Number of modes per unit volume of the cavity per unit frequency interval is given by: Mode Density, N
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1 SMES404 - LASER PHYSCS (LECTURE 5 on /07/07) Number of modes per uni volume of he aviy per uni frequeny inerval is given by: 8 Mode Densiy, N (.) Therefore, energy densiy (per uni freq. inerval); U 8h e h Ploing U v agains ν;.(.) Uni of energy densiy: J/m U T Eqn.. is imporan o derive Einsein Coeffiiens. Einsein Coeffiiens Exied aom in free spae has a lifeime of abou 0-8 s. Equivalen o he average number of 0 8. Sponaneous ransiion from an exied sae lower sae per seond. Transiion Probabiliy : The larger he ransiion rae, he greaer he probabiliy of ransiion. Probabiliy of sponaneous emission is alled Einsein A Coeffiien. A (.) Definiion: s he probabiliy per uni ime per aom ha he exied aom will deay sponaneously o he lower sae. ν
2 A s s For N aom N A ransiions per uni ime. Einsein B Coeffiiens Eg: B and B absorpion B B simulaed emission B = is he absorpion oeffiien. = is he probabiliy per aom per uni ime per uni radiaion densiy per uni frequeny inerval ha absorpion from () (). Suppose ha -level aoms are in an enlosure ( a box), energy densiy (in he freq ν o ν+dν ) = ρ(ν)dν. Then, probabiliy per uni ime ha an aom will absorb a phoon and be exied () (). So, he simulaed absorpion per aom per uni ime = B ρ(ν) for N aoms = N B ρ(ν ) [ransiion per uni ime per uni freq inerval] Einsein B (for simulaed emission) B The probabiliy per aom per seond per uni radiaion energy densiy per uni frequeny inerval. For N aoms, (per uni volume): Transiions from () () = N B ρ(ν )
3 Relaionship beween A and B oeffiiens. Consider he ase where he sysem is in hermal equilibrium, Toal energy of he sysem mus remain onsan ( no. of phoons absorbed per seond mus be equal o he oal no. emied, simulaed and sponaneous) Thus; N B N A N B.(.4) absorpion spon an eous simulaed Solving for ρ(ν), N A N B N..(.5) From Bolzmann disribuion, a hermal equilibrium, B N h e.(.6) N E Subsiue (.6) ino (.5), E -E = hν A..(.7) h e B B Noe: This radiaion densiy ρ(ν) emied a hermal equilibrium for -level sysem has o be idenial o blakbody radiaion densiy (Plank Law) given as; U 8h e h.(.8) Therefore, for hem o be equal beween (.7) & (.8), hese relaionship mus hold, B B B..(.9) A B 8 h (.0) E
4 Noe: (.9) shows ha he probabiliy of simulaed emission is equal o he probabiliy of simulaed absorpion. (.0) Raio R of he rae of sponaneous emission o he rae of simulaed emission under hermal equilibrium. A R B Using (.7) ino above, R e h (.) 4 Subsiue 5 0 Hz Green ligh. 5 R 0 spon an eous simulaed A hermal equilibrium, i is quie impossible o ge he simulaed emission in visible region. Bu, for mirowave 9 0 Hz R Simulaed emission is dominan proess, direionaliy in a mirowave. For lasing aion, in visible region he populaion has o be invered populaion inversion or negaive emperaure. Why negaive emperaure? N h e ; o make N N, he erm h has o be posiive, herefore N his is possible when T(-T) so ha; h h k T 4
5 Populaion inversion ; a siuaion whereby he sysem is no longer in hermal equilibrium and simulaed emission of visible ligh beomes possible. 4.0 Threshold Condiion. he minimum populaion differene N -N needed o subsain laser aion. A laser onsis of an amplifying medium (ha is being invered) in he form of gas, liquid and solid plaed beween wo mirrors. Fully refleing Parially refleing oupu Amplifying medium Losses proess in an opial resonaor: (i) Transmission, absorpion and saering by mirrors (ii) Absorpion wihin he amplifying medium due o oher energy levels. 5 reabsorpion 4 h E 5 E h 5
6 (iii) Saering by opial inhomogeneiies wihin he amplifying medium is imporan in solid-sae lasers (impossible o produe perfe rysals) (iv) Diffraion losses by he mirrors. All his losses an be inluded in a parameer, phoon. phoon : lifeime of a phoon exising wihin he laser aviy. Deriving he hreshold ondiion- for lasing aion. -level sysem; ρ(ν) E h E E h E Simulaed Emission rae per aom in sae ; W From (.0) B ' B from;. J s m J m. m s ' 8h W A (4.) A = he probabiliy per uni ime of a sponaneous emission A (relaed o he lifeime of he upper laser level) W ' (4.) 8h 8h 8h *W = ransiion rae per aom. For N aoms; W ' N N 8h..(4.) 6
7 Toal absorpion rae (for N aoms in ground sae); W ' N Consider he medium below; N 8h (beause B =B =B) 0 (x) (x+dx) N x dx d N and dx d x x+dx Rae of hange of inensiy wih ime = d Or, i.e = differene beween simulaed emission and absorpion raes. = hange of energy densiy per uni ime x. d h For absorpion, N N 8h d h N N d 8h h N N d 8h..(4.4) f N N (hermal equilibrium) ; or are negaive. d dx As a resul of (N -N ) is negaive..: sauahayalangi:. 7
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