396 Femtosecond Pulse Amplification
The basic design principles of amplifiers have already been established for ps
and ns pulse amplification. The pulses to be increased in energy are sent through
a medium which provides the required gain factor (Figure 7.1). However, on a
femtosecond time scale, new design methods are required to (a) keep the pulse
duration short and (b) prevent undesired nonlinear effects caused by the extremely
high intensities of amplified fs pulses. A popular technique to circumvent the
problems associated with high peak powers is to use dispersive elements to stretch
the pulse duration to the ps scale, prior to amplification.
Femtosecond pulse amplification is a complex issue because of the interplay
of linear and nonlinear optical processes. The basic physical phenomena relevant
to fs amplification are discussed individually in the next sections.
7.2. FUNDAMENTALS
7.2.1. Gain Factor and Saturation
It is usually desired to optimize the amplifier to achieve the highest possi-
ble gain coefficient for a given pump energy. To simplify our discussion let us
assume that the pump inverts uniformly the part of the gain medium (Fig. 7.1)
that is traversed by the pulse to be amplified (single pulse). A longitudinal geom-
etry is often used when pumping the gain medium with a laser of good beam
quality. Transverse pumping is used for high gain amplifiers such as dyes, or
when pumping with low coherence sources such as semiconductor laser bars.
We discuss next the case of transverse pumping. To achieve uniform inversion
with transverse optical pumping, we have to choose a certain concentration
¯
N
of the (amplifying) particles which absorb the pump light and a certain focusing
of the pump. The focusing not only determines the transverse dimensions of the
x
y
z
Pump laser
Pump
Gain
a
b
L
1
2
0
Figure 7.1 Light pulse amplification.
Fundamentals 397
pumped volume, x = a, but also controls the saturation coefficient s
p
and the
depth y = b of the inverted region. The saturation parameter s
p
= W
p0
/W
s
was
defined as the ratio of incident pump pulse density and saturation energy density
[see Eq. (3.59)].
In practice, the pump energy is set by equipment availability and other exper-
imental considerations. Therefore, to change s
p
, we have to change the focusing.
Note that here the total number of excited particles corresponding to the num-
ber of absorbed pump photons remains constant. To illustrate the effect of the
pump focusing for transverse pumping let us determine the depth distribution
of the gain coefficient for various pump conditions. For simplicity, we assume
a three-level system for the amplifier where |0→|2 is the pump transition
and |1→|0is the amplifying transition. The relaxation between |2and |1
is to be much shorter than the pumping rate. Starting from the rate equations
for a two-level system Eqs. (3.51) and (3.52), it can easily be shown that the
system of rate equations for the photon flux density of the pump pulse F
p
and
the occupation number densities N
i
=
¯
Nρ
ii
(i = 0, 1, 2) reads now:
∂
∂t
N
0
(y, t) =−σ
02
N
0
(y, t)F
p
(y, t) (7.1)
∂
∂y
F
p
(y, t) =−σ
02
N
0
(y, t)F
p
(y, t) (7.2)
and
N
1
(y, t) =
¯
N −N
0
(y, t) (7.3)
where σ
02
is the interaction (absorption) cross-section of the transition |0→|2.
The coefficient of the small signal gain, a
g
, is proportional to the occupation
number difference of levels |1 and |0:
a
g
= σ
10
(N
1
− N
0
)L = σ
10
N
10
L (7.4)
where L is the amplifier length. With the initial conditions N
0
(y,0)= N
(e)
0
(y) =
¯
N
(all particles are in the ground state) we find from Eqs. (7.1), (7.2), and (7.3) for
the inversion density N
10
after interaction with the pump:
N
10
(y) =
¯
N
1 −
2
1 − e
a
(
1 − e
s
p
)
(7.5)
398 Femtosecond Pulse Amplification
Absorption
1.5
1
0.5
0
0.5
1
1.5
Gain
6
5
4
3
2
102
Normalized pump depth
Normalized inversion density
345
1
Figure 7.2 Inversion density N
10
as a function of normalized depth y/
a
, where
a
=|σ
02
¯
N|
−1
,
for different saturation parameters s
p
= W
p0
/W
s
.
where a =−σ
02
¯
Ny is the coefficient of the small signal absorption for the
pump. Figure 7.2 shows some examples of the population inversion distribution
for different intensities of the pump pulse.
In the limit of zero saturation the penetration depth is roughly given by the
absorption length
a
=|σ
02
¯
N|
−1
defined as the propagation length at which the
pulse intensity drops to 1/e of its original value. If the pump density is large
enough to saturate the transition 0 → 2 the penetration depth becomes larger
and, moreover, a region of almost constant inversion (gain) is built.
Given a uniformly pumped volume, the system needs to be optimized for
maximum energy amplification of the signal pulse. Using Eq. (3.57) the energy
gain factor achieved at the end of the amplifier can be written as:
1
G
e
=
W(L)
W
0
=
ω
2σ
10
W
0
ln
1 − e
a
g
1 − e
2σ
10
W
0

=
1
s
ln
6
1 − e
a
g
1 − e
s
7
. (7.6)
1
Note that for the amplification, the relations found for the two-level system hold if we assume
that during the amplification process no other transitions occur. This is justified in most practical
situations.

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