120 Femtosecond Optics
where cos
2
β
(ω
) = 1 −[2πc/(ω
d) +sin β]
2
. The third derivative can be
written as
d
3
d
3
ω
=−
3λ
2πc cos
2
β
(ω
)
cos
2
β
(ω
) +
λ
d
λ
d
+ sin β
d
2
d
2
ω
.
(2.114)
To decide when the third term in the expansion [as defined in Eq. (1.167)] of the
phase response of the grating needs to be considered we evaluate the ratio
R
G
=
b
3
( − ω
)
3
b
2
( − ω
)
2
=
(ω
)
3
(ω
)
| − ω
|≈
ω
p
ω
1 +
λ
/d(λ
/d + sin β)
1 − (λ
/d − sin β)
2
(2.115)
where the spectral width of the pulse ω
p
was used as an average value for
| − ω
|.
Obviously it is possible to minimize (or tune) the ratio of second- and third-
order dispersion by changing the grating constant and the angle of incidence. For
instance, with ω
p
/ω
= 0. 05, λ
/d = 0. 5 and β = 0
o
we obtain R
G
≈ 0. 07.
Let us next compare Eq. (2.112) with Eq. (2.76), which related GVD to angu-
lar dispersion in a general form. From Eq. (2.108) we obtain for the angular
dispersion of a grating
dβ
d
ω
=−
2πc
ω
2
d cos β
(2.116)
If we insert Eq. (2.116) in the general expression linking GVD to angular
dispersion, Eq. (2.76), and remember that L = b/ cos β
, we also obtain
Eq. (2.112).
2.5.7. Grating Pairs for Pulse Compressors
For all practical purpose, a pulse propagating from grating G
1
to G
2
can be
considered as having traversed a linear medium of length L characterized by a
negative dispersion. We can write Eq. (2.112) in the form of:
d
2
d
2
ω
= k
L =−
λ
2πc
2
λ
d
2
1
cos
2
β
(ω
)
L. (2.117)
Elements with Angular Dispersion 121
Referring to Table 1.2, a bandwidth-limited Gaussian pulse of duration τ
G0
,
propagating through a dispersive medium characterized by the parameter k
,
broadens to a Gaussian pulse of duration τ
G
τ
G
= τ
G0
-
1 +
L
L
d
2
, (2.118)
with a linear chirp of slope:
¨ϕ =
2L/L
d
1 + (L/L
d
)
2
1
τ
2
G0
(2.119)
where the parameter L
d
relates both to the parameters of the grating and to the
minimum (bandwidth-limited) pulse duration:
L
d
=
τ
2
G0
2|k
|
=
πc
2
d
2
r
λ
3
τ
2
G0
. (2.120)
Conversely, a pulse with a positive chirp of magnitude given by Eq. (2.119) and
duration corresponding to Eq. (2.118) will be compressed by the pair of grat-
ings to a duration τ
G0
. A pulse compressor following a pulse stretcher is used in
numerous amplifications systems and will be dealt with in Chapter 7. The “com-
pressor” is a pair of gratings with optical path L, designed for a compression
ratio τ
G
/τ
G0
= L/L
d
.
11
The ideal compressor of length L will restore the ini-
tial (before the stretcher) unchirped pulse of duration τ
0
. To a departure x from
the ideal compressor length L, corresponds a departure from the ideal unchirped
pulse of duration τ
0
:
τ
G
= τ
G0
-
1 +
x
2
L
2
d
. (2.121)
This pulse is also given a chirp coefficient (cf. Table 1.2) ¯a = x/L
d
.
In most compressors, the transverse displacement of the spectral components
at the output of the second grating can be compensated by using two pairs of
gratings in sequence or by sending the beam once more through the first grating
pair. As with prisms, the overall dispersion then doubles. Tunability is achieved
by changing the grating separation b. Unlike with prisms, however, the GVD is
always negative. The order of magnitude of the dispersion parameters of some
typical devices is compiled in Table 2.2.
11
In all practical cases with a pair of gratings, (L/L
d
)
2
1.
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