Nonlinear Fiber Optics - 4 ed. Agrawal

72 Chapter 3. Group-Velocity Dispersion
4σ < T B [44]; for a Gaussian pulse, at least 95% of the pulse energy remains within the
bit slot when this condition is satisfied. The limiting bit rate is obtained using 4Bσ < 1.
Assuming σ 0 ≪ σ, this condition becomes
4BL|D|σ λ < 1. (3.4.2)
As an illustration, consider the case of multimode semiconductor lasers [28] for which
σ λ ≈ 2 nm. If the system is operating near λ = 1.55 μm using standard fibers,
D ≈ 16 ps/(km-nm). With these parameter values, Eq. (3.4.2) requires BL < 8 (Gb/s)-
km. For a 100-km-long fiber, GVD restricts the bit rate to relatively low values of only
80 Mb/s. However, if the system is designed to operate near the zero-dispersion wavelength
(occurring near 1.3 μm) such that |D| < 1 ps/(km-nm), the BL product increases
to beyond 100 (Gb/s)-km.
Modern fiber-optic communication systems operating near 1.55 μm reduce the
GVD effects using dispersion-shifted fibers designed such that the minimum-loss wavelength
and the zero-dispersion wavelengths nearly coincide. At the same time, they use
lasers designed to operate in a single longitudinal mode such that the source spectral
width is well below 100 MHz [28]. Under such conditions, V ω ≪ 1 in Eq. (3.3.16). If
we neglect the β 3 term and set C = 0, Eq. (3.3.16) can be approximated by
σ =[σ 2 0 +(β 2 L/2σ 0 ) 2 ] 1/2 . (3.4.3)
A comparison with Eq. (3.4.1) reveals a major difference: Dispersion-induced broadening
now depends on the initial width σ 0 . In fact, σ can be minimized by choosing an
optimum value of σ 0 . The minimum value of σ is found to occur for σ 0 =(|β 2 |L/2) 1/2
and is given by σ =(|β 2 |L) 1/2 . The limiting bit rate is obtained by using 4Bσ < 1or
the condition
4B(|β 2 |L) 1/2 < 1. (3.4.4)
The main difference from Eq. (3.4.2) is that B scales as L −1/2 rather than L −1 . Figure
3.10 compares the decrease in the bit rate with increasing L by choosing D = 16 ps/(kmnm)
and σ λ = 0 and 1 nm. Equation (3.4.4) was used in the case σ λ = 0.
For a lightwave system operating exactly at the zero-dispersion wavelength, β 2 =0
in Eq. (3.3.16). Assuming V ω ≪ 1 and C = 0, the pulse width is given by
σ =[σ 2 0 + 1 2 (β 3L/4σ 2 0 ) 2 ] 1/2 . (3.4.5)
Similar to the case of Eq. (3.4.3), σ can be minimized by optimizing the input pulse
width σ 0 . The minimum value of σ 0 is found to occur for σ 0 =(|β 3 |L/4) 1/3 . The
limiting bit rate is obtained