7.2. PRECOMPENSATION SCHEMES 281
whereA ̃( 0 ,ω)is the Fourier transform ofA( 0 ,t).
Dispersion-induced degradation of the optical signal is caused by the phase factor
exp(iβ 2 zω^2 / 2 ), acquired by spectral components of the pulse during its propagation in
the fiber. All dispersion-management schemes attempt to cancel this phase factor so
that the input signal can be restored. Actual implementation can be carried out at the
transmitter, at the receiver, or along the fiber link. In the following sections we consider
the three cases separately.
7.2 Precompensation Schemes
This approach to dispersion management modifies the characteristics of input pulses at
the transmitter before they are launched into the fiber link. The underlying idea can be
understood from Eq. (7.1.4). It consists of changing the spectral amplitudeA ̃( 0 ,ω)of
the input pulse in such a way that GVD-induced degradation is eliminated, or at least
reduced substantially. Clearly, if the spectral amplitude is changed as
A ̃( 0 ,ω)→A ̃( 0 ,ω)exp(−iω^2 β 2 L/ 2 ), (7.2.1)whereLis the fiber length, GVD will be compensated exactly, and the pulse will retain
its shape at the fiber output. Unfortunately, it is not easy to implement Eq. (7.2.1)
in practice. In a simple approach, the input pulse is chirped suitably to minimize the
GVD-induced pulse broadening. Since the frequency chirp is applied at the transmitter
before propagation of the pulse, this scheme is called the prechirp technique.
7.2.1 Prechirp Technique
A simple way to understand the role of prechirping is based on the theory presented in
Section 2.4 where propagation of chirped Gaussian pulses in optical fibers is discussed.
The input amplitude is taken to be
A( 0 ,t)=A 0 exp[
−
1 +iC
2(
t
T 0) 2 ]
, (7.2.2)
whereCis the chirp parameter. As seen in Fig. 2.12, for values ofCsuch thatβ 2 C<0,
the input pulse initially compresses in a dispersive fiber. Thus, a suitably chirped pulse
can propagate over longer distances before it broadens outside its allocated bit slot.
As a rough estimate of the improvement, consider the case in which pulse broadening
by a factor of up to
√
2 is acceptable. By using Eq. (2.4.17) withT 1 /T 0 =√
2, the
transmission distance is given by
L=
C+
√
1 + 2 C^2
1 +C^2
LD, (7.2.3)
whereLD=T 02 /|β 2 |is the dispersion length. For unchirped Gaussian pulses,C= 0
andL=LD. However,Lincreases by 36% forC=1. Note also thatL<LDfor
large values ofC. In fact, the maximum improvement by a factor of
√
2 occurs for