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

116 2.1.6 Mode-locking techniques [Ref. p. 134
the requirements on the saturable absorber and we can obtain ultrashort pulses even in the 10-fs
regime with semiconductor saturable absorbers that have much longer recovery times. With the
same modulation depth, one can obtain almost the same minimal pulse duration as with a fast
saturable absorber, as long as the absorber recovery time is roughly less than ten times longer
than the final pulse width. In addition, high dynamic range autocorrelation measurements showed
excellent pulse pedestal suppression over more than seven orders of magnitude in 130-fs pulses of
a Nd:glass laser [95Kop1] and very similar to or even better than KLM pulses in the 10-fs pulse
width regime [97Jun2]. Even better performance can be expected if the saturable absorber also
shows a negative refractive-index change coupled with the absorption change as is the case for
semiconductor materials [96Kae].
With a slow saturable absorber as a starting and stabilizing mechanism for soliton mode-locking,
there remains a time window with net round-trip gain behind the pulse, where the loss of the still
saturated absorber is smaller than the total loss for the pulse that is balancing the saturated gain.
At a first glance it may seem that the discussion in the last section, Sect. 2.1.6.6, with slow saturable
absorbers would apply here as well. However, there is another limiting effect which usually becomes
more effective in soliton mode-locked lasers: The dispersion causes the background to temporally
broaden and thus permanently loosing the energy in those parts which drift into the time regions
with net loss.
We can describe soliton mode-locking by the Haus’s master equation formalism, where we take
into account GVD, SPM and a slow saturable absorber q (T,t) that recovers slower than the pulse
duration (see Fig. 2.1.5c) [95Kae1, 96Kae]:
∑
]
ΔA i =
[−iD ∂2
∂t 2 +iδ L |A (T,t)| 2 ∂
A (T,t)+
[g 2 ]
− l + D g
∂t 2 − q (T,t) A (T,t)
i
=0. (2.1.71)
This differential equation can be solved analytically using soliton perturbation theory. The first
bracket term of this differential equation determines the nonlinear Schrödinger equation for which
the soliton pulse is a stable solution for negative GVD (i.e. D 0):
( ) t
A (z,t) =A 0 sech e −iφs(z) . (2.1.72)
τ
This soliton pulse propagates without distortion through a medium with negative GVD and positive
SPM because the effect of SPM exactly cancels that due to dispersion. The FWHM soliton pulse
duration is given by τ p =1.7627 · τ and the time–bandwidth product by Δ ν p τ p =0.3148. φ s (z) is
the phase shift of the soliton during propagation along the z-axis
φ s (z) = 1 2 kn 2I 0 z ≡ 1 2 φ nl (z) , (2.1.73)
where I 0 is the peak intensity inside the SPM medium (i.e. typically the gain medium). Thus, the
soliton pulse experiences during propagation a constant phase shift over the full beam profile. For
a given negative dispersion and an intracavity pulse energy E p we obtain a pulse duration
τ p =1.7627 2 |D|
δ L E p
, (2.1.74)
for which the effects of SPM and GVD are balanced with a stable soliton pulse.
This soliton pulse looses energy due to gain dispersion and losses in the cavity. Gain dispersion
and losses can be treated as perturbation to the nonlinear Schrödinger equation for which a soliton is
a stable solution (i.e. second bracket term in (2.1.71)). This lost energy, called continuum in soliton
perturbation theory, is initially contained in a low-intensity background pulse, which experiences
negligible SPM, but spreads in time due to GDD (Fig. 2.1.17a). In soliton mode-locking a stable
Landolt-Börnstein
New Series VIII/1B1