50 CHAPTER 2. OPTICAL FIBERS
condition is not always satisfied in practice. To account for the source spectral width,
we must treat the optical field as a stochastic process and consider the coherence prop-
erties of the source through the mutual coherence function [22]. Appendix C shows
how the broadening factor can be calculated in this case. When the source spectrum is
Gaussian with the RMS spectral widthσω, the broadening factor is obtained from [56]
σ^2
σ 02=
(
1 +
Cβ 2 L
2 σ 02) 2
+( 1 +Vω^2 )(
β 2 L
2 σ 02) 2
+( 1 +C^2 +Vω^2 )^2(
β 3 L
4√
2 σ 03) 2
, (2.4.23)
whereVωis defined asVω= 2 σωσ 0. Equation (2.4.23) provides an expression for
dispersion-induced broadening of Gaussian input pulses under quite general condi-
tions. We use it in the next section to find the limiting bit rate of optical communication
systems.
2.4.3 Limitations on the Bit Rate
The limitation imposed on the bit rate by fiber dispersion can be quite different depend-
ing on the source spectral width. It is instructive to consider the following two cases
separately.
Optical Sources with a Large Spectral Width
This case corresponds toVω 1 in Eq. (2.4.23). Consider first the case of a lightwave
system operating away from the zero-dispersion wavelength so that theβ 3 term can
be neglected. The effects of frequency chirp are negligible for sources with a large
spectral width. By settingC=0 in Eq. (2.4.23), we obtain
σ^2 =σ 02 +(β 2 Lσω)^2 ≡σ 02 +(DLσλ)^2 , (2.4.24)whereσλis the RMS source spectral width in wavelength units. The output pulse
width is thus given by
σ=(σ 02 +σD^2 )^1 /^2 , (2.4.25)
whereσD≡|D|Lσλprovides a measure of dispersion-induced broadening.
We can relateσto the bit rate by using the criterion that the broadened pulse should
remain inside the allocated bit slot,TB= 1 /B, whereBis the bit rate. A commonly used
criterion isσ≤TB/4; for Gaussian pulses at least 95% of the pulse energy then remains
within the bit slot. The limiting bit rate is given by 4Bσ≤1. In the limitσD σ 0 ,
σ≈σD=|D|Lσλ, and the condition becomes
BL|D|σλ≤^14. (2.4.26)This condition should be compared with Eq. (2.3.6) obtained heuristically; the two
become identical if we interpret∆λas 4σλin Eq. (2.3.6).
For a lightwave system operating exactly at the zero-dispersion wavelength,β 2 = 0
in Eq. (2.4.23). By settingC=0 as before and assumingVω 1, Eq. (2.4.23) can be
approximated by
σ^2 =σ 02 +^12 (β 3 Lσω^2 )^2 ≡σ 02 +^12 (SLσλ^2 )^2 , (2.4.27)