Page 280 - High Power Laser Handbook

248 So l i d - S t at e La s e r s Thin-Disc Lasers 249

limit without extensive numerical calculations, some simplifications
are useful. We will assume for the further calculations that the den-
sity of excited ions N and the temperature T are constant in the whole
2
active region. Additionally, we will assume that all fluorescence is
emitted at the laser wavelength. Therefore also γ is constant and we
l
can calculate
N exp(γ s )−1
dΦ(, )sin ϑ d ϑ df 2 max (10.26)
fϑ
=
4 πτ γ
If we further assume that we are in the center of the active region, the
dependency to f will vanish and we get
N π
Φ= 2 ∫ (exp( s γ ) −1)sin ϑ dϑ (10.27)
2τγ max
0
The integration will not produce an analytic result, but it is useful to
assume a constant s max and we will get

N
Φ= 2 (exp( s γ ) −1) (10.28)
τγ max
and

N = & Q − N 2 − N 2 (exp( s γ ) − γ 1)
2 τ τγ max
(10.29)
N
= Q − 2 exp( s γ )
τ max
This effect can be expressed as a reduced lifetime τ ASE :
τ ASE = τ − exp( γs max ) (10.30)

This approach was used to find a scaling limit for the maximum
output power P max , assuming as maximum integration distance the
diameter of the active region, that is, s max = r 2 . The reduced lifetime
p
in this case can also be written as 35
 r 2 
τ = τ  − exp p g (10.31)
ASE  h 
 
with r the radius of the pump spot and g the single pass gain in the
p
disc. Based on these assumptions, the maximum power is:
l
l 2 27 σ (, T)(⋅ − 1 f (,l TTf) (l ,)) C 2
T
P = p ⋅ ⋅ em l abs l em p ⋅
max l 64 exp( ) 2 2 πh c b 3
l
(10.32)