54 CHAPTER 2. OPTICAL FIBERS
provides the frequency response and is called thetransfer function. In general,|H(f)|
falls off with increasingf, indicating that the high-frequency components of the input
signal are attenuated by the fiber. In effect, the optical fiber acts as abandpass filter.
Thefiber bandwidth f3dBcorresponds to the frequencyf=f3dBat which|H(f)|is
reduced by a factor of 2 or by 3 dB:
|H(f3dB)/H( 0 )|=^12. (2.4.37)Note thatf3dBis the optical bandwidth of the fiber as the optical power drops by 3 dB
at this frequency compared with the zero-frequency response. In the field of electrical
communications, the bandwidth of a linear system is defined as the frequency at which
electrical power drops by 3 dB.
Optical fibers cannot generally be treated as linear with respect to power, and Eq.
(2.4.35) does not hold for them [60]. However, this equation is approximately valid
when the source spectral width is much larger than the signal spectral width (Vω 1).
In that case, we can consider propagation of different spectral components indepen-
dently and add the power carried by them linearly to obtain the output power. For a
Gaussian spectrum, the transfer functionH(f)is found to be given by [61]
H(f)=(
1 +
if
f 2)− 1 / 2
exp[
−
(f/f 1 )^2
2 ( 1 +if/f 2 )]
, (2.4.38)
where the parametersf 1 andf 2 are given by
f 1 =( 2 πβ 2 Lσω)−^1 =( 2 π|D|Lσλ)−^1 , (2.4.39)
f 2 =( 2 πβ 3 Lσω^2 )−^1 =[ 2 π(S+ 2 |D|/λ)Lσλ^2 ]−^1 , (2.4.40)and we used Eqs. (2.3.5) and (2.3.13) to introduce the dispersion parametersDandS.
For lightwave systems operating far away from the zero-dispersion wavelength
(f 1
f 2 ), the transfer function is approximately Gaussian. By using Eqs. (2.4.37)
and (2.4.38) withf f 2 , the fiber bandwidth is given by
f3dB=(2ln2)^1 /^2 f 1 ≈ 0. 188 (|D|Lσλ)−^1. (2.4.41)If we useσD=|D|Lσλ from Eq. (2.4.25), we obtain the relationf3dBσD≈ 0. 188
between the fiber bandwidth and dispersion-induced pulse broadening. We can also get
a relation between the bandwidth and the bit rateBby using Eqs. (2.4.26) and (2.4.41).
The relation isB≤ 1. 33 f3dBand shows that the fiber bandwidth is an approximate
measure of the maximum possible bit rate of dispersion-limited lightwave systems. In
fact, Fig. 2.13 can be used to estimatef3dBand its variation with the fiber length under
different operating conditions.
For lightwave systems operating at the zero-dispersion wavelength, the transfer
function is obtained from Eq. (2.4.38) by settingD=0. The use of Eq. (2.4.37) then
provides the following expression for the fiber bandwidth
f3dB=√
15 f 2 ≈ 0. 616 (SLσλ^2 )−^1. (2.4.42)