404 Femtosecond Pulse Amplification
gain can be reached and thus correspondingly shorter pulses can be amplified.
This was demonstrated by Glownia et al. [11] and Szatmari et al. [12] who
succeeded in amplifying 150–200 fs pulses in XeCl.
7.2.3. Amplified Spontaneous Emission (ASE)
So far we have neglected one severe problem in (fs) pulse amplification,
namely amplified spontaneous emission (ASE), which mainly results from the
pump pulses being much longer than the fs pulses to be amplified. As a conse-
quence of the medium being inverted before (and after) the actual amplification
process, spontaneous emission traveling through the pumped volume can con-
tinuously be amplified and can therefore reach high energies. ASE reduces the
available gain and decreases the ratio of signal (amplified fs pulse) to background
(ASE), or even can cause lasing of the amplifier, preventing amplification of the
seed pulse. For these reasons the evolution of ASE and its suppression has to be
considered thoroughly in constructing fs pulse amplifiers. Here we shall illus-
trate the essential effects on basis of a simple model (illustrated in Fig. 7.5), and
compare the small signal gain (for the signal pulse) with and without ASE [1].
For simplicity, let us assume that the ASE starts at z = 0 and propagates toward
the exit of the amplifier while being amplified. The photon flux of the ASE is
thus given by:
F
ASE
(z, t) = F
ASE
(0) exp
z
0
σ
ASE
N
1
(z
, t) −N
0
(z
, t)
dz
(7.11)
0
ASE
(0)
ASE
(L)
L
Signal
Pump
ASE
Time
Without signal
With signal
Time
IntensityInversion
Z
σ
ASE
(b)(a)
Figure 7.5 (a) Geometry of unidirectional ASE evolution. (b) Temporal behavior of pump pulse,
ASE, signal pulse and inversion. (Adapted from [1].)
Fundamentals 405
where σ
ASE
is the emission cross section which, with reference to Fig. 7.1,
describes the transition between levels |1 and |0. With strong pumping the
ASE will follow the pump intensity almost instantaneously, and after a cer-
tain time a stationary state is reached in which the population numbers do not
change. This means that additional pump photons are transferred exclusively to
ASE while leaving the population inversion unchanged. Under these conditions
the rate equations for the occupation numbers read:
dN
0
(z, t)
dt
=−σ
02
N
0
(z)F
p
(t) +σ
10
N
1
(z)F
ASE
(z, t) = 0 (7.12)
and
N
0
(z) + N
1
(z) =
¯
N. (7.13)
Combination of Eq. (7.11) with Eqs. (7.12) and (7.13) yields an integral equation
for the gain coefficient a(z) for a signal pulse that has propagated a length z in
the amplifier:
a(z) =
z
0
σ
10
¯
N
F
p
/F
ASE
(0) −e
a(z
)
F
p
/F
ASE
(0) +e
a(z
)
dz
. (7.14)
In the absence of ASE the gain coefficient is:
a = σ
10
¯
Nz. (7.15)
The actual gain in the presence of ASE is reduced to G
a
= exp[a(z)] from the
larger small signal gain in the ideal condition (without ASE) of G
i
= exp(σ
10
¯
Nz).
The ASE at z = 0 can be estimated from:
F
ASE
(0) =
η
F
ω
ASE
4σ
ASE
T
10
(7.16)
where = d
2
/4L
2
is the solid angle spanning the exit area of the amplifier from
the entrance, T
10
is the fluorescence life time, and η
F
is the fluorescence quan-
tum yield. Figure 7.6 shows the result of a numerical evaluation of Eq. (7.14).
Note that a change in small signal gain at constant F
p
can be achieved by chang-
ing either the amplifier length or the concentration
¯
N. As can be seen at high
small signal gain the ASE drastically reduces the gain available to the signal
pulse. In this region a substantial part of the pump energy contributes to the
build up of ASE.
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