124 Femtosecond Optics
d
2
/d
2
is achieved for z =−f and z
=−f
. Because the sign of the angular
dispersion is changed by the telescope we have to tilt the second grating to
recollimate the beam. For the folded geometry we can use a mirror instead of a
roof prism.
In summary, the use of telescopes in connection with grating or prism pairs
allows us to increase or decrease the amount of GVD as well as to change the sign
of the GVD. As will be discussed later, interesting applications of such devices
include the recompression of pulses after long optical fibers and extreme pulse
broadening (>1000) before amplification [40,41]. A more detailed discussion of
this type of dispersers, including the effects of finite beam size, can be found
in Martinez [33].
2.6. WAVE-OPTICAL DESCRIPTION OF
ANGULAR DISPERSIVE ELEMENTS
Because our previous discussion of pulse propagation through prisms, grat-
ings, and other elements was based on ray–optical considerations, it failed to give
details about the influence of a finite beam size. These effects can be included by
a wave-optical description which is also expected to provide new insights into
the spectral, temporal, and spatial field distribution behind the optical elements.
We will follow the procedure developed by Martinez [33], and use the char-
acteristics of Gaussian beam propagation, i.e., remain in the frame of paraxial
optics.
First, let us analyze the effect of a single element with angular dispersion as
sketched in Figure 2.32. The electric field at the disperser can be described by
a complex amplitude
˜
U(x, y, z, t) varying slowly with respect to the spatial and
0
z
y
x
d
z
y
x
0
Figure 2.32 Interaction of a Gaussian beam with a disperser.
Wave-Optical Description of Angular Dispersive Elements 125
temporal coordinate:
E(x, y, z, t) =
1
2
˜
U(x, y, z, t)e
i(ω
t−k
z)
+ c. c. (2.125)
Using Eq. (1.185) the amplitude at the disperser can be written as
˜
U(x, y, t) =
˜
E
0
(t) exp
−
ik
2˜q(d)
(x
2
+ y
2
)
=
˜
U(x, t) exp
−ik
y
2
2˜q(d)
(2.126)
where ˜q is the complex beam parameter, d is the distance between beam waist
and disperser, and
˜
E
0
is the amplitude at the disperser. Our convention shall
be that x and y refer to coordinates transverse to the respective propagation
direction z. Further, we assume the disperser to act only on the field distribution
in the x direction, so that the field variation with respect to y is the same as for
free space propagation of a Gaussian beam. Hence, propagation along a distance
z changes the last term in Eq. (2.126) simply through a change of the complex
beam parameter ˜q. According to Eq. (1.186) this change is given by
˜q(d + z) =˜q(d) +z. (2.127)
To discuss the variation of
˜
U(x, t) it is convenient to transfer to frequencies
¯
and spatial frequencies ρ applying the corresponding Fourier transforms
˜
U(x,
¯
) =
∞
−∞
˜
U(x, t)e
−i
¯
t
dt (2.128)
and
˜
U(ρ,
¯
) =
∞
−∞
˜
U(x,
¯
)e
−iρx
dx. (2.129)
A certain spatial frequency spectrum of the incident beam means that it contains
components having different angles of incidence. Note that
¯
is the variable
describing the spectrum of the envelope (centered at
¯
= 0), while =
¯
+ω
is the actual frequency of the field. In terms of Fig. 2.32 this is equivalent to a
certain angular distribution γ. The spatial frequency ρ is related to γ through
γ =
ρ
k
. (2.130)
For a plane wave,
˜
U(ρ,
¯
) exhibits only one nonzero spatial frequency component
which is at ρ = 0. The disperser not only changes the propagation direction
126 Femtosecond Optics
(γ
0
→ θ
0
) but also introduces a new angular distribution θ of beam components
which is a function of the angle of incidence γ and the frequency
¯
θ = θ(γ, )
=
∂θ
∂γ
γ
0
γ +
∂θ
∂
ω
¯
= αγ + β
¯
. (2.131)
The quantities α and β are characteristics of the disperser and can easily be
determined, for example, from the prism and grating equations.
12
By means
of Eq. (2.130) the change of the angular distribution γ → θ can also be
interpreted as a transformation of spatial frequencies ρ into spatial frequencies
ρ
= θk
where
ρ
= αk
γ + k
β
¯
= αρ +k
β
¯
. (2.132)
Just behind the disperser we have an amplitude spectrum
˜
U
T
(ρ
,
¯
) given by
˜
U
T
(ρ
,
¯
) = C
1
˜
U
1
α
ρ
−
k
α
β
¯
,
¯
(2.133)
where C
1
and further constants C
i
to be introduced are factors necessary for
energy conservation that shall not be specified explicitly. In spatial coordinates
the field distribution reads
˜
U
T
(x,
¯
) =
∞
−∞
˜
U
T
(ρ
,
¯
)e
iρ
x
dρ
= C
1
∞
−∞
˜
U
1
α
ρ
−
k
α
β
¯
,
¯
e
iρ
x
dρ
= C
1
∞
−∞
˜
U(ρ,
¯
)e
iαρ x
e
ik
β
¯
x
d(αρ)
= C
2
e
ik
β
¯
x
˜
U(α x,
¯
). (2.134)
12
For a Brewster prism adjusted for minimum deviation we find α = 1 and β =−(λ
2
/πc)(dn/dλ).
The corresponding relations for a grating used in diffraction order m are α = cos γ
0
/cos θ
0
and
β =−mλ
2
/(2πcd cos θ
0
).
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