Shaping through Spectral Filtering 451
Compression factor
a
10
5
L
opt
/L
D
0.3
0.2
0.1
10
8
10
7
10
6
nA (m
2
)
6 J 60 J 600 J
Figure 8.10 Plot of compression parameters after chirping in bulk SQ1 (fused silica) for different
pulse energies and τ
P0
= 60 fs as a function of the beam cross section at the sample input (location of
the beam waist). −·−·: Compression factor, —- parameter of the optimum quadratic compressor (a ≈
3b
2
/τ
2
P0
), −−− normalized optimum sample length (L
D
≈ 3cm). (Adapted from Petrov et al. [32].)
substantially to the continuum generation. There have been successful experi-
ments to compress continuum pulses produced by high-power fs pulses [35].
8.2. SHAPING THROUGH SPECTRAL
FILTERING
On a ps and longer time scale optical pulses can be shaped directly by elements
of which the transmission is controlled externally. An example is a Pockels cell
placed between crossed polarizers and driven by an electrical pulse. The transients
of this pulse determine the time scale on which the optical pulse can be shaped.
The advantage of this technique is the possibility of producing a desired optical
transmission by synthesizing a certain electrical pulse, as demonstrated in the
picosecond scale by Haner and Warren [2]. The speed limitations of electronics
have so far prevented the application of this technique to the fs scale.
A technique best suited for the shortest pulses consists of manipulating the
pulse spectrum in amplitude and phase. This technique was originally introduced
for ps light pulses [36–38] and later successfully applied and improved for fs
optical pulses by Thurston et al. [39] and Weiner et al. [8]. The corresponding
experimental arrangement is shown in Figure 8.11.
The pulse to be shaped is spectrally dispersed using a grating or a pair of
prisms. The spectrum is propagated through a mask which spectrally filters
452 Pulse Shaping
ffff
Grating Grating
Lens
Lens
Mask
“Blue” “Red”
Output pulse
(no mask)
Input pulse
(with m
ask)
Figure 8.11 Spectral filtering in a dispersion-free grating-lens combination. (Adapted from
Weiner et al. [8].)
the pulse. The spectral components are recollimated into a beam by a second
grating or pair of prisms. In the arrangement of Fig. 8.11, the two-grating–two-
lens combination has zero GVD, as can be verified easily by setting z
= z = 0
in Eq. (2.124). Each spectral component is focused at the position of the mask
(the usual criterion for resolution applies). Because, to a good approximation,
the frequency varies linearly in the focal plane of the lens, a variation of the
complex transmission across the mask causes a transfer function of the form:
˜
H() = R()e
−i()
(8.39)
where R() represents the amplitude transmission and () the phase change
experienced by a spectral component at frequency . These masks can be
produced by microlithographic techniques.
1
A pure phase filter, for example,
could consist of a transparent material of variable thickness. If we neglect the
effects caused by the finite resolution, the field at the device output is:
˜
E
out
(t) = F
−1
{
˜
E
out
()}, (8.40)
1
Another option is to use pixelated liquid-crystal arrays whose complex transmission can be
controlled by applying voltages individually to each pixel.
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