12 Fundamentals
with the spectral phase given by:
φ() =−
1
2
arctan(a) +
aτ
G
2
4(1 +a
2
)
2
. (1.36)
It can be seen from Eq. (1.35) that the spectral intensity is the Gaussian:
S(ω
+ ) =
|η|
2
πE
2
0
τ
2
G
√
1 + a
2
exp
−
2
τ
G
2
2(1 +a
2
)
(1.37)
with a FWHM given by:
ω
p
= 2πv
p
=
1
τ
G
8ln2(1+ a
2
) (1.38)
For the pulse duration–bandwidth product we find
v
p
τ
p
=
2ln2
π
1 + a
2
(1.39)
Obviously, the occurrence of chirp (a = 0) results in additional spectral compo-
nents which enlarge the spectral width and lead to a duration–bandwidth product
exceeding the Fourier limit (2 ln 2/π ≈ 0. 44) by a factor
√
1 + a
2
, consistent
with Eq. (1.30). We also want to point out that the spectral phase given by
Eq. (1.36) changes quadratically with frequency if the input pulse is linearly
chirped. Although this is exactly true for Gaussian pulses as can be seen from
Eq. (1.36), it holds approximately for other pulse shapes. In the next section, we
will develop a concept that allows one to discuss the pulse duration–bandwidth
product from a more general point of view, which is independent of the actual
pulse and spectral profile.
1.1.4. Wigner Distribution, Second-Order Moments,
Uncertainty Relations
Wigner Distribution
The Fourier transform as defined in Section 1.1.1 is a widely used tool in
beam and pulse propagation. In beam propagation, it leads directly to the far
field pattern of a propagating beam (Fraunhofer approximation) of arbitrary trans-
verse profile. Similarly, the Fourier transform leads directly to the pulse temporal
Characteristics of Femtosecond Light Pulses 13
profile, following propagation through a dispersive medium, as we will see at the
end of this chapter. The Fourier transform gives a weighted average of the spectral
components contained in a signal. Unfortunately, the exact spatial or temporal
location of these spectral components is hidden in the phase of the spectral field.
There has been therefore a need for new two-dimensional representation of the
waves in either the plane of space–wave vector, or time–angular frequency. Such
a function was introduced by Wigner [3] and applied to quantum mechanics.
The same distribution was applied to the area of signal processing by Ville [4].
Properties and applications of the Wigner distribution in quantum mechanics and
optics are reviewed in two recent books by Schleich [5] and Cohen [6]. A clear
analysis of the close relationship between quantum mechanics and optics can be
found in Praxmeir and W
´
okiewicz [7]. The Wigner distribution of a function
˜
E(t)
is defined by
1
:
W
E
(t, ) =
∞
−∞
˜
E
t +
s
2
˜
E
∗
t −
s
2
e
−is
ds
=
1
2π
∞
−∞
˜
E
+
s
2
˜
E
∗
−
s
2
e
its
ds. (1.40)
One can see that the definition is a local representation of the spectrum of the
signal, because:
∞
−∞
W
E
(t, )dt =
˜
E()
2
(1.41)
and
∞
−∞
W
E
(t, )d = 2π
˜
E(t)
2
. (1.42)
The subscript E refers to the use of the instantaneous complex electric field
˜
E
in the definition of the Wigner function, rather than the electric field envelope
˜
E = E exp[iω
t + iϕ(t)] defined at the beginning of this chapter. There is a
simple relation between the Wigner distribution W
E
of the instantaneous field
˜
E,
1
t and are conjugated variables as in Fourier transforms. The same definitions can be made in
the space–wave vector domain, where the variables are then x and k.
14 Fundamentals
and the Wigner distribution W
E
of the real envelope amplitude E:
W
E
(t, ) =
∞
−∞
E
t +
s
2
e
i[ω
(t+s/2)+ϕ(t+s/2)]
× E
∗
t −
s
2
e
−i[ω
(t−s/2)+ϕ(t−s/2)]
e
−is
ds
=
∞
−∞
E
t +
s
2
E
∗
t −
s
2
e
−i[−(ω
+˙ϕ(t))]s
ds
= W
E
{t, [ −(ω
+˙ϕ)]}. (1.43)
We will drop the subscript E and E for the Wigner function when the distinction
is not essential.
The intensity and spectral intensities are directly proportional to frequency
and time integrations of the Wigner function. In accordance with Eqs. (1.21)
and (1.27):
1
2
√
µ
0
/
∞
−∞
W
E
(t, )d = I(t) (1.44)
1
2
√
µ
0
/
∞
−∞
W
E
(t, )dt = S(). (1.45)
Figure 1.4 shows the Wigner distribution of an unchirped Gaussian pulse [(a), left]
versus a Gaussian pulse with a quadratic chirp [(b), right]. The introduction of a
quadratic phase modulation leads to a tilt (rotation) and flattening of the distribu-
tion. This distortion of the Wigner function results directly from the relation (1.43)
applied to a Gaussian pulse. We have defined in Eq. (1.33) the phase of the linearly
chirped pulse as ϕ(t) =−at
2
/τ
2
G
.IfW
unchirp
is the Wigner distribution of the
unchirped pulse, the linear chirp transforms that function into:
W
chirp
= W
unchirp
t, −
2at
τ
2
G
, (1.46)
hence the tilt observed in Fig. 1.4. Mathematical tools have been developed
to produce a pure rotation of the phase space (t, ). We refer the interested
reader to the literature for details on the Wigner distribution and in particular
on the fractional Fourier transform [8,9]. It has been shown that such a rotation
describes the propagation of a pulse through a medium with a quadratic dispersion
(index of refraction being a quadratic function of frequency) [10].

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