Page 220 - High Power Laser Handbook

188 So l i d - S t at e La s e r s Zigzag Slab Lasers 189

8.2.2 Scaling Laws
Under steady-state operating conditions, in which the gain medium
is volumetrically pumped and simultaneously cooled from the sur-
face, the temperature gradients in the gain medium are the ultimate
limitation to power scaling. Figure 8.3 shows a simplified graphical
representation of the cooling geometry for a slab and a cylindrical rod
of thickness t and diameter d, respectively. The functional depen-
dence of ∆T under uniform heat deposition for a slab is given by

2
/
k
T
∆= Qt 8 (8.1)
For a rod it is
k
/
T
2
∆= Qd 16 (8.2)
where Q is the volumetric heat density and k is the thermal conduc-
tivity. For propagation down the axis of a gain medium of length L,
this center-to-edge temperature difference results in optical path dif-
ference (OPD) ∆z across the aperture of the gain medium:
dn
L
z
∆= ∆T (8.3)
dT
where dn/dT is the coefficient of index change with temperature.
To the first order, the parabolic wavefront curvature introduced
by this OPD can be approximated as a thermally induced lens of focal
length:

f = d /( 8 ∆ z) (8.4)
2

d t

2
∆T = Qd /8k
2
∆T = Qd /16k
Cylindrical rod Rectangular slab
Figure 8.3 Temperature gradients in a uniformly heated cylindrical rod and slab.