"Introduction". In: Fiber-Optic Communication Systems

48 CHAPTER 2. OPTICAL FIBERS

The spectral half-width (at 1/eintensity point) is given by

∆ω 0 =( 1 +C^2 )^1 /^2 T 0 −^1. (2.4.14)

In the absence of frequency chirp (C=0), the spectral width satisfies the relation
∆ω 0 T 0 =1. Such a pulse has the narrowest spectrum and is calledtransform-limited.
The spectral width is enhanced by a factor of( 1 +C^2 )^1 /^2 in the presence of linear chirp,
as seen in Eq. (2.4.14).
The pulse-propagation equation (2.4.9) can be easily solved in the Fourier domain.
Its solution is [see Eq. (2.4.6)]

A(z,t)=

1

2 π

∫∞

−∞

A ̃( 0 ,ω)exp

(

i

2

β 2 zω^2 +

i

6

β 3 zω^3 −iωt

)

dω, (2.4.15)

whereA ̃( 0 ,ω)is given by Eq. (2.4.13) for the Gaussian input pulse. Let us first con-
sider the case in which the carrier wavelength is far away from the zero-dispersion
wavelength so that the contribution of theβ 3 term is negligible. The integration in Eq.
(2.4.15) can be performed analytically with the result

A(z,t)=

A 0

√

Q(z)

exp

[

−

( 1 +iC)t^2

2 T 02 Q(z)

]

, (2.4.16)

whereQ(z)= 1 +(C−i)β 2 z/T 02. This equation shows that a Gaussian pulse remains
Gaussian on propagation but its width, chirp, and amplitude change as dictated by the
factorQ(z). For example, the chirp at a distancezchanges from its initial valueCto
becomeC 1 (z)=C+( 1 +C^2 )β 2 z/T 02.
Changes in the pulse width withzare quantified through the broadening factor
given by

T 1

T 0

=

[(

1 +

Cβ 2 z

T 02

) 2

+

(

β 2 z

T 02

) 2 ] 1 / 2

, (2.4.17)

whereT 1 is the half-width defined similar toT 0. Figure 2.12 shows the broadening
factorT 1 /T 0 as a function of the propagation distancez/LD, whereLD=T 02 /|β 2 |is
thedispersion length. An unchirped pulse (C=0) broadens as[ 1 +(z/LD)^2 ]^1 /^2 and
its width increases by a factor of

√

2atz=LD. The chirped pulse, on the other hand,
may broaden or compress depending on whetherβ 2 andChave the same or opposite
signs. Forβ 2 C>0 the chirped Gaussian pulse broadens monotonically at a rate faster
than the unchirped pulse. Forβ 2 C<0, the pulse width initially decreases and becomes
minimum at a distance
zmin=

[

|C|/( 1 +C^2 )

]

LD. (2.4.18)

The minimum value depends on the chirp parameter as

T 1 min=T 0 /( 1 +C^2 )^1 /^2. (2.4.19)

Physically, whenβ 2 C<0, the GVD-induced chirp counteracts the initial chirp, and the
effective chirp decreases until it vanishes atz=zmin.