Characteristics of Femtosecond Light Pulses 9
Normalized frequency
Spectral intensity
–2 –1 0 1 2
1.2
1.0
0.8
0.6
0.4
0.2
0
(a) (b)
Normalized time
Pulse intensity
–2 –1 0 1 2
1.0
0.8
0.6
0.4
0.2
0
1.2
p
Figure 1.3 Temporal pulse profiles and the corresponding spectra (normalized).
———— Gaussian pulse E (t) ∝ exp [−1. 385(t/τ
p
)
2
]
––––––– Sech pulse E(t) ∝ sech [1. 763(t/τ
p
)]
········· Lorentzian pulse E (t) ∝[1 +1. 656(t/τ
p
)
2
]
−1
——— Asymm. sech pulse E (t) ∝[exp(t/τ
p
) +exp(−3t/τ
p
)]
−1
and Parseval’s theorem [2]:
∞
−∞
|
˜
E
+
(t)|
2
dt =
1
2π
∞
0
|
˜
E
+
()|
2
d (1.26)
from which follows |η|
2
=
0
cn/π. The complete expression for the spectral
intensity [from Eq. (1.24)] is thus:
S() =
0
cn
4π
˜
E( − ω
)
2
. (1.27)
Figure 1.3 gives examples of typical pulse shapes and the corresponding spectra.
The complex quantity
˜
E
+
will be used most often throughout the book to
describe the electric field. Therefore, to simplify notations, we will omit the
superscript + whenever this will not cause confusion.
1.1.3. Pulse Duration and Spectral Width
Unless specified otherwise, we define the pulse duration τ
p
as the full width
at half maximum (FWHM) of the intensity profile, |
˜
E(t)|
2
, and the spectral width
ω
p
as the FWHM of the spectral intensity |
˜
E()|
2
. Making that statement
is an obvious admission that other definitions exist. Precisely because of the
10 Fundamentals
difficulty of asserting the exact pulse shape, standard waveforms have been
selected. The most commonly cited are the Gaussian, for which the temporal
dependence of the field is:
˜
E(t) =
˜
E
0
exp{−(t/τ
G
)
2
} (1.28)
and the secant hyperbolic:
˜
E(t) =
˜
E
0
sech(t/τ
s
). (1.29)
The parameters τ
G
= τ
p
/
√
2 ln 2 and τ
s
= τ
p
/1. 76 are generally more convenient
to use in theoretical calculations involving pulses with these assumed shapes than
the FWHM of the intensity, τ
p
.
Because the temporal and spectral characteristics of the field are related to
each other through Fourier transforms, the bandwidth ω
p
and pulse duration τ
p
cannot vary independently of each other. There is a minimum duration–bandwidth
product:
ω
p
τ
p
= 2πv
p
τ
p
≥ 2πc
B
. (1.30)
c
B
is a numerical constant on the order of 1, depending on the actual pulse
shape. Some examples are shown in Table 1.1. The equality holds for pulses
without frequency modulation (unchirped) which are called “bandwidth limited”
or “Fourier limited.” Such pulses exhibit the shortest possible duration at a given
Table 1.1
Examples of standard pulse profiles. The spectral values given are for unmodulated
pulses. Note that the Gaussian is the shape with the minimum product of mean
square deviation of the intensity and spectral intensity.
Shape Intensity τ
p
Spectral ω
p
c
B
τ
p

p
profile I(t) FWHM profile S() FWHM MSQ
Gauss e
−2(t/τ
G
)
2
1.177τ
G
e
−
τ
G
2
2
2.355/τ
G
0.441 0.5
Sech sech
2
(t/τ
s
) 1.763τ
s
sech
2
πτ
s
2
1.122/τ
s
0.315 0.525
Lorentz [1 +(t/τ
L
)
2
]
−2
1.287τ
L
e
−2||τ
L
0.693/τ
L
0.142 0.7
Asym.
e
t/τ
a
+ e
−3t/τ
a
−2
1.043τ
a
sech
πτ
a
2
1.677/τ
a
0.278
sech
Square 1 for |t/τ
r
|≤1, τ
r
sinc
2
(τ
r
) 2.78/τ
r
0.443 3.27
0 elsewhere
Characteristics of Femtosecond Light Pulses 11
spectral width and pulse shape. We refer the reader to Section 1.1.4, for a more
general discussion of the uncertainty relation between pulse and spectral width
based on mean square deviations (MSQ).
The shorter the pulse duration, the more difficult it becomes to assert its
detailed characteristics. In the femtosecond domain, even the simple concept
of pulse duration seems to fade away in a cloud of mushrooming definitions.
Part of the problem is that it is difficult to determine the exact pulse shape. For
single pulses, the typical representative function that is readily accessible to the
experimentalist is the intensity autocorrelation:
A
int
(τ) =
∞
−∞
I(t)I(t − τ)dt (1.31)
The Fourier transform of the correlation (1.31) is the real function:
A
int
() =
˜
I()
˜
I
∗
() (1.32)
where the notation
˜
I() is the Fourier transform of the function I(t) and should
not be confused with the spectral intensity S(). The fact that the autocorrelation
function A
int
(τ) is symmetric, hence its Fourier transform is real, [2] implies that
little information about the pulse shape can be extracted from such a measure-
ment. Furthermore, the intensity autocorrelation (1.31) contains no information
about the pulse phase or coherence. This point is discussed in detail in Chapter 9.
Gaussian Pulses
Having introduced essential pulse characteristics, it seems convenient to
discuss an example to which we can refer to in later chapters. We choose a
Gaussian pulse with linear chirp. This choice is one of analytical convenience:
the Gaussian shape is not the most commonly encountered temporal shape.
The electric field is given by
˜
E(t) = E
0
e
−(1+ia)(t/τ
G
)
2
(1.33)
with the pulse duration
τ
p
=
√
2ln2 τ
G
. (1.34)
Note that with the definition (1.33) the chirp parameter a is positive for a
downchirp (dϕ/dt =−2at/τ
2
G
). The Fourier transform of (1.33) yields
˜
E() =
E
0
√
πτ
G
4
√
1 + a
2
exp
i −
2
τ
G
2
4(1 +a
2
)
(1.35)

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