406 Femtosecond Pulse Amplification
10
Small signal gain
10
3
10
5
10
7
10
2
10
3
10
4
10
5
10
6
10
7
Small signal gain with ASE
1.000E07
1,000,000
100,000
10,000
1,000
100
10
1
Figure 7.6 Comparison of small signal gain with and without ASE for different values of the
normalized pump intensity F
p
/F
ASE
(0).
One solution to the problem of gain reduction because of ASE is the seg-
mentation of the amplifier in multiple stages. To understand the nature of this
improvement, let us compare a single- and a two-stage amplifier. With a nor-
malized pump power F
p
/F
ASE
(0) = 10
4
and G
i
= 10
6
, we expect a small signal
amplification of about 10
4
in the single-stage amplifier (Fig. 7.6). In contrast, we
obtain a gain of about 10
3
in one cell and thus 10
6
in the whole device when we
pump two cells by the same intensity, and each has half the length of the original
cell. Another advantage of multistage amplifiers is the possibility to place filters
between the individual stages and thus to reduce further the influence of ASE.
If saturable absorbers are used this may also lead to a favorable steepening of
the leading pulse edge. Moreover, in multistage arrangements the beam size and
pump power can be adjusted to control the saturation, taking into account the
increasing pulse energy. For a more quantitative discussion of the interplay of
ASE and signal pulse amplification as well as for the amplifier design, see, for
example, Penzkofer and Falkenstein [13] and Hnilo and Martinez [14].
7.3. NONLINEAR REFRACTIVE INDEX EFFECTS
7.3.1. General
As discussed in previous chapters, the propagation and amplification of an
intense pulse will induce changes of the index of refraction in the traversed
Nonlinear Refractive Index Effects 407
medium. As a result, the optical pathlength through the amplifier varies along
the beam and pulse profile, leading possibly to SPM and self-lensing. The origins
of this pulse induced change in refractive index can be
(a) saturation in combination with off-resonant amplification (absorption) or
(b) nonresonant nonlinear refractive index effects in the host material.
The nonlinearity is somewhat more complex in semiconductor amplifiers,
because it is related to the dependence of the index of refraction on the car-
rier density (which is a function of current, light intensity, and wavelength). The
nonlinearities are nevertheless large and can contribute to significant spectral
broadening in semiconductor amplifiers [15].
While SPM leads to changes in the pulse spectrum, self-lensing modifies the
beam profile. Being caused by the same change in index, both effects occur simul-
taneously, unless the intensity of the input beam does not vary transversely to the
propagation direction. Such a “flat” beam profile can be obtained by expanding
the beam and filtering out the central part with an almost constant intensity.
It will often be desirable to exploit SPM in the amplifier chain for pulse
compression, while self-focusing should be avoided. There are a number of
successful attempts to achieve spectral broadening (to be exploited in subsequent
pulse compression) through SPM in a dye amplifier [16, 17] and semiconductor
amplifier [15]. We will elaborate on this technique toward the end of this chapter.
At the same time, self-focusing should be avoided, because it leads to instabilities
in the beam parameters such as filamentation or even to material damage.
We proceed with some estimates of the SPM that occurs in amplifiers. We
derived in Chapter 3 an expression [Eq. (3.68)] for the change in instantaneous
frequency with time because of gain depletion:
δω(t) =−
(ω
− ω
10
)T
2
2
e
−a
− 1
e
−a
− 1 +e
W(t)/W
s
I(t)
W
s
. (7.17)
The contribution from the nonlinear refractive index ¯n
2
of the host material is
given by:
δω(t) =−k
¯n
2
z
0
∂
∂t
I(z
, t)dz
(7.18)
A comparison of the functional behavior of the frequency modulation because
of saturation [Eq. (7.17)] and because of the nonlinear index [Eq. (7.18)] is
shown in Figure 7.7. The nonresonant refractive index change always results in
up-chirp at the pulse center while the sign of the chirp because of gain saturation
depends on the sign of the detuning (ω
−ω
10
). The corresponding refractive index
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