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

100 2.1.5 Pulse propagation in dispersive media [Ref. p. 134
For example, at a fused quartz prism spacing of 40 cm, the total dispersion that is produced by a
double pass through the prism pairs amounts to −862 fs 2 and a third-order dispersion of −970 fs 3 at
a center wavelength of 800 nm, assuming zero prism insertion into the beam (Table 2.1.9). We can
reduce the negative dispersion by moving the prisms into the laser beam: Each additional millimeter
of prism insertion produces a positive dispersion of 101 fs 2 but only a third-order dispersion of
78 fs 3 per cavity round trip. Thus, the prism pairs can generally only be used to compensate for
either second- or third-order dispersion. With fused quartz prisms alone pulses as short as 10 fs
can be produced with a Ti:sapphire laser [93Asa, 96Flu1]. Slightly shorter pulses can be achieved
if the Ti:sapphire laser is operated at a center wavelength of approximately 850 nm, where nearzero
second- and third-order dispersion can be obtained for a Ti:sapphire crystal of about 2 mm
thickness. In this case 8.5 fs have been generated [94Zho]. Another dispersion compensation is then
required for shorter pulse generation. Therefore, the chirped mirrors were designed to show the
inverse higher-order group delay dispersion of the dispersion of the prism pair plus laser crystal
to eliminate the higher-order dispersion and obtain a slightly negative but constant group delay
dispersion required for an ideal soliton pulse.
2.1.5.2.4 Chirped mirrors
Dielectric Bragg mirrors with regular quarter-wave layer stacks have a fairly small dispersion when
operated well within their reflection bandwidth, but increasing dispersion at the edges of this
range. Modified designs can be used to obtain well-controlled dispersion over a large wavelength
range. One possibility already discussed is the use of a GTI structure (see Sect. 2.1.5.2.1). Another
broad range of designs is based on the concept of the chirped mirrors [94Szi] (Fig. 2.1.16d): If the
Bragg wavelength is appropriately varied within a Bragg mirror design, longer wavelengths can be
reflected deeper in the mirror structure, thus acquire a larger phase change, which leads to negative
dispersion. However, the straight-forward implementation of this idea leads to strong oscillations
of the GDD (as a function of frequency) which render such simple designs useless. These oscillations
can be greatly reduced by numerical optimizations which introduce complicated (and not
analytically explainable) deviations from the simple chirp law. A great difficulty is that the figure
of merit to optimize is a complicated function of many layer thickness variables, which typically
has a large number of local maxima and minima and thus is quite difficult to optimize. Nevertheless,
refined computing algorithms have lead to designs with respectable performance, which were
realized with precision growth of dielectric mirrors. Such mirrors can compensate the dispersion
in Ti 3+ :sapphire lasers for operation with pulse durations well below 10 fs. Figure 2.1.16d shows
a typical chirped mirror structure which schematically shows the path of a long-wavelength (i.e.
1000 nm) and short-wavelength (i.e. 650 nm) beam. This results in negative dispersion, because the
long wavelengths are made to penetrate deeper into the mirror structure than short wavelengths.
Figure 2.1.16d also shows the standing-wave electric field patterns in a chirped mirror structure
versus wavelength. The negative dispersion of the mirrors is clearly illustrated by the dependence
of penetration depth on wavelength for the range of 650 to 1050 nm. The highly transmissive region
around 500 nm is used for pumping the Ti:sapphire laser through these chirped mirrors. According
to [94Szi], chirping means that the Bragg wavelength λ B is gradually increased along the mirror,
producing a negative Group Delay Dispersion (GDD). However, no analytical explanation of the
unwanted oscillations typically observed in the group delay and GDD of such a simple-chirped
mirror was given. These oscillations were minimized purely by computer optimization.
The original chirped mirror design was further refined by the Double-Chirped Mirror (DCM)
[97Kae, 99Mat] concept which takes into account the impedance matching problem which occurs
at the air–mirror interface and the grating structure in the mirror. An exact coupled-mode analysis
[97Mat] was used to develop a double-chirp technique. The impedance matching concept allowed a
much better insight to the design limitations and allowed for the first time for an analytical design
with custom-tailored dispersion characteristics which required only minor numerical optimization
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