"Introduction". In: Fiber-Optic Communication Systems

46 CHAPTER 2. OPTICAL FIBERS

width is dominated by the spectrum of the optical source. In general, the extent of
pulse broadening depends on the width and the shape of input pulses [56]. In this
section we discuss pulse broadening by using the wave equation (2.2.16).

2.4.1 Basic Propagation Equation

The analysis of fiber modes in Section 2.2.2 showed that each frequency component of
the optical field propagates in a single-mode fiber as

E ̃(r,ω)=xˆF(x,y)B ̃( 0 ,ω)exp(iβz), (2.4.1)

wherexˆis the polarization unit vector,B ̃( 0 ,ω)is the initial amplitude, andβis the
propagation constant. The field distributionF(x,y)of the fundamental fiber mode can
be approximated by the Gaussian distribution given in Eq. (2.2.44). In general,F(x,y)
also depends onω, but this dependence can be ignored for pulses whose spectral width
∆ωis much smaller thanω 0 —a condition satisfied by pulses used in lightwave systems.
Hereω 0 is the frequency at which the pulse spectrum is centered; it is referred to as the
carrier frequency.
Different spectral components of an optical pulse propagate inside the fiber accord-
ing to the simple relation

B ̃(z,ω)=B ̃( 0 ,ω)exp(iβz). (2.4.2)

The amplitude in the time domain is obtained by taking the inverse Fourier transform
and is given by

B(z,t)=

1

2 π

∫∞

−∞

B ̃(z,ω)exp(−iωt)dω. (2.4.3)

The initial spectral amplitudeB ̃( 0 ,ω)is just the Fourier transform of the input ampli-
tudeB( 0 ,t).
Pulse broadening results from the frequency dependence ofβ. For pulses for which
∆ω ω 0 , it is useful to expandβ(ω)in a Taylor series around the carrier frequency
ω 0 and retain terms up to third order. In this quasi-monochromatic approximation,

β(ω)=n ̄(ω)

ω

c

≈β 0 +β 1 (∆ω)+

β 2

2

(∆ω)^2 +

β 3

6

(∆ω)^3 , (2.4.4)

where∆ω=ω−ω 0 andβm=(dmβ/dωm)ω=ω 0. From Eq. (2.3.1)β 1 = 1 /vg, where
vgis the group velocity. The GVD coefficientβ 2 is related to the dispersion parameter
Dby Eq. (2.3.5), whereasβ 3 is related to the dispersion slopeSthrough Eq. (2.3.13).
We substitute Eqs. (2.4.2) and (2.4.4) in Eq. (2.4.3) and introduce aslowly varying
amplitude A(z,t)of the pulse envelope as

B(z,t)=A(z,t)exp[i(β 0 z−ω 0 t)]. (2.4.5)

The amplitudeA(z,t)is found to be given by

A(z,t)=

1

2 π

∫∞

−∞

d(∆ω)A ̃( 0 ,∆ω)

×exp

[

iβ 1 z∆ω+

i

2

β 2 z(∆ω)^2 +

i

6

β 3 z(∆ω)^3 −i(∆ω)t

]

, (2.4.6)