Nonlinear Fiber Optics - 4 ed. Agrawal

58 Chapter 3. Group-Velocity Dispersion
broadens monotonically at a rate faster than that of the unchirped pulse (the dashed
curves). The reason is related to the fact that the dispersion-induced chirp adds to the
input chirp because the two contributions have the same sign.
The situation changes dramatically for β 2 C < 0. In this case, the contribution of the
dispersion-induced chirp is of a kind opposite to that of the input chirp. As seen from
Figure 3.2(b) and Eq. (3.2.20), C 1 becomes zero at a distance ξ = |C|/(1+C 2 ), and the
pulse becomes unchirped. This is the reason why the pulse width initially decreases in
Figure 3.2(a) and becomes minimum at that distance. The minimum value of the pulse
width depends on the input chirp parameter as
T1 min T 0
=
(1 +C 2 . (3.2.21)
) 1/2
Since C 1 = 0 when the pulse attains its minimum width, it becomes transform-limited
such that Δω 0 T min
1 = 1, where Δω 0 is the input spectral width of the pulse.
3.2.3 Hyperbolic Secant Pulses
Although pulses emitted from many lasers can be approximated by a Gaussian shape,
it is necessary to consider other pulse shapes. Of particular interest is the hyperbolic
secant pulse shape that occurs naturally in the context of optical solitons and pulses
emitted from some mode-locked lasers. The optical field associated with such pulses
often takes the form
( ) T
U(0,T )=sech exp
T 0
(− iCT 2
2T 2
0
)
, (3.2.22)
where the chirp parameter C controls the initial chirp similarly to that of Eq. (3.2.15).
The transmitted field U(z,T ) is obtained by using Eqs. (3.2.5), (3.2.6), and (3.2.22).
Unfortunately, it is not easy to evaluate the integral in Eq. (3.2.5) in a closed form for
non-Gaussian pulse shapes. Figure 3.3 shows the intensity and chirp profiles calculated
numerically at z = 2L D and z = 4L D for initially unchirped pulses (C = 0). A comparison
of Figures 3.1 and 3.3 shows that the qualitative features of dispersion-induced
broadening are nearly identical for the Gaussian and “sech” pulses. The main difference
is that the dispersion-induced chirp is no longer purely linear across the pulse.
Note that T 0 appearing in Eq. (3.2.22) is not the FWHM but is related to it by
T FWHM = 2ln(1 + √ 2)T 0 ≈ 1.763T 0 . (3.2.23)
This relation should be used if the comparison is made on the basis of FWHM. The
same relation for a Gaussian pulse is given in Eq. (3.2.8).
3.2.4 Super-Gaussian Pulses
So far we have considered pulse shapes with relatively broad leading and trailing edges.
As one may expect, dispersion-induced broadening is sensitive to the steepness of pulse
edges. In general, a pulse with steeper leading and trailing edges broadens more rapidly