Page 464 - Engineering Electromagnetics, 8th Edition

446 ENGINEERING ELECTROMAGNETICS

pulse envelope of width T with a Gaussian envelope whose width is τ. Thus, in
general, the pulse width at location z will be

2
T + ( τ) 2 (94)
T =
EXAMPLE 12.11
2
An optical fiber link is known to have dispersion β 2 = 20 ps /km. A Gaussian light
pulse at the input of the fiber is of initial width T = 10 ps. Determine the width of
the pulse at the fiber output if the fiber is 15 km long.
Solution. The pulse spread will be
β 2 z (20)(15)
τ = = = 30 ps
T 10
So the output pulse width is

2
2
(10) + (30) = 32 ps
T =
An interesting by-product of pulse broadening through chromatic dispersion is
that the broadened pulse is chirped. This means that the instantaneous frequency
of the pulse varies monotonically (either increases or decreases) with time over the
pulse envelope. This again is just a manifestation of the broadening mechanism, in
which the spectral components at different frequencies are spread out in time as they
propagate at different group velocities. We can quantify the effect by calculating the
group delay, τ g ,asa function of frequency, using (92). We obtain:
z dβ
τ g = = z = (β 1 + (ω − ω 0 )β 2 ) z (95)
ν g dω
This equation tells us that the group delay will be a linear function of frequency
and that higher frequencies will arrive at later times if β 2 is positive. We refer to
the chirp as positive if the lower frequencies lead the higher frequencies in time
[requiring a positive β 2 in (95)]; chirp is negative if the higher frequencies lead in time
(negative β 2 ). Figure 12.15 shows the broadening effect and illustrates the chirping
phenomenon.

D12.6. For the fiber link of Example 12.11, a 20-ps pulse is input instead of
the 10-ps pulse in the example. Determine the output pulsewidth.

Ans. 25 ps

As a final point, we note that the pulse bandwidth, ω,was found to be 1/T .
This is true as long as the Fourier transform of the pulse envelope is taken, as was
done with (86) to obtain (87). In that case, E 0 was taken to be a constant, and so the
only time variation arose from the carrier wave and the Gaussian envelope. Such a