Page 463 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 445

them out in time. In this process, a new temporal pulse envelope is constructed whose
width is based fundamentally on the spread of propagation delays of the different
spectral components. By determining the delay difference between the peak spectral
component and the component at the spectral half-width, we construct an expression
for the new temporal half-width. This assumes, of course, that the initial pulse width
is negligible in comparison, but if not, we can account for that also, as will be shown
later on.
To evaluate (88), we need more information about the ω-β curve. If we assume
that the curve is smooth and has fairly uniform curvature, we can express β(ω)asthe
first three terms of a Taylor series expansion about the carrier frequency, ω 0 :

. 1 2
β(ω) = β(ω 0 ) + (ω − ω 0 )β 1 + (ω − ω 0 ) β 2 (89)
2
where

β 0 = β(ω 0 )

dβ
β 1 = (90)
dω
ω 0
and
2
d β
β 2 = (91)
dω 2
ω 0
Note that if the ω-β curve were a straight line, then the first two terms in (89) would
precisely describe β(ω). It is the third term in (89), involving β 2 , that describes the
curvature and ultimately the dispersion.
Noting that β 0 , β 1 , and β 2 are constants, we take the first derivative of (89) with
respect to ω to find
dβ
= β 1 + (ω − ω 0 )β 2 (92)
dω
We now substitute (92) into (88) to obtain
β 2 z
τ = [β 1 + (ω b − ω 0 )β 2 ] z − [β 1 + (ω 0 − ω 0 )β 2 ] z = ωβ 2 z = (93)
T
where ω = (ω b −ω 0 ) = 1/T . β 2 ,as defined in Eq. (91), is the dispersion parameter.
2
Its units are in general time /distance, that is, pulse spread in time per unit spectral
bandwidth, per unit distance. In optical fibers, for example, the units most commonly
2
2
used are picoseconds /kilometer (psec /km). β 2 can be determined when we know
how β varies with frequency, or it can be measured.
If the initial pulse width is very short compared to τ, then the broadened pulse
width at location z will be simply τ.If the initial pulse width is comparable to τ,
then the pulse width at z can be found through the convolution of the initial Gaussian