2.4. DISPERSION-INDUCED LIMITATIONS 51
where Eq. (2.3.13) was used to relateβ 3 to the dispersion slopeS. The output pulse
width is thus given by Eq. (2.4.25) but nowσD≡|S|Lσλ^2 /
√
- As before, we can relate
σto the limiting bit rate by the condition 4Bσ≤1. WhenσD σ 0 , the limitation on
the bit rate is governed by
BL|S|σλ^2 ≤ 1 /
√
8. (2.4.28)
This condition should be compared with Eq. (2.3.14) obtained heuristically by using
simple physical arguments.
As an example, consider the case of a light-emitting diode (see Section 3.2) for
whichσλ≈15 nm. By usingD=17 ps/(km-nm) at 1.55μm, Eq. (2.4.26) yields
BL<1 (Gb/s)-km. However, if the system is designed to operate at the zero-dispersion
wavelength,BLcan be increased to 20 (Gb/s)-km for a typical valueS= 0 .08 ps/(km-
nm^2 ).
Optical Sources with a Small Spectral Width
This case corresponds toVω 1 in Eq. (2.4.23). As before, if we neglect theβ 3 term
and setC=0, Eq. (2.4.23) can be approximated by
σ^2 =σ 02 +(β 2 L/ 2 σ 0 )^2 ≡σ 02 +σD^2. (2.4.29)A comparison with Eq. (2.4.25) reveals a major difference between the two cases. In
the case of a narrow source spectrum, dispersion-induced broadening depends on the
initial widthσ 0 , whereas it is independent ofσ 0 when the spectral width of the optical
source dominates. In fact,σcan be minimized by choosing an optimum value ofσ 0.
The minimum value ofσis found to occur forσ 0 =σD=(|β 2 |L/ 2 )^1 /^2 and is given
byσ=(|β 2 |L)^1 /^2. The limiting bit rate is obtained by using 4Bσ≤1 and leads to the
condition
B
√
|β 2 |L≤^14. (2.4.30)The main difference from Eq. (2.4.26) is thatBscales asL−^1 /^2 rather thanL−^1. Figure
2.13 compares the decrease in the bit rate with increasing forσλ=0, 1, and 5 nmL
usingD=16 ps/(km-nm). Equation (2.4.30) was used in the caseσλ=0.
For a lightwave system operating close to the zero-dispersion wavelength,β 2 ≈ 0
in Eq. (2.4.23). UsingVω 1 andC=0, the pulse width is then given by
σ^2 =σ 02 +(β 3 L/ 4 σ 02 )^2 / 2 ≡σ 02 +σD^2. (2.4.31)Similar to the case of Eq. (2.4.29),σcan be minimized by optimizing the input pulse
widthσ 0. The minimum value ofσoccurs forσ 0 =(|β 3 |L/ 4 )^1 /^3 and is given by
σ=(^32 )^1 /^2 (|β 3 |L/ 4 )^1 /^3. (2.4.32)The limiting bit rate is obtained by using the condition 4Bσ≤1, or
B(|β 3 |L)^1 /^3 ≤ 0. 324. (2.4.33)The dispersive effects are most forgiving in this case. Whenβ 3 = 0 .1ps^3 /km, the bit
rate can be as large as 150 Gb/s forL=100 km. It decreases to only about 70 Gb/s