290 CHAPTER 7. DISPERSION MANAGEMENT
(a) (b)Figure 7.5: (a) Schematic of a DCF made using a higher-order mode (HOM) fiber and two long-
period gratings (LPGs). (b) Dispersion spectrum of the DCF. (After Ref. [47];©c2001 IEEE;
reprinted with permission.)
7.5 Optical Filters
A shortcoming of DCFs is that a relatively long length (>5 km) is required to com-
pensate the GVD acquired over 50 km of standard fiber. This adds considerably to the
link loss, especially in the case of long-haul applications. For this reason, several other
all-optical schemes have been developed for dispersion management. Most of them
can be classified under the category ofoptical equalizing filters. Interferometric filters
are considered in this section while the next section is devoted to fiber gratings.
The function of optical filters is easily understood from Eq. (7.1.4). Since the GVD
affects the optical signal through the spectral phase exp(iβ 2 zω^2 / 2 ), it is evident that
an optical filter whose transfer function cancels this phase will restore the signal. Un-
fortunately, no optical filter (except for an optical fiber) has a transfer function suitable
for compensating the GVD exactly. Nevertheless, several optical filters have provided
partial GVD compensation by mimicking the ideal transfer function. Consider an op-
tical filter with the transfer functionH(ω). If this filter is placed after a fiber of length
L, the filtered optical signal can be written using Eq. (7.1.4) as
A(L,t)=1
2 π∫∞−∞A ̃( 0 ,ω)H(ω)exp(
i
2β 2 Lω^2 −iωt)
dω, (7.5.1)By expanding the phase ofH(ω)in a Taylor series and retaining up to the quadratic
term,
H(ω)=|H(ω)|exp[iφ(ω)]≈|H(ω)|exp[i(φ 0 +φ 1 ω+^12 φ 2 ω^2 )], (7.5.2)whereφm=dmφ/dωm(m= 0 , 1 ,...)is evaluated at the optical carrier frequencyω 0.
The constant phaseφ 0 and the time delayφ 1 do not affect the pulse shape and can be
ignored. The spectral phase introduced by the fiber can be compensated by choosing an
optical filter such thatφ 2 =−β 2 L. The pulse will recover perfectly only if|H(ω)|= 1
and the cubic and higher-order terms in the Taylor expansion in Eq. (7.5.2) are negli-
gible. Figure 7.6 shows schematically how such an optical filter can be combined with
optical amplifiers such that both fiber losses and GVD can be compensated simultane-
ously. Moreover, the optical filter can also reduce the amplifier noise if its bandwidth
is much smaller than the amplifier bandwidth.