Photon Physics. Week 5 5/03/2013

Transcription

1 Photon Physics Week 5 5/3/213 1

2 Rate equations including pumping dn 2 = R 2 N * σ 21 ( ω L ω ) I L N 2 R 2 2 dn 1 = R 1 + N * σ 21 ( ω L ω ) I L N 1 + N 2 A 21 ss solution: dn 2 = dn 1 = N 2 = R 2 N * σ 21 ( ω L ω ) I L absorption stimulated emission spontaneous emission N 1 = R 1 + N * σ 21 ( ω L ω ) I L + N 2 A 21 R 1 1 N * = N 2 g 2 N 1 = R 2 N * σ 21 ( ω L ω ) I L g 2 R 1 N * = g 2 N * σ 21 ( ω L ω ) I L g 2 R 2 1 ( g 2 )A σ 21 ( ω L ω ) I L R 2 N * σ 21 ω L ω R 1 I L [ ] g 2 [ ( g 2 )A 21 ] + g 2 A 21 2

3 Steady-state inversion N * = [ ] ( g 2 )R 1 [ ( g 2 )A 21 ] R 2 1 ( g 2 )A σ 21 ( ω L ω ) I L + g 2 For I L =, N * = N * A 21 [ ] g 2 = R 2 1 g 2 R 1 together with the following relations: recovery time R = + ( g 2 ) ( g 2 )A 21 we can rewite: N * = I L N * 1+ I L ω L ω ( ω L ω ) = saturation intensity R σ 21 ω L ω N* depends on I L, leads to gain saturation when I L = the inversion is ½ of N*() 3

4 Significance of R rate equations: dn 2 dn 1 = R 2 N * σ 21 ( ω L ω ) I L N 2 = R 1 + N * σ 21 ( ω L ω ) I L N 1 + N 2 A 21 rate equations can be rewitten using: dn 2 dn 1 = R 2 N * R = R 1 + N * R and in terms of N*: I L N 2 I L N 1 + N 2 A 21 dn * net pumping rate ( ω L ω ) = stimulated processes = R 2 g 2 R 1 1+ g 2 N * 4 R R σ 21 ω L ω spontaneous processes I L N 2 + g 2 N 1 + N 2 A 21

5 Measurement of saturated gain coefficient by weak probe Homogeneous broadening α I L ( ) = N * ( I L )σ 21 ( ) N from before N * ( I L ) * ( ) = = N * α I L 1+ I L ω L ω σ 21 = α ( ) 1+ I L ω L ω 1+ I L ω L ω 5

6 Measurement of saturated gain coefficient by strong probe α I ( ) = N * ( I)σ 21 ( ) = N * α I ( ) = α 1+ I σ I α I L ( ) = α 1+ I L ω L ω strong probe weak probe power broadening strong probe=pump probe + pump at ω L weak probe only 6

7 Beam growth in steady-state, homogeneously broadened amplifier I(,ω) 1 I = α I ( )I( ω,) = α ( ) 1+ I ( ) I = α I radiation in narrow band around ω rearrange: 1+ I I integrate: I I 1 I + 1 I = 1 I + 1 I = α di = α d solution: ln I I + I I( ) = α 7

8 Beam growth in steady-state, homogeneously broadened amplifier ln I I + I I( ) = α I << I >> I( ) I( )exp( α ) I( ) I( ) + α = I( ) + N * ω 8 R

9 Inhomogeneous broadening In absence of saturation, inversion for atoms with centre frequencies between and : ΔN 2 ( ω c ω )δω c g 2 ΔN 1 ( ω c ω )δω c = N * g D ( ω c ω )δω c ω c total inversion (all atoms) ω c + δω c irradiation with intense beam at ω L each group of atoms interacts with beam independently: N ω c * = N * I L g D ( ω c ω )δω c 1+ I L ( ω L ω c ) 9

10 Inhomogeneous broadening-weak probe gain contribution from atoms with centre frequencies between ω c and ω c + δω c : N * ( )g D ( ω c ω )δω c σ 1+ I L ( ω L ω c ) 21 ( ω ω c ) total gain, integrate over distribution of centre frequencies: : α D I L ( ) = N * ( )g D ω c ω 1+ I L ω L ω c σ 21( ω ω c ) true inhomogeneous broadening, Δω D >>Δω H : probe ω far away from ω L : N * α D I L dω c g D ( ω c ω )σ 21 ω ω c = α D dω c unsaturated gain profile probe ω close to ω L : strong saturation, hole burning Δω hole = Δω [ H I ( ) ] 1

