2.4. DISPERSION-INDUCED LIMITATIONS 47
whereA ̃( 0 ,∆ω)≡B ̃( 0 ,ω)is the Fourier transform ofA( 0 ,t).
By calculating∂A/∂zand noting that∆ωis replaced byi(∂A/∂t)in the time do-
main, Eq. (2.4.6) can be written as [31]
∂A
∂z+β 1∂A
∂t+
iβ 2
2∂^2 A
∂t^2−
β 3
6∂^3 A
∂t^3= 0. (2.4.7)
This is the basic propagation equation that governs pulse evolution inside a single-mode
fiber. In the absence of dispersion (β 2 =β 3 =0), the optical pulse propagates without
change in its shape such thatA(z,t)=A( 0 ,t−β 1 z). Transforming to a reference frame
moving with the pulse and introducing the new coordinates
t′=t−β 1 z and z′=z, (2.4.8)theβ 1 term can be eliminated in Eq. (2.4.7) to yield
∂A
∂z′+
iβ 2
2∂^2 A
∂t′^2−
β 3
6∂^3 A
∂t′^3= 0. (2.4.9)
For simplicity of notation, we drop the primes overz′andt′in this and the following
chapters whenever no confusion is likely to arise.
2.4.2 Chirped Gaussian Pulses
As a simple application of Eq. (2.4.9), let us consider the propagation of chirped Gaus-
sian pulses inside optical fibers by choosing the initial field as
A( 0 ,t)=A 0 exp[
−
1 +iC
2(
t
T 0) 2 ]
, (2.4.10)
whereA 0 is the peak amplitude. The parameterT 0 represents the half-width at 1/e
intensity point. It is related to the full-width at half-maximum (FWHM) of the pulse
by the relation
TFWHM= 2 (ln 2)^1 /^2 T 0 ≈ 1. 665 T 0. (2.4.11)
The parameterCgoverns the frequency chirp imposed on the pulse. A pulse is said to
be chirped if its carrier frequency changes with time. The frequency change is related
to the phase derivative and is given by
δω(t)=−
∂φ
∂t=
C
T 02
t, (2.4.12)whereφis the phase ofA( 0 ,t). The time-dependent frequency shiftδωis called the
chirp. The spectrum of a chirped pulse is broader than that of the unchirped pulse. This
can be seen by taking the Fourier transform of Eq. (2.4.10) so that
A ̃( 0 ,ω)=A 0(
2 πT 02
1 +iC) 1 / 2
exp[
−
ω^2 T 02
2 ( 1 +iC)