Chapter 9 Threshold Requirements. Looking at Loss again Laser medium R1R1 R2R2 d Loss/length =...
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Transcript of Chapter 9 Threshold Requirements. Looking at Loss again Laser medium R1R1 R2R2 d Loss/length =...
Chapter 9
Threshold Requirements
Looking at Loss again
R1
R2
d
Loss/length = 1
Gain/length = (or )
I
I e-(1+)d
IR2 e-(1+)d
IR2R1 e-2(1+)d
All self-sustaining oscillators require
gain to exceed losses
= h r B/c (n1-n2) = -
Laser
medium
Gain Threshold
R2R1 e2( -1)d 1
Eq 9.1
Rewrite this as:
1 - ln[R2R1]/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 R2R1 e2( -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
tc = 2d
c (21 d - ln [R1R2])
Eq 9.3
Population Difference Threshold
n2-n1
8r2 tsp
c3 g(r) tc
Since achieving n2>n1 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 tc = 2.8ns, and therefore we find by substitution into Eq. 9.4 that:
n2-n1 2 x 1023 m-3
Typical Cr atom densities in ruby (about 1% replacement of Al)
are around 1.6 x 1025 m-3.
Resonator Transfer Function in presence of gain
1
0
2
1
0
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.
Rabs = n1c3A g(r) u(T)/(8 h r3)
= n1 W12
E1
E2
n1
n2
u(T)
Rem = n2c3A g(r) u(T)/(8 h r3)
= n2 W21
N.B. W12 = W21
n1 = W21 (n2-n1) + A n2
.
W12 n1
W21 n2
A n2
Solution for 2 level system
n2/n1 = W21/(W21 + A)
Cannot possibly work!
Three Level Atom
Narrow Band Radiation at p with density u.
E1
E3
n1 = W31 (n3-n1) + A31 n3 + A21 n2 + W21 (n2-n1)
.
W13 n1
W31 n3
A31 n3
A32 n3
hvp
A21 n2
W21 n2
W12 n1
Three Level Solution
n
N
=
W31-A21
W31+A21
1
1+ W12
3W31+2A32
A32 (W31+A21)
Below threshold essentially no laser light: W12 = 0
In this case pop difference, and hence gain, are indep. of W12
Intensity depends exponentially on distance
Lets look at saturation in more detail in 4 level laser
Example: pump power for Ruby
At threshold:
n
N
W31-A21
W31+A21
0
PD
is nearly
zero
No. of pumped atoms/sec/volume ~ W31 n1
Power (per unit volume) will be:
P = W31 n1 h p = A21 N/2 h p
For Ruby, A21 =1/ = 1/(3ms), N ~ 1.6 x 1019 cm-3
Pump wavelength ~ 480nm (frequency ~ 6 x 1014 Hz)
P = 1100 Wcm-3
Electrical thresholds are about 16 times higher than this
Four Level Atom
E1
E4
n1 = W41 (n4-n1) + A41 n4 + A21 n2
.
W14 n1
W41 n4
A41 n4
A43 n4
hvp
A21 n2
W32 n3
E3
A21 n2
E2
W23 n2
hvL
Smart: avoid using the ground state for one of the laser levels
If A21 > A43 then get PI for smallest pumping intensities
Four level solution
n
N
=
W31-A21
W31+A21
1
1+ W12
3W31+2A32
A32 (W31+A21)
n
N
=
W41
W41+A32
1
1+ W32
2W41+ A21
A21 (W41+A32)
Viz. 3 level:
Small signal behaviour
(below and near threshold)
Nd:YAG example
NB. 2 errors in notes
p. 125 - should be N ~ 6 x 1025 m-3 (0.5% doping)
p. 124 - should be Gain coeff/m on vertical axis of Fig 10.3
~ 1064nm, ~ 195 GHz, ~ 230s
N ~ 6 x 1025 m-3 , n0 ~ 1.82, p ~ 400 THz
R1 ~ 100%, R2 ~ 80%, l ~ 7cm 1 = 0
n ~ 2.5 x 1021 m-3
W41 ~ 0.2 s-1
P ~ W41 n1 h vp ~ W41 N h vp ~ 3.2 W/cm3
Gain dependence on output
W41 ~ 0.2 s-1
W41 ~ 0.4 s-1
W41 ~ 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 )
Collisions
Doppler Broadening