6. The Interaction of Light and Matter

Transcription

1 6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd atoms & th forcd oscillator - Th Rotating Wav Approximation - Complx Lorntzian: 1/(-j)

2 Diffrnt matrials rspond to light diffrntly Ths diffrncs can aris from diffrnt chmical compositions, or vn from diffrnt sizs and shaps of th matrials. A vial of nanomtr-sizd gold sphrs in watr gold nuggts W would lik to dvlop an undrstanding of ths phnomna by starting from a microscopic dscription of th light-mattr intraction. Th Big Qustion: How is a light wav changd by propagating in th prsnc of a bunch of atoms?

3 Light xcits atoms, which mit light that adds (or subtracts) with th input light. An atom is xcitd (causd to vibrat) by incidnt light. Incidnt light Emittd light Transmittd light An xcitd atom vibrats at th frquncy of th light that xcitd it and r-mits th nrgy as light of that frquncy. Th crucial issu is th rlativ phas of th incidnt light and this r-mittd light. For xampl, if ths two wavs ar ~18 dgrs out of phas, th bam will b attnuatd.

4 Adding complx amplituds Whn two wavs add togthr with th sam complx xponntials, w add th complx amplituds, E + E '. Constructiv Dstructiv "Quadratur phas" ±9 intrfrnc: intrfrnc: intrfrnc: j j Amplification Attnuation Slowr phas vlocity

5 That s not obvious, so hr s an xampl Suppos ths two light wavs ar co-propagating: E E Acos kxt 1 E Acos kxt9 W comput th rsulting wav by suprposing (adding) ths two. First, convrt to complx notation: 1 R A jkxt jkxt j E R A Thn, th nt fild is th sum of th two light wav E filds: j jkx t E R 1 A nt Acos kxt45 sam wav vctor and frquncy, but phas shiftd! jkx t j 4 jkx t j A A R 1 R

6 Modling th intraction of light with mattr Our gnral approach is as follows: 1. W know that light xrts forcs on lctrons: F = E. W can figur out how th lctrons mov in rspons: F = ma 3. Ths moving lctrons produc a fild that must b includd in a modifid wav quation which w can thn solv. Two cass of intrst: 1. Each lctron is bound to its atom. Som lctrons ar fr to mov away from thir atoms. Insulators Conductors

7 Insulators: th Forcd Oscillator modl Considr an lctron on a spring with position x (t), and drivn by an incidnt light wav, E xp(j t): nuclus lctron E applid (t) Th forcs on th lctron ar: 1. Th rstoring forc of th spring: kx. Th forc xrtd by th lctric fild: E This modl was first proposd by Hndrik Lorntz in th 189 s as a way to xplain th mission of light by atoms. Hndrik Lorntz

8 Lorntz s Forcd Oscillator modl Us Nwton s Law, F = ma, to writ down an quation of motion: t d x jt ma t m kx t E dt Dfin a nw constant to charactriz th spring: nuclus d x t E jt x t dt m lctron E applid (t) k m th rsonanc frquncy of th spring this is th frquncy of th light wav W gnrally nvision that th frquncy of th light wav can b varid, but th rsonant frquncy of th atom is a fixd constant.

9 Th Forcd Oscillator modl: solution Th standard way to solv a DE: guss th solution, and s if it works. Plug into th quation: xp x t A j t d x Axp jt dt E A A jt jt jt m Our guss is indd a solution, but only if th factor A satisfis: d x t E jt x t dt m A Thus th solution is: E m E m j t x t So th lctron oscillats at th frquncy of th incidnt light wav (), but with an amplitud that dpnds on th light wav s frquncy.

10 Amplitud and phas rspons How dos th amplitud (and phas) of th motion of th charg dpnd on th frquncy of th lctric fild? x t E 1 m amplitud of x angular frquncy ngativ x = phas of 18 phas of x rlativ to th phas of th incidnt light wav angular frquncy Qustion: what if th light wav oscillats at frquncy =?

11 Th Dampd Forcd Oscillator A dampd forcd oscillator is a harmonic oscillator xprincing a sinusoidal forc and friction. It is not ralistic to ignor friction. W must add a frictional drag trm, proportional to th vlocity of th lctron: Th solution is: E jt x t dx m dt d x t dx t dt m dt ( E / m ) x () t ( j ) complx! jt Th lctron oscillats at th frquncy of th incidnt light wav (), but with an amplitud and a phas that both dpnd on. nw trm

12 Why w includd th damping factor, Atoms spontanously dcay to th ground stat aftr a tim. Also, th vibration of a mdium is th sum of th vibrations of all th atoms in th mdium. Atom #1 Atom # Atom #3 Sum: tim t Collisions dphas th vibrations, causing cancllation of th total mdium vibration, typically xponntially. Th tim constant of this xponntial dcay is th invrs of th damping factor: 1

13 Th lin shap: a complx Lorntzian Considr: 1 1 j ( )( ) j Assuming this bcoms: 1 ( ) j 1 1 ( ) j W can now rwrit x (t) in a mor compact form, using th following dfinition: a complx Lorntzian Our solution thn bcoms: x () t / m 1 j E jt th applid fild

14 Plots of th complx Lorntzian Amplitud Phas 1 1 j tan j 1 / phas (radians)

15 Plots of th complx Lorntzian Ral Imaginary 1 j j Imaginary (vn) componnt width Ral (odd) componnt

16 Dampd Forcd Oscillator Solution for Light-drivn atoms Th forcd-oscillator rspons is sinusoidal, with a frquncydpndnt strngth that's approximatly a complx Lorntzian: ( / m ) x () t () E t ( j) 1 E( t) m ( j) (rmmbr: < ) Whn <<, th lctron vibrats 18 out of phas with th light wav. Whn =, th lctron vibrats 9 out of phas with th light wav. Whn >>, th lctron vibrats in phas with th light wav.

17 A Java applt to illustrat th ida, using a mass on a spring

18 Anothr look at th approximation w just mad This is known as th rotating wav approximation. How good is it? This shows a rsonanc cntrd at = 75 nm, with = 1 13 rad/sc. imaginary part linar scal wavlngth (microns) log(imaginary part) log scal xact approx wavlngth (microns)

19 R-mittd light from an xcitd atom intrfrs with original light bam Th r-mittd light may intrfr constructivly, dstructivly, or, mor gnrally, somwhat out of phas with th original light wav. Unfortunatly, it is not so simpl as knowing th phas of x(t), which w hav just discussd. W nd to know th phas of th r-mittd light. Incidnt light Emittd light Transmittd light W nvision that th important fild is th total lctric fild, E(t) = E original (t) + E r-mittd (t) Maxwll's Equations allow us to solv slf-consistntly for th total fild, E(t).

20 Maxwll's Equations in a Mdium P Th inducd polarization,, is th rspons of th atoms to th applid fild. It contains th ffct of th mdium: B E E t E B B t P whr is givn by: Pt () Nx() t P t and w v just sn a way to dtrmin this! N = numbr of lctrons pr m 3 oscillating in rspons to E-fild.