2.1 Ultrafast solid-state lasers - ETH - the Keller Group

118 2.1.6 Mode-locking techniques [Ref. p. 134
In soliton mode-locking, the dominant pulse-formation process is assumed to be soliton formation.
Therefore, the pulse has to be a soliton for which the negative GVD is balanced with
the SPM inside the laser cavity. The pulse duration is then given by the simple soliton solution
(2.1.74). This means that the pulse duration scales linearly with the negative group delay dispersion
inside the laser cavity (i.e. τ p ∝|D|). In the case of an ideal fast saturable absorber, an
unchirped soliton pulse is only obtained at a very specific dispersion setting (2.1.69), whereas for
soliton mode-locking an unchirped transform-limited soliton is obtained for all dispersion levels as
long as the stability requirement against the continuum is fulfilled. This fact has been also used to
experimentally confirm that soliton mode-locking is the dominant pulse-formation process and not
a fast saturable absorber such as KLM [97Aus]. Higher-order dispersion only increases the pulse
duration, therefore it is undesirable and is assumed to be compensated.
Solitons alone in the cavity are not stable. The continuum pulse is much longer and therefore
experiences only the gain at line center, while the soliton exhibits an effectively lower average gain
due to its larger bandwidth. Thus, the continuum exhibits a higher gain than the soliton. After
a sufficient build-up time, the continuum would actually grow until it reaches lasing threshold,
destabilizing the soliton. However, we can stabilize the soliton by introducing a relatively slow
saturable absorber into the cavity. This absorber is fast enough to add sufficient additional loss for
the growing continuum that spreads in time during its build-up phase so that it no longer reaches
lasing threshold.
The break-up into two or even three pulses can be explained as follows: Beyond a certain
pulse energy, two soliton pulses with lower power, longer duration, and narrower spectrum will
be preferred, since their loss introduced by the limited gain bandwidth decreases so much that
the slightly increased residual loss of the less saturated saturable absorber cannot compensate for
it. This results in a lower total round-trip loss and thus a reduced saturated or average gain for
two pulses compared to one pulse. The threshold for multiple pulsing is lower for shorter pulses,
i.e. with spectra which are broad compared to the gain bandwidth of the laser. A more detailed
description of multiple pulsing is given elsewhere [98Kae]. Numerical simulations show, however,
that the tendency for pulse break-up in cases with strong absorber saturation is found to be
significantly weaker than expected from simple gain/loss arguments [01Pas1].
To conclude, soliton shaping effects can allow for the generation of significantly shorter pulses,
compared to cases without SPM and dispersion. The improvement is particularly large for absorbers
with a relatively low modulation depth and when the absorber recovery is not too slow. In this
regime, the pulse shaping is mainly done by the soliton effects, and the absorber is only needed to
stabilize the solitons against growth of the continuum. The absorber parameters are generally not
very critical. It is important not only to adjust the ratio of dispersion and SPM to obtain the desired
soliton pulse duration (2.1.74), but also to keep their absolute values in a reasonable range where
the nonlinear phase change is in the order of a few hundred mrad per round trip (i.e. significantly
larger than acceptable in cases without negative dispersion). Soliton formation is generally very
important in femtosecond lasers, which has been already recognized in colliding pulse mode-locked
dye lasers. However, no analytic solution was presented for the soliton pulse shortening. It was
always assumed that for a stable solution the mode-locking mechanism without soliton effects
has to generate a net gain window as short as the pulse (Fig. 2.1.5a and b). In contrast to these
cases, in soliton mode-locking we present an analytic solution based on soliton perturbation theory,
where soliton pulse shaping is clearly assumed to be the dominant pulse formation process, and
the saturable absorber required for a stable solution is treated as a perturbation. Then, the net
gain window can be much longer than the pulse (Fig. 2.1.5c). Stability of the soliton against the
continuum then determines the shortest possible pulse duration. This is a fundamentally different
mode-locking model than previously described. We therefore refer to it as soliton mode-locking,
emphasizing the fact that soliton pulse shaping is the dominant factor.
Landolt-Börnstein
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