Pulse Propagation 33
1.2.4. Gaussian Pulse Propagation
For a more quantitative picture of the influence that GVD has on the pulse
propagation we consider the linearly chirped Gaussian pulse of Eq. (1.33)
˜
E(t, z = 0) = E
0
e
−(1+ia)(t/τ
G0
)
2
= E
0
e
−(t/τ
G0
)
2
e
iϕ(t,z=0)
entering the sample. To find the pulse at an arbitrary position z, we multiply
the field spectrum, Eq. (1.35), with the propagator exp
−i
1
2
k
2
z
as done in
Eq. (1.109), to obtain
˜
E(, z) =
˜
A
0
e
−x
2
e
iy
2
(1.119)
where
x =
τ
2
G0
4(1 +a
2
)
(1.120)
and
y(z) =
aτ
2
G0
4(1 +a
2
)
−
k
z
2
. (1.121)
˜
A
0
is a complex amplitude factor which we will not consider in what follows and
τ
G0
describes the pulse duration at the sample input. The time dependent electric
field that we obtain by Fourier transforming Eq. (1.119) can be written as
˜
E(t, z) =
˜
A
1
exp
⎧
⎪
⎨
⎪
⎩
−
1 + i
y(z)
x
⎛
⎜
⎝
t
4
x
[x
2
+ y
2
(z)]
⎞
⎟
⎠
2
⎫
⎪
⎬
⎪
⎭
. (1.122)
Obviously, this describes again a linearly chirped Gaussian pulse. For the “pulse
duration” (note τ
p
=
√
2ln2 τ
G
) and phase at position z we find
τ
G
(z) =
,
4
x
[x
2
+ y
2
(z)] (1.123)
and
ϕ(t, z) =−
y(z)
4[x
2
+ y
2
(z)]
t
2
. (1.124)
34 Fundamentals
Let us consider first an initially unchirped input pulse (a = 0). The pulse duration
and chirp parameter develop as:
τ
G
(z) = τ
G0
-
1 +
z
L
d
2
(1.125)
∂
2
∂t
2
ϕ(t, z) =
1
τ
2
G0
2z/L
d
1 + (z/L
d
)
2
. (1.126)
We have defined a characteristic length:
L
d
=
τ
2
G0
2
k
.
(1.127)
For later reference let us also introduce a so-called dispersive length defined as
L
D
=
τ
2
p0
|k
|
(1.128)
where for Gaussian pulses L
D
≈ 2. 77L
d
. Bandwidth-limited Gaussian pulses
double their length after propagation of about 0. 6L
D
. For propagation lengths
z L
d
the pulse broadening of an unchirped input pulse as described by
Eq. (1.125) can be simplified to
τ
G
(z)
τ
G0
≈
z
L
d
=
2|k
|
τ
2
G0
z. (1.129)
It is interesting to compare the result of Eq. (1.125) with that of Eq. (1.62),
where we used the second moment as a measure for the pulse duration. Because
the Gaussian is the shape for minimum uncertainty [Eq. (1.57)], and because
d
2
φ/d
2
=−k
z, Eq. (1.125) is equivalent to
t
2
=t
2
0
+ 4
(k
)
2
z
2
t
2
0
.
If the input pulse is chirped (a = 0) two different behaviors can occur depending
on the relative sign of a and k
. In the case of opposite sign, y
2
(z) increases
Pulse Propagation 35
monotonously resulting in pulse broadening, cf. Eq. (1.123). If a and k
have
equal sign y
2
(z) decreases until it becomes zero after a propagation distance
z
c
=
τ
2
G0
a
2|k
|(1 + a
2
)
. (1.130)
At this position the pulse reaches its shortest duration
τ
G
(z
c
) = τ
Gmin
=
τ
G0
√
1 + a
2
(1.131)
and the time-dependent phase according to Eq. (1.124) vanishes. From here on
the propagation behavior is that of an unchirped input pulse of duration τ
Gmin
,
that is, the pulse broadens and develops a time-dependent phase. The larger the
input chirp (|a|), the shorter the minimum pulse duration that can be obtained
[see Eq. (1.131)]. The underlying reason is that the excess bandwidth of a chirped
pulse is converted into a narrowing of the envelope by chirp compensation, until
the Fourier limit is reached. The whole procedure including the impression of
chirp on a pulse will be treated in Chapter 8 in more detail.
There is a complete analogy between the propagation (diffraction) effects of
a spatially Gaussian beam and the temporal evolution of a Gaussian pulse in a
dispersive medium. For instance, the pulse duration and the slope of the chirp
follow the same evolution with distance as the waist and curvature of a Gaussian
beam, as detailed at the end of this chapter. A linearly chirped Gaussian pulse in
a dispersive medium is completely characterized by the position and (minimum)
duration of the unchirped pulse, just as a spatially Gaussian beam is uniquely
defined by the position and size of its waist. To illustrate this point, let us consider
a linearly chirped pulse whose “duration” τ
G
and chirp parameter a are known at
a certain position z
1
. The position z
c
of the minimum duration (unchirped pulse)
is found again by setting y = 0 in Eq. (1.121):
z
c
= z
1
+
τ
2
G
2k
a
1 + a
2
= z
1
+ a
τ
2
Gmin
2k
. (1.132)
The position z
c
is after z
1
if a and k
have the same sign
2
; before z
1
if they have
opposite sign. All the temporal characteristics of the pulse are most conveniently
defined in terms of the distance L = z − z
c
to the point of zero chirp, and the
minimum duration τ
Gmin
. This is similar to Gaussian beam propagation where
2
For instance, an initially downchirped (a > 0) pulse at z = z
c
will be compressed in a medium
with positive dispersion (k
> 0).
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