis (i.. f ) vr n prid T / f scillatin: T 1 f f ( t) (6) T 0 Ntic, fr xampl, that th tim avrag f Cs ( t) vr n prid is ½. j ω t W als will b using indistinctly Cs (ωt) r. Fr th lattr it is assumd that w will tak th ral part aftr th calculatins in rdr t btain a prpr (classical) physical intrprtatin. Frm xprssin (5) w calculat an acclratin, a' j t' x (7) Whs avrag is, 4 ' ( 1/ ) ω x a. (8) 3

7 Rplacing ths valus in xprssin (4) givs, P q x 1 c 3 4 Ttal avrag pwr mittd by a charg q scillating with amplitud x (9) whr x and ar th amplitud and angular frquncy f th charg s scillatins, rspctivly. Ntic, th largr th amplitud x, th highr th pwr mittd by th charg. 3. Elctrmagntic radiatin damping T calculat th amplitud f scillatin x, lt s mdl th atmic xcitatin with a frcd harmnic scillatr. Cnsidr th quatin f mtin fr th charg q, attachd t a mchanically frictinlss spring, d x m m dx j t kx q E (10) Hr th trm dx m (11) accunts fr th dissipatin nrgy du t th missin f lctrmagntic radiatin by th acclratd charg. (Th trm is chsn prprtinally t th vlcity just t facilitat th slutin f th quatin (10)). A statinary slutin f (11) is givn by (as dmnstratd in th nxt sctin), whr x j jω t [ x ] (1) 4

8 and x x ( ω) ( q / m ) E 1/ ( ) 1 tan (13) (14) Exprssin (13) indicats that th amplitud f scillatin x (and hnc th acclratin) f th charg dpnds n th xtrnal incidnt radiatin s frquncy. Cmparisn btwn xprssins (9) and (11) If xprssin (10) is ging t b usd t mdl th ptical xcitatins f an atm, dx th trm m in (10), that charactrizs th dissipatin f nrgy f th mchanical scillatr, shuld b cmpatibl with q x xprssin (9), P 1 dissipatd by th scillatr. 3 c Hw t xprss mathmatically this cmpatibility? 4, that givs th nrgy Stratgy: Find th dissipatin pwr assciatd t th trm Onc fund, it shuld b qual t pwr q x P 1 dx m. 3 c 4 Th pwr dissipat by a mchanical scillatr, mdlld by (10), is givn by, 5

9 Frc x vlcity = [ m dx dx ]( ) using (1) = [ m ( j x) ]( ( j x) ) = m γω x. Th magnitud f th avrag f will b < Frc x vlcity > = (15) ( 1/ ) mω x This is th xprssin that must b cmpatibl with xprssin (9). Accrdingly, if th mchanical scillatr mdl (10) is t b usd t dscrib th ptical xcitatin, it must ccur that, q x 4 ( 1/ ) mω x = P 3 1 c This allws t idntify 3c q ω 3 r, rarranging trms, q 4 mc 6 mc lctrmagntic radiatin damping (16) d x m m d x m dx b dx j t kx q E jω t kx q E 6

10 Light-mattr intractin will thrfr b mdlld by a harmnic frcd dampd scillatr Light Mattr Extrnal xcitatin f driving frquncy ω f ω f ω, k

11 Slving th frcd harmnic scillatr quatin, Eq. 1, in cmplx variabl

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13 - ω ο ω f ω ο ω f ω f ω variabl driving frquncy scillatr natural frquncy (fixd valu)

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15 Graphic analysis f th slutins (phasrs) Vlcity phasr z dz / Ntic, sinc φ is always ngativ, th POSITION PHASOR always lags th FORCE PHASOR

16 ω f variabl driving frquncy ω scillatr natural frquncy (fixd valu)

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