Chapter 9 Threshold Requirements

Looking at Loss again Laser medium R1R1 R2R2 d Loss/length =  1 Gain/length =  −  or  I I e -(  1+  d IR 2 e -(  1+  d IR 2 R 1 e -2(  1+  d All self-sustaining oscillators require gain to exceed losses  = h  r B/c (n 1 -n 2 ) = - 

Gain Threshold R 2 R 1 e 2(  -  1  d ≥ 1 Eq 9.1 Rewrite this as:  ≥  1 - ln[R 2 R 1 ]/2d Cavity stuff Atom stuff

Cavity Lifetime Lets make another defn so we can simplify: ‘ cavity lifetime’ (also called cold cavity lifetime or photon lifetime). This is the time the intensity decreases to 1/e of its starting value when there is no gain in the cavity In one round trip intensity decreases by R 2 R 1 e 2(  -  1  d which takes 2d/c seconds (assumed cavity and gain material length are the same) Assume intensity decrease can be approx. described as e -t/tc where tc is cold cavity lifetime t c = 2d c (2  1 d - ln [R 1 R 2 ]) Eq 9.3

Population Difference Threshold n 2 -n 1 ≥ 8π  r 2 t sp c 3 g(  r ) t c Since achieving n 2 >n 1 is really hard, what should a smart person do?

Example Ruby rod from before except lets use a 5cm long rod Between two 90% reflectivity mirrors From Eq. 9.3 we get t c = 2.8ns, and therefore we find by substitution into Eq. 9.4 that: n 2 -n 1 ≥ 2 x m -3 Typical Cr atom densities in ruby (about 1% replacement of Al) are around 1.6 x m -3.

Resonator Transfer Function in presence of gain What happens after lots of round trips?

Go to Phet Simulation Talking about Gain Also, Talk about two level situation

Chapter 10 Pumping Processes and Rate Equations

Two Level Atom Narrow Band Radiation at  r with density u, hit some atoms. R abs = n 1 c 3 A g(  r ) u(T)/(8 π h  r 3 ) = n 1 W 12 E1E1 E2E2 n1n1 n2n2 u(T) R em = n 2 c 3 A g(  r ) u(T)/(8 π h  r 3 ) = n 2 W 21 N.B. W 12 = W 21 n 1 = W 21 (n 2 -n 1 ) + A n 2. W 12 n 1 W 21 n 2 A n 2

Solution for 2 level system n 2 /n 1 = W 21 /(W 21 + A) Cannot possibly work!

Three Level Atom Narrow Band Radiation at  p with density u. E1E1 E3E3 n 1 = W 31 (n 3 -n 1 ) + A 31 n 3 + A 21 n 2 + W 21 (n 2 -n 1 ). W 13 n 1 W 31 n 3 A 31 n 3 A 32 n 3 hv p A 21 n 2 W 21 n 2 W 12 n 1

Three Level Solution ΔnΔn N = W 31 -A 21 W 31 +A W 12 3W 31 +2A 32 A 32 (W 31 +A 21 ) Below threshold essentially no laser light: W 12 = 0 In this case pop difference, and hence gain, are indep. of W 12  Intensity depends exponentially on distance Lets look at saturation in more detail in 4 level laser

Example: pump power for Ruby At threshold: ΔnΔn N  W 31 -A 21 W 31 +A 21  0 PD is nearly zero No. of pumped atoms/sec/volume ~ W 31 n 1 Power (per unit volume) will be: P = W 31 n 1 h  p = A 21 N/2 h  p For Ruby, A 21 =1/  = 1/(3ms), N ~ 1.6 x cm -3 Pump wavelength ~ 480nm (frequency ~ 6 x Hz) P = 1100 Wcm -3 Electrical thresholds are about 16 times higher than this

Four Level Atom E1E1 E4E4 n 1 = W 41 (n 4 -n 1 ) + A 41 n 4 + A 21 n 2. W 14 n 1 W 41 n 4 A 41 n 4 A 43 n 4 hv p A 21 n 2 W 32 n 3 E3E3 A 21 n 2 E2E2 W 23 n 2 hv L Smart: avoid using the ground state for one of the laser levels If A 21 > A 43 then get PI for smallest pumping intensities

Four level solution ΔnΔn N = W 31 -A 21 W 31 +A W 12 3W 31 +2A 32 A 32 (W 31 +A 21 ) ΔnΔn N = W 41 W 41 +A W 32 2W 41 + A 21 A 21 (W 41 +A 32 ) Viz. 3 level: Small signal behaviour (below and near threshold)

Nd:YAG example NB. 2 errors in notes p should be N ~ 6 x m -3 (0.5% doping) p should be Gain coeff/m on vertical axis of Fig 10.3  ~ 1064nm, Δ  ~ 195 GHz,  ~ 230µs N ~ 6 x m -3, n 0 ~ 1.82,  p ~ 400 THz R 1 ~ 100%, R 2 ~ 80%, l ~ 7cm  1 = 0 Δ n ~ 2.5 x m -3 W 41 ~ 0.2 s -1 P ~ W 41 n 1 h v p ~ W 41 N h v p ~ 3.2 W/cm 3

Gain dependence on output W 41 ~ 0.2 s -1 W41 ~ 0.4 s -1 W 41 ~ 0.1 s -1 Threshold

No. of photons in mode

Power vs Current in LDs

Line Broadening Chapter 11: Not for examination

Radiation from a bunch of excited atoms  FWHM = 1/(2 π  )

Collisions

Doppler Broadening