2. Laser physics - basics

Transcription

1 . Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation

2 What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light" could man anything from microwavs to x-rays Essntial lmnts:. A lasr mdium - a collction of atoms, molculs, tc.. A pumping procss - puts nrgy into th lasr mdium 3. Optical fdback - provids a mchanism for th light to intract (possibly many tims) with th lasr mdium

3 Th two-lvl atom Quantum nrgy lvls xcitd stat ground stat Absorption: promots an lctron from th ground to th xcitd stat Emission: drops th lctron back to th ground stat "spontanous mission" - th dcay of an xcitd stat to th ground stat with th corrsponding mission of a photon Consrvation of nrgy: E xcitd -E ground = E photon

4 Thr things can occur Absorption Promots molcul to a highr nrgy stat Dcrass th numbr of photons Spontanous Emission Molcul drops from a high nrgy stat to a lowr stat Incrass th numbr of photons This is th only on that dos NOT rquir a photon in th initial stat Stimulatd Emission Molcul drops from a high nrgy stat to a lowr stat Th prsnc of on photon stimulats th mission of a scond on

5 Rlaxation of th two-lvl atom An atom in th xcitd stat can rlax to th ground stat by: spontanous mission: rat is rad any of a varity of non-radiativ pathways: rat = nr All of ths procsss ar singl-atom procsss; ach atom acts indpndntly of all th othrs. Thus, volution of th xcitd stat population only dpnds on th numbr of atoms in th xcitd stat: dn N N N rad nr 0 0 = total spontanous rlaxation rat from stat to stat 0

6 A collction of two-lvl atoms "Stimulatd transitions" - a collctiv procss involving many two-lvl atoms stimulatd absorption: light inducs a transition from 0 to stimulatd mission: light inducs a transition from to 0 In th mission procss, th mittd photon is idntical to th photon that causd th mission! Stimulatd transitions: liklihood dpnds on th numbr of photons around

7 How did it all bgin? Rayligh-Jans law (circa 900): nrgy dnsity of a radiation fild u() = 8 kt/c 3 Not: th units of this xprssion ar corrct. Strictly spaking, u() is an nrgy dnsity pr unit bandwih, such that th intgral u d givs an answr with units of nrgy/volum. Total nrgy radiatd from a black body: uh-oh th "ultraviolt catastroph" Solution: quantum mchanics Tim-dpndnt prturbation thory u d As a rsult of a prturbation h(t), a systm in quantum stat maks a transition to quantum stat with probability givn by: t Notation : ' ie t P h t' ' E

8 Ky xampl: suppos w subjct a two-lvl systm, initially in stat, to a harmonic prturbation, of th form: ht 0 t 0 A0 sint t 0 Transition probability to stat is: 4A 0 Harmonic prturbation t A0 i t' it' it' P 0 sin t / 4A 0 i t/ i t/ sin t/ sin t/ (and suppos that th frquncy of th prturbation,, is clos to ) ' P P Not that Absorption and stimulatd mission ar qually likly! RWA

9 Einstin A and B cofficints Considr a radiation fild and a collction of two-lvl systms, in thrmal quilibrium with ach othr. stimulatd mission probability: proportional to th numbr of atoms in uppr stat N, and also to th numbr of photons spontanous mission probability: proportional to N, but dos not dpnd on th photon dnsity! Not: this is th W AN Bu N sam as rad stimulatd absorption probability: proportional to th numbr of atoms in lowr stat N, and also to th numbr of photons spontanous absorption: thr is no such thing W Bu N Quantum mchanics says that ths two cofficints must b qual! But: in thrmal quilibrium, th upward and downward transition rats must balanc: W W

10 Einstin A and B cofficints N ABu N Bu Equat ths two rats: But Boltzmann's Law tlls us that (in quilibrium) N N Rcognizing that E E = h, w solv for u(): u A B h kt This must corrspond to th Rayligh-Jans rsult in th classical limit (h 0), which implis: 3 A B Sinc A = rad, w can now solv for B also: 8 h 3 c E E kt Also has units of nrgy dnsity pr unit bandwih 3 B radc 3 8 h

11 Transition rats Our xprssion for th downward transition rat is now: But sinc u W AN Bu N B AN u A A B w thrfor hav h W AN kt Bos-Einstin distribution h kt In othr words, W is proportional to: + th numbr of photons. It is asy to s that th upward transition rat, W, is proportional to th numbr of photons: W B u N AN h kt

12 Rat quation analysis stimulatd g spontanous spontanous mission: proportional to initial stat population stimulatd transitions: proportional to initial stat population proportional to photon dnsity n p th sam for upwards, downwards transitions dn N Kn N Kn g N P P g dn g dn N Kn N N g P g Not: th constant K is simply givn by h B, whr B is th Einstin B cofficint

13 Rat quation analysis, continud dn N Kn N N g P g mittd photons go in all dirctions mittd photons go only into th dirction of th incidnt light So, photon numbr varis according to: dn P Kn N N Kn N P g P Numbr of photons grows xponntially if N > N g A LASER!

14 Rat quation analysis, part 3 In thrmal quilibrium: N N N EkT g g Population invrsion is impossibl in quilibrium. In a stady-stat situation: dn 0 N Kn N N g P g KnP N N N Kn g g g P Population invrsion is impossibl in stady-stat.

15 So how do you mak a lasr? 3 Pump 0 Four-lvl systm Non-radiativ dcay Lasing transition Non-radiativ dcay Stps and : Combin to giv an ffctiv pumping rat for lvl : R p Stp 3: stimulatd transitions du to n p spontanous dcay rat: Stp 4: spontanous dcay rat: 0 dn dn dn 0 R p N 0N N R Kn p( N N) p + Kn p( N N) 0N

16 Th four-lvl modl Stady-stat solution: P 0 N NN 0 KnP R Population invrsion (i.., N < 0) is assurd if 0 > (vn if n p = 0, and vn if R p is small) A ncssary condition for lasing Othr ncssary conditions: a rsonant cavity - provids fdback nt gain pr round trip > nt loss pr round trip - thrshold

17 Saturation in th four-lvl atom R P 0 N 0 KnP 0 RP 0 Wsig population invrsion whn > small signal invrsion is proportional to th pump rat invrsion lvl drops whn W sig > th charactristic intnsity for this ffct is indpndnt of pump rat R p "gain saturation" Not: W sig is proportional to n P and thrfor to th intnsity of th light in th mdium. Thus, w can dfin a saturation intnsity I sat such that: W sig I I sat N 0 N/N Kn p / W sig