11 Inhomogeneous broadening-weak probe 11

12 Inhomogeneous broadening- strong probe = α D I L α I D N * ( )g D ω c ω 1+ I L ω L ω c I σ 21( ω ω c ) ω dω c strong inhomogeneous broadening, Δω D >>Δω H : α D I ( ) = N * ( )g D ( ) α D I ( ) N * ( )g D ( ) α D ( ) 1+ I ( ) α I D σ 21 ω ω c 1+ I ω ω c σ 21 ( x) 1+ I x dω c 12 dx constant If homogeneous shape is Lorentian

13 Beam growth in steady-state, inhomogeneously broadened amplifier I(,ω) 1 I = α D I ( )I ω, I = α D 1+ I = α D ( ) I ( ) I 1+ I 1+ I ( ) I I ( ω, ) di = α D d = α D I << I >> I( ) I( )exp( α D ) I( ) I( ) I sα D 13

14 Pulsed amplifier, homogeneous broadening rate equations: dn 2,t dn 1,t = R 2 ( t) N *,t = R 1 ( t) + N *,t change in em energy ρ(,t) t I,t σ 21 N 2,t I,t σ 21 N,t 1 + N 2 (,t)a 21 I,t Aδ = [ I(,t) I( + δ,t) ]A + N * σ 21.. Aδ 1 c I(,t) t + I(,t) = N * σ 21 I(,t) 14

15 Pulsed amplifier, homogeneous broadening 1 c I(,t) t + I(,t) = N * σ 21 I(,t) change of variables to traveling-wave coordinates = t c I = t + 1 c = N * σ 21 I(, ) t and = t 15

16 Pulsed amplifier, homogeneous broadening assumptions to simplify problem: no pumping during amplification stimulated emission dominates rate out of upper level dn 2,t dn 1,t = R 2 ( t) N *,t = R 1 ( t) + N *,t I,t σ 21 N 2,t I,t σ 21 N 1,t + N 2 (,t)a 21 2 idealied cases: a) rapid decay out of lower level N 1 =, N*=N 2 b) slow decay out of N 1 sum of N 1 and N 2 populations constant N * I = βσ 21 N * (,)I, = σ 21 N * (, )I(,) β = 1 1+ g 2 case a) case b) 16

17 Pulsed amplifier, homogeneous broadening N * I = βσ 21 N * (,)I, = σ 21 N * (, )I(,) N * σ 21 = βσ 21 I integrate over : I out l ( ) = I in ( )exp N * σ 21 d = G( )I in ln[ G( ) ] Γ s = βσ 21 [ I in ( ) ] = 1 Γ s I out eliminate I in : eliminate I out : Γ out fluence ( ) = I out ( ) d = Γ s ln G 1 G 1 Γ in d = Γ s ln 1 1 G 1 1 G( ) = I in ( ) 17

18 Pulsed amplifier, homogeneous broadening Eqn for Γ in can be rewritten: Γ s is varied between curves: G() = G G [ G 1]exp Γ in Γ s output pulse has same shape (Gaussian) as input pulse I out ( ) = I in l exp N * σ 21 d = G( )I in 18 distorted output pulse, because of reduced gain in tail of pulse due to inversion reduction by front of pulse

19 Pulsed amplifier, homogeneous broadening Γ in ( ) = I in ( ) d = Γ s ln 1 1 G 1 1 G( ) rearrange to write expression for G() G( ) = G Γ s G [ G 1]exp Γ in Γ out ( ) = I out ( ) d = Γ s ln G 1 G 1 = Γ ln G exp Γ ( ) in 1 s Γ s +1 for Γ in ( ) Γ s Γ max out ( ) = Γ s lng + Γ in ( ) Γ max extr ( ) = Γ max out ( ) Γ in ( ) = Γ s lng =. σ 21 N * = βσ 21 β. N * energy per transition x inversion 19

20 Amplifier design rules for Γ in Γ max extr ( ) Γ s ( ) = Γ max out Γ max out ( ) = Γ s lng + Γ in ( ) ( ) Γ in ( ) = Γ s lng =. σ 21 N * = βσ 21 β. N * energy per transition x inversion For linear operation: high saturation intensity or fluence For max extraction: high N but small-signal gain low low σ, high 2