Page 272 - High Power Laser Handbook

240 So l i d - S t at e La s e r s Thin-Disc Lasers 241

with

ZT()  2 πh c 
f (, l T) = 2 exp  vac (10.12)
abs ZT() l kT 
1  B 
and Z , Z the partition functions of the lower and upper laser level,
1
2
respectively.
Similarly, one can calculate for a given absorption cross section
σ T () at pump wavelength l the absorption coefficient
abs p
ασ () − σ T ()(1 + f ( TN (10.13)
=
TN
))
abs 0 abs em 2
with
ZT()  2πh c 
f () = em T 1 exp −   l vac  (10.14)

kT
ZT()
2  pB 
With this absorption coefficient, a thickness of the disc h, and a pump
power density E , we can calculate the number of absorbed pump
p
photons per volume and time:
E l 1 [ − −exp( α hM )]
Q = p p p (10.15)
2 πh c vac h
if we use M pump beam passes through the disc.
p
We can also calculate the gain at the laser wavelength for one pass
through the disc:
T N −σ
g = h[ σ em laser (1 + f ()) 2 emlaser abs TN ] (10.16)
)
f (
,
0
,
abs
Because energy extraction is only possible if g > 0, we can simi-
larly define the maximum extractable energy per area as:
2πh c
TN −
[(
H = vac h 1 + f ()) f (TTN) ] (10.17)
extractable l abs 2 abs 0
laser
These formulas ensure the correct handling of thermal popula-
tion, bleaching, and saturation effects.
10.5.8 Coupled Quasi-Static Numerical Model
For the coupled model, the disc is discretized in finite elements in
radial, azimuthal and axial direction. From the equation of motion
(Eq. [10.10]), we can derive the formula for the density N of the
2
3+
excited Yb ions in the quasi-static limit in each element:
l P l ME N 2
pV
r r
l
0
,
2 π c h vac + 2 π c h vac σ em laser [ Nf abs − N 1( + 2 2 abs )]− f τ − D ASE = N 0
(10.18)