122 Femtosecond Optics
Table 2.2
Values of second-order dispersion for typical devices.
Device λ
(nm) ω
(fs
−1
)
(fs
2
)
Fused silica (L
g
= 1 cm) 620 3.04 535
800 2.36 356
Brewster prism 620 3.04 −760
pair, fused silica
L = 50 cm 800 2.36 −523
Grating pair 620 3.04 −9.3 ×10
4
b = 20 cm; β = 0
◦
d = 1.2 µm 800 2.36 −3 ×10
5
The choice between gratings and prism for controllable dispersion is not
always a simple one. Prisms pairs have lower losses than gratings (the total trans-
mission through a grating pair usually does not exceed 80%), and are therefore
the preferred intracavity dispersive element. Gratings are often used in amplifier
chain where extremely high compression and stretching ratios are desired, which
implies a small L
d
. It should be noted however that L
d
is not only determined by
the properties of the prism or grating, but is also proportional to τ
2
G0
as shown
by Eq. (2.120). Therefore, prisms stretcher-compressors are also used in medium
power amplifiers for sub-20 fs pulses. The disadvantage of prisms is that the
beam has to be transmitted through glass, which, for high power pulses, is a
nonlinear medium.
2.5.8. Combination of Focusing and Angular
Dispersive Elements
A disadvantage of prism and grating sequences is that for achieving large GVD
the length L between two diffraction elements becomes rather large, cf. Eq. (2.76).
As proposed by Martinez [40] the GVD of such devices can be considerably
increased (or decreased) by using them in connection with focusing elements
such as telescopes. Let us consider the optical arrangement of Figure 2.31, where
a telescope is placed between two gratings, so that its object focal point is at
the grating G
1
. According to Martinez, the effect of a telescope can then be
understood as follows. Neglecting lens aberrations, the optical beam path between
focal points F and F
is independent of α, α
. Therefore, the length which has to
be considered for the dispersion reduces from L to
z
= L −FF
= L −2( f +f
). (2.122)
Elements with Angular Dispersion 123
L
G
1
fz
ff
G
2
f
z
F
F
Figure 2.31 Combination of a grating pair and a telescope. In the particular arrangement shown,
the object focal point F is at the grating. In general grating 1 can be a distance z away from F. Note:
z and z
can be positive as well as negative depending on the relative positions of the gratings and
focal points.
In addition, the telescope introduces an angular magnification of M = f /f
, which
means that the angular dispersion of G
1
is magnified by M to M(dα/d). For the
second grating to produce a parallel output beam its dispersion must be M times
larger than that of G
1
. With Eq. (2.76) the overall dispersion is now given by
d
2
d
2
ω
=−
ω
c
dα
d
ω
2
z
M
2
=−
ω
c
dα
d
ω
2
6
L − 2( f + f
)
7
f
f
2
.
(2.123)
Because it is not practicable to use a second grating of higher dispersion, one
can fold the arrangement by means of a roof prism as is done in standard grating
compressors. Also, it is not necessary that the object focal point of the telescope
coincide with the diffraction point Q at the grating. A possible separation z
between Q and F has then to be added to z
M
2
to obtain the overall dispersion
from Eq. (2.123):
d
2
d
2
ω
=−
ω
c
dα
d
ω
2
z
M
2
+ z
. (2.124)
Equation (2.124) suggests another interesting application of telescopic systems.
For z
M
2
+ z < 0 the dispersion changes sign. The largest amount of positive
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