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

Ref. p. 134] 2.1 Ultrafast solid-state lasers 113
coupled cavity mode-locking techniques. KLM is very close to an ideal fast saturable absorber,
where the modulation depth is produced either by the decreased losses because of self-focusing
through a hard aperture [91Kel, 91Sal1, 91Neg] or by increased gain in the laser as a result of an
increased overlap of the laser mode with the pump mode in the laser crystal [93Pic]. Only in the
ultrashort pulse regime of a few optical cycles more complicated space–time coupling occurs and
wavelength-dependent effects start to limit further pulse reduction.
Besides the tremendous success of KLM, there are some significant limitations for practical or
“real-world” ultrafast lasers. First, the cavity is typically operated near one end of its stability
range, where the Kerr-lens-induced change of the beam diameter is large enough to sustain modelocking.
This results in a requirement for critical cavity alignment where mirrors and the laser
crystal have to be positioned to an accuracy of several hundred microns typically. Once the cavity is
correctly aligned, KLM can be very stable and under certain conditions even self-starting. However,
self-starting KLM lasers in the sub-50-fs regime have not yet been demonstrated without any
additional starting mechanisms. This is not surprising, since in a 10-fs Ti:sapphire laser with a
100 MHz repetition rate, the peak power changes by 6 orders of magnitude when the laser switches
from cw to pulsed operation. Therefore, nonlinear effects that are still effective in the sub-10-fs
regime are typically too small to initiate mode-locking in the cw-operation regime. In contrast,
if self-starting is optimized, KLM tends to saturate in the ultrashort pulse regime or the large
Self-Phase Modulation (SPM) will drive the laser unstable.
For an ideal fast saturable absorber, Haus et al. [92Hau] developed an analytic solution for the
pulse width starting with the following master equation:
∑
ΔA i =
i
[
g
(
1+ 1 Ω 2 g
∂ 2 )
]
∂t 2 − l + γ A |A| 2 A (T,t) = 0 (2.1.63)
not taking into account GVD and SPM. The solution is an unchirped sech 2 -pulse shape
( ) t
P (t) =P 0 sech 2 τ
(2.1.64)
with a FWHM pulse width:
τ p =1.7627 · τ =1.7627 4D g
γ A E p
, (2.1.65)
where P 0 is the peak power of the pulse, E p is the intracavity pulse energy and D g is the gain
dispersion of the laser medium (2.1.32).
The shortest possible pulses can be obtained when we use the full modulation depth of the
fast saturable absorber. We only obtain an analytic solution if we assume an ideal fast absorber
that saturates linearly with the pulse intensity (2.1.18) over the full modulation depth. This is
clearly a strong approximation because (2.1.18) only holds for weak absorber saturation. For a
maximum modulation depth and this linear approximation we then can assume that γ A P 0 = q 0 .
For a sech 2 -shaped pulse (2.1.64), the pulse energy is given by
∫
E p = I (t)dt =2τP 0 . (2.1.66)
The minimal pulse width for a fully saturated ideal fast absorber then follows from (2.1.65):
τ p,min = 1.7627 √ 2g
. (2.1.67)
Ω g q 0
This occurs right at the stability limit when the filter loss due to gain dispersion is equal to the
residual loss a soliton undergoes in an ideal fast saturable absorber (2.1.19):
Landolt-Börnstein
New Series VIII/1B1