Page 307 - High Power Laser Handbook

276 So l i d - S t at e La s e r s Heat-Capacity Lasers 277

of its position relative to the ground state, is effectively 0. The parti-
tion functions Z and Z in Eq. (11.6) are defined as
2
3
ZT() = ∑ e − E / kT (11.7)
ia
i
a
where a labels the sublevels of level i. One finds that Z is, to a good
32
approximation,

Z () = T 11 4 exp (−∆ E / kT)] (11.8)
+
/
[
32 32
where ∆E = 0.13 eV.
32
The equations describing the oscillator can then be given as
( ))( +
TN T ( )−
∂NT () N (s ( ) 0 ) s ( )TN T I + I − )
0
2 = 0 Rz t) − k NT ( ) − 21 2 12 1 L L
0
(,
∂t hc p T 2 h ν
L
∂I ± n ∂I ±
() s
± L + L = = ()TN T − {[s ()NT ± + )](I ± ) aI }}(L / L )
0
±
T
− I
(
∂z c ∂t 21 2 12 1 L n L slab cav
(11.9)
where k is the total decay rate out of the upper laser level (i.e.,
T
k = T k + F k ASE , where k is the fluorescence decay rate and k ASE is the
F
decay rate due to ASE, see Sec. 11.3.2). The noise term that initiates
±
the lasing process is given by I , and any distributed loss in the sys-
n
tem is given by the parameter a. The above set of equations is closed
by noting that N + 0 N = 2 N, where N is the total Nd concentration.
The initial/boundary conditions are
=
Nt 0) = It 0) = 0
0
=
±
(
(
2 L

=
t =
=
=
=
+
(
−
Iz 0, t) = R Iz 0,) annd Iz L , ) RI zL ,)
t
t
+
(
−
(
(
L 1 L L cav 2 L cav
(11.10)
where R is the high-reflector reflectivity and R is the output-coupler
2
1
reflectivity.
As an example of the results obtained with these calculations, we
show in Fig. 11.10 the spatially averaged gain coefficient, the output
intensity, and the output fluence as a function of time for a four-slab
Nd:YAG system.
The presence of relaxation oscillations is readily apparent in
Fig. 11.10, as is the clamping of the gain at threshold after a steady
state has been reached. For this case, the output fluence is approxi-
2
mately 1 J/cm . For an active region 100 cm in area, this represents
2
an output energy/pulse of about 100 J, or an average power of 20 kW
at a 200-Hz repetition rate.