Page 276 - High Power Laser Handbook

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

lϑ
and HR(, ) (highly reflective). The AR face is on the top and the HR
face is at the bottom.
The AR face is typically the interface between the crystalline laser
medium and air (or vacuum); it is antireflective for normal incidence
at the laser and pump wavelengths. Thus it is reasonable to assume
that the AR face is nonreflective for angles smaller than the critical
angle of total reflection ϑ = arcsin (1/ n), with n the index of refraction
tr
of the active medium, and that it is ideally reflecting for larger angles.
In addition, we can assume HR(, )lϑ ≠ 0 for all ϑ due to technical
limitations of the coating design.
In spherical coordinates, Eq. (10.22) transforms to:

s
b max
(,
ϑ
dΦ fϑ)sin ϑϑf l ∫ N s fϑ(, ,) g s f(, ,, )ds (10.23)
d d
=
l
πτ
l
2
4
0
Taking into account the multiple reflections at the faces of the cyl-
inder, s can be expressed by:
max
R − 2 sin f 2 −ρ ρ cos f
s max = sin ϑ (10.24)
lϑ
if AR(, ) HR( ,) ≠ 0 .
lϑ
To account for losses at the AR and HR faces, the gain coefficient
can be modified to:
lϑ
lϑ))
γ lϑ = γ l + ln(AR ( ,)HR (, cos ϑ (10.25)
,
2
h
if AR(, )lϑ HR( ,) ≠ 0 .
lϑ
10.5.11 Time Resolved Numerical Model
Based on these considerations, it is possible to develop a numerical
model of the interaction of amplified spontaneous emission and
34
excitation, a more detailed description can be found in literature .
For this, we discretize the problem in ρ, ϑ, f , and l . Neglecting the
variation of excitation and gain in axial direction and assuming rota-
%
tional symmetry, only the radial variation of Nr () and % γ ()r
l
2
remains.
The temporal development of the excitation can easily be calcu-
lated by integrating the differential Eq. (10.19) with implicit methods.
For each time step, the source Q and the photon flux densities dΦ (ˆ s)
l
are calculated based on the distribution of excitation from the previous
time step. Because the typical time constant of the excitation is in the
order of the spontaneous lifetime (several hundreds of microseconds),
this is adequate for time steps of a few microseconds.