Page 275 - High Power Laser Handbook

244 So l i d - S t at e La s e r s Thin-Disc Lasers 245

First of all, it is necessary to calculate the photon flux density.
r
r
With an excitation density Ns() and a gain coefficient γ ()s the pho-
2
l
r
ton flux density arriving at the point s = 0, coming from a volume
element dV at a distance s = r s in the direction ˆ s = r s s, is
/
r
r Ns () 1 r
dΦ () = b l 2 τ 4 s π 2 gs dV (10.20)
()
s
l
l
with the spectral distribution of the fluorescence b , fulfilling
l
b ∫ l l d = 1 and with an amplification of the photon flux density of


r
gs() exp ∫ s γ ( ˆ % ss ds % )  (10.21)
=
l l
  0  
The entire photon flux density at wavelength l from direction ˆ s
can be calculated as
s max
b
(ˆ )(ˆ )
d Φ l s (ˆ) = d Ω τ l ∫ N ss gssds (10.22)
l
2
0
using dV = sdΩ.
2
The maximum integration distance s max depends on the analyzed
geometry. The thin-disc is a cylindrical volume of height (thickness) h
and radius R , with the faces of the cylinder orientated horizontally
(cf. Fig. 10.13). No reflection from the lateral surface is taken into
account; with reflections from the lateral surface, no maximum
integration distance could be defined. The reflectivity of the faces of
the cylinder will be given by the functions AR(, )lϑ (antireflective)
R
S max
S

ρ
h
r
θ
ϕ

Figure 10.13 Geometry of the thin-disc with radius R and thickness h,
illustrating the relations between the maximum integration distance S max and
the radial coordinates r and ρ.