Nonlinear Fiber Optics - 4 ed. Agrawal

60 Chapter 3. Group-Velocity Dispersion
Intensity
1
0.8
0.6
0.4
0.2
4
z/L D
= 0
2
(a)
0
−6
−10 −5 0 5 10 −20 0 20
Time, T/T 0
Time, T/T 0
Frequency Chirp
6
4
2
0
−2
−4
z/L D
= 0
2
(b)
4
Figure 3.4: Normalized (a) intensity |U| 2 and (b) frequency chirp δωT 0 as a function of T /T 0
for a super-Gaussian pulse at z = 2L D and 4L D . Dashed lines show the input profiles at z = 0.
Compare with Figure 3.1 where the case of a Gaussian pulse is shown.
component is directly related to its separation from the central frequency ω 0 , a wider
spectrum results in a faster rate of pulse broadening.
For complicated pulse shapes such as those seen in Figure 3.4, the FWHM is not a
true measure of the pulse width. The width of such pulses is more accurately described
by the root-mean-square (RMS) width σ defined as [8]
σ =[〈T 2 〉−〈T 〉 2 ] 1/2 , (3.2.26)
where the angle brackets denote averaging over the intensity profile as
∫ ∞
〈T n −∞
〉 =
T n |U(z,T )| 2 dT
∫ ∞
−∞ |U(z,T )|2 dT . (3.2.27)
The moments 〈T 〉 and 〈T 2 〉 can be calculated analytically for some specific cases. In
particular, it is possible to evaluate the broadening factor σ/σ 0 analytically for super-
Gaussian pulses using Eqs. (3.2.5) and (3.2.24) through (3.2.27) with the result [17]
[
σ
=
σ 0
1 + Γ(1/2m)
Γ(3/2m)
Cβ 2 z
T 2
0
+ m 2 (1 +C 2 )
Γ(2 − 1/2m)
Γ(3/2m)
( ) ]
β2 z 2 1/2
T0
2 , (3.2.28)
where Γ(x) is the gamma function. For a Gaussian pulse (m = 1) the broadening factor
reduces to that given in Eq. (3.2.19).
To see how pulse broadening depends on the steepness of pulse edges, Figure 3.5
shows the broadening factor σ/σ 0 of super-Gaussian pulses as a function of the propagation
distance for values of m ranging from 1 to 4. The case m = 1 corresponds
to a Gaussian pulse; the pulse edges become increasingly steeper for larger values of
m. Noting from Eq. (3.2.25) that the rise time is inversely proportional to m, itisevident
that a pulse with a shorter rise time broadens faster. The curves in Figure 3.5 are