772
Pulse Propagation in Optical Fiber
E.1
Propagation of Chirped Gaussian Pulses
Mathematically, a chirped Gaussian pulse at
z
=
0
is described by the equation
G(t)
=±
±
A
0
e
−
1
+
iκ
2
²
t
T
0
³
2
e
−
iω
0
t
´
=
A
0
e
−
1
2
²
t
T
0
³
2
cos
µ
ω
0
t
+
κ
2
¶
t
T
0
·
2
¸
.
(E.6)
The peak amplitude of the pulse is
A
0
. The parameter
T
0
determines the width of the
pulse. It has the interpretation that it is the halfwidth of the pulse at the
1
/e
intensity
point. (The intensity of a pulse is the square of its amplitude.) The
chirp factor
κ
determines the degree of chirp of the pulse. From (E.4), the phase of this pulse is
φ(t)
=
ω
0
t
+
κt
2
2
T
2
0
.
The instantaneous angular frequency of the pulse is the derivative of the phase and
is given by
d
dt
µ
ω
0
t
+
κ
2
t
2
T
2
0
¸
=
ω
0
+
κ
T
2
0
t.
We deﬁne the
chirp factor
of a Gaussian pulse as
T
2
0
times the derivative of its
instantaneous angular frequency. Thus the chirp factor of the pulse described by
(E.6) is
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 Spring '09
 Boussert
 Derivative, Physical quantities, Fundamental physics concepts, chirped Gaussian pulse

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