Elements with Angular Dispersion 117
Angle of incidence
8
45% loss
8
6
6
4
4
2
2
40
°
50
°
60
°
70
°
–4
°
0
°
4
°
–4
°
0
°
4
°
Reflection loss (%)
(10
–9
nm
–2
)
1
L
d
2
P
d
2
–
Figure 2.29 Dispersion (solid lines) and reflection losses (dash–dotted lines) of a two-prism
sequence (SQ1—fused silica) as a function of the angle of incidence on the first prism surface.
Symmetric beam path through the prism at the central wavelength is assumed. Curves for three dif-
ferent apex angles (−4
◦
,0
◦
,4
◦
) relative to α = 68. 9
◦
(apex angle for a Brewster prism at 620 nm)
are shown. The tic marks on the dashed lines indicate the angle of incidence and the dispersion where
the reflection loss is 4.5%. (Adapted from Petrov et al. [31]).
the prisms to determine the phase factor at any frequency and angle of inci-
dence [30, 31, 37–39]. The more complex studies revealed that the GVD and
the transmission factor R [as defined in Eq. (2.71)] depend on the angle of inci-
dence and apex angle of the prism. In addition, any deviation from the Brewster
condition increases the reflection losses. An example is shown in Fig. 2.29.
2.5.6. GVD Introduced by Gratings
Gratings can produce larger angular dispersion than prisms. The resulting
negative GVD was first utilized by Treacy [28] to compress pulses of a Nd:glass
laser. In complete analogy with prisms, the simplest practical device consists of
two identical elements arranged as in Figure 2.30 for zero net angular dispersion.
The dispersion introduced by a pair of parallel gratings can be determined by
tracing the frequency dependent ray path. The optical path length
ACP between
A and an output wavefront
PP
o
is frequency dependent and can be determined
118 Femtosecond Optics
A
b
G
2
P
0
G
1
P
l
C
0
C
d
d
Figure 2.30 Two parallel gratings produce GVD without net angular dispersion. For convenience
a reference wavefront is assumed so that the extension of
PP
0
intersects G
1
at A.
with help of Fig. (2.30) to be:
ACP =
b
cos(β
)
6
1 + cos(β
+ β)
7
(2.107)
where β is the angle of incidence, β
is the diffraction angle for the frequency
component and b is the normal separation between G
1
and G
2
. If we restrict our
consideration to first-order diffraction, the angle of incidence and the diffraction
angle are related through the grating equation
sin β
− sin β =
2πc
d
(2.108)
where d is the grating constant. The situation with gratings is however different
than with prisms, in the sense that the optical path of two parallel rays out of
grating G
1
impinging on adjacent grooves of grating G
2
will see an optical path
difference
CP −
C
0
P
0
of mλ, m being the diffraction order. Thus, as the angle
β
changes with wavelength, the phase factor ACP/c increments by 2mπ each
time the ray
AC passes a period of the ruling of G
2
[28]. Because only the relative
phase shift across
PP
0
matters, we may simply count the rulings from the (virtual)
intersection of the normal in A with G
2
. Thus, for first-order diffraction (m = 1),
Elements with Angular Dispersion 119
we find for ():
() =
c
ACP() − 2π
b
d
tan(β
). (2.109)
The group delay is given by:
d
d
=
b
c
1 + cos(β +β
)
cos β
+
b
c cos
2
β
sin β
6
1 + cos(β +β
)
7
−cos β
sin(β +β
)
dβ
d
−
2π
d
b
cos
2
β
dβ
d
=
b
c
1 + cos(β +β
)
cos β
. (2.110)
In deriving the last equation, we have made use of the grating equation sin β
−
sin β = 2πc/(d). Equation (2.110) shows a remarkable property of gratings,
namely that the group delay is simply equal to the phase delay. The carrier to
envelope delay is zero. The second-order derivative, obtained by differentiation
of Eq. (2.110), is:
d
2
d
2
=
b
c
1
cos
2
β
sin β
6
1 + cos(β +β
)
7
− cos β
[sin(β +β
)]
dβ
d
=
−4π
2
bc
3
d
2
cos
3
β
, (2.111)
where we have again made use of the grating equation. Evaluating this expression
at the central frequency ω
, and using wavelengths instead of frequencies:
d
2
d
2
ω
=−
λ
2πc
2
λ
d
2
b
cos
3
β
(λ
)
.
(2.112)
In terms of the distance L = b/cos β
between the gratings along the ray at
= ω
:
d
2
d
2
ω
=−
λ
2πc
2
λ
d
2
L
cos
2
β
(ω
)
. (2.113)
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