Chapter 10. Laser Oscillation : Gain and Threshold

Transcription

1 Chaper 0. aser Osillaio : Gai ad hreshold Deailed desripio of laser osillaio 0. Gai <Plae wave propaai i vauum> Cosider a quasi-moohromai plae wave of frequey propaai i he + direio ; u A he rae a whih eleromaei eery passes hrouh a plae of ross-seioal area A a is A +D he flux differee ; [ D ] A A D oliear Opis ab. Haya Uiv.

2 oliear Opis ab. Haya Uiv. his differee ives he rae a whih eleromaei eery leaves he volume AD, A A u D D 0 u / : Equaio of oiuiy <Plae wave propaai i medium> he hae of eleromaei eery due o he medium should be osidered. => he rae of hae of upperlower-sae populaio desiy due o boh absorpio ad simulaed emissio a => eery flux added o he field : h

3 oliear Opis ab. Haya Uiv : / BS h S A S B h S B h 8 o deeeraies deeeraies iluded Defie, ai oeffiie, 8 S A

4 oliear Opis ab. Haya Uiv. emporal seady sae : 0 d d e 0 : valid oly for low iesiy sauraio effe, = * >0 whe : populaio iversio * f, 8 a S A he ai oeffiie is ideial, exep for is si, o he absorpio oeffiie.

5 0.3 Feedbak praie, ~0.0 m - if he leh of aive medium is m. => A spoaeously emied phoo a oe ed of he aive medium leads o a oal of e 0.0 x 00 =e =.7 phoos emeri a he oher ed. => he oupu of suh a laser is obviously o very impressive. => Refleive mirrors a he eds of he aive medium : Feedbak. 0.4 hreshold a laser here is o oly a irease i he umber of aviy phoos beause of simulaed emissio, bu also a derease beause of loss effes. oss effes : saeri, absorpio, diffraio, oupu oupli order o susai laser osillaio he simulaed amplifiaio mus be suffiie o overome he losses. oliear Opis ab. Haya Uiv.

6 <hreshold odiio> Absorpio ad saeri wihi he ai medium is quie small ompared wih he loss ourri a he mirrors of he laser. => Cosider oly he losses assoiaed wih he mirrors. r s where, r : refleio oeffiie : rasmissio oeffiie s : fraioal loss A he mirror a = ad 0 ; r 0 r 0 oliear Opis ab. Haya Uiv.

7 oliear Opis ab. Haya Uiv. seady sae or CW operaio, : osa d d d d e 0 e Amplifiaio odiio : afer oe roud rip mus be hiher ha he iiial ] [ ] 0 [ ] [ ] [ 0 0 e r r e e r r r r e e r r e r r

8 hreshold ai, l r r l r r * he ase of hih refleiviy, rr, ad l- x -x r r hih refleivi ies, r r 0.9 * f he disribued losses losses o assoiaed wih he mirrors are iluded, l r r a where, a : effeive loss per ui leh oliear Opis ab. Haya Uiv.

9 Example He-e laser, =50 m, r =0.998, r = l[ ] m 4 m hreshold populaio iversio ; A 8 S D 8 AS 6 A.4 0 se a 63.8 m 6 ~400K, M e 0 amu D m D m.4 0 se aoms/m 9 D very small! M 0 / se MH 500 MH oliear Opis ab. Haya Uiv.

10 oliear Opis ab. Haya Uiv. 0.5 Rae Equaios for Phoos ad Populaios ime-depede pheomea? Gai erm simulaed emissio or absorpio may lasers here is very lile ross variaio of eiher or wih. => 0 * l< l : ai medium leh, : aviy leh d d l d d 0.5.4b

11 oss erm oupu oupli, absorpio/saeri a he mirrors By he oupu oupli, a fraio -r r of iesiy is los per roud rip ime /. d d r r From 0.5.4b, 0.5.8, ad pu d l r r d or phoo umber, dq d l l q q l q r r q q : oal iesiy oliear Opis ab. Haya Uiv.

12 oliear Opis ab. Haya Uiv. f =, r r l r r S A l d d 8 Coupled equaios for he lih ad he aoms i he laser aviy iludi pumpi effe : r r l d d K A d d A d d

13 0.7 hree-evel aser Sheme wo-level laser sheme is o possible. elei, i 7.3. => A : a ahieve he populaio iversio hree-level laser sheme P : pumpi rae d P d pumpi d d3 d P d d d pumpi pumpi pumpi => d P d d P d d l r r d oliear Opis ab. Haya Uiv.

14 <hreshold pumpi> for he seady-sae operaio i ear hreshold he umber of aviy phoos is small eouh ha simulaed emissio may be omied from Eq ii Seady-sae : 0 P Ad, P : oal populaio is oserved P P P P # Posiive seady-sae populaio iversio : P Pmi Pwr h # hreshold pumpi power : 3P h 3P V P h 3 oliear Opis ab. Haya Uiv.

15 0.8 Four-evel aser Sheme ower laser level is o he roud level : he depleio of he lower laser level obviously ehaes he populaio iversio. Populaio rae equaios : d d d d d d 0 P 0 P oliear Opis ab. Haya Uiv.

16 0 osa Seady-sae soluios : P P Populaio iversio : P P 0 P P 0 0P f 0 0 0, 0 0 0P P P P P P # Posiive iversio : oliear Opis ab. Haya Uiv. P 0 lower level deays more rapidly ha he upper level.

17 0.9 Compariso of Pumpi Requiremes for hree ad Four-evel asers hreshold pumpi rae, P => 0.8.7, => D P hree level laser D D P four level laser D P P four levellaser hreelevellaser D D hreshold pumpi power, Pwr => Pwr V Pwr V h hreelevellaser four level laser 3 30 h D Pwr Pwr / V / V four levellaser hreelevellaser 30D 3 oliear Opis ab. Haya Uiv.

18 0. Small-Sial Gai ad Sauraio For he hree-level laser sheme, seady-sae soluio iludi he simulaed emissio erm ; P P Gai oeffiie assumi =, P P f : A lare phoo umber, ad herefore a lare simulaed emissio rae, eds o equalie he populaios ad. his ase, he ai is said o be sauraed. <Mirosopi view of he ai sauraio> As he aviy phoo umber ireased, he simulaed absorpio as well as he simulaed emissio ireased. => he lower level s absorpio rae is exaly equal o he upper level s emissio rae i he exreme limi. he ai is ero. oliear Opis ab. Haya Uiv.

19 <Small sial ai / Sauraio flux> P P 0 / sa [ / P ] Where we defie, Small sial ai as P 0 P Sauraio flux as sa P * he larer he deay raes, he larer he sauraio flux. * he sauraio flux Simulaed emissio rae : he averae of he upper- ad lower-level deay raes. oliear Opis ab. Haya Uiv.

20 <Gai widh> Whe he absorpio lieshape is oreia, / * Small sial ai widh : D 0..3 => 0 sa / / where, P 0 P : ie-eer small sial ai * Power-broadeed ai widh : D sa / oliear Opis ab. Haya Uiv.

21 P 4 sa P A : ie-eer sauraio flux is direly proporioal o he rasiio liewidh. oliear Opis ab. Haya Uiv.

22 0. Spaial Hole Buri mos laser, we have sadi waves raher ha raveli waves. => Caviy sadi wave field is he sum of wo opposiely propaai raveli wave fields ; E, E E 0 os si k E [si k si k] 0, E, where, E, E0 si k he ime-averaed square of he eleri field ives a field eery desiy : => oal phoo flux, [ ]si k h [ ]si h k E oliear Opis ab. Haya Uiv.

23 0..3 => 0 sa [ / ]si k oliear Opis ab. Haya Uiv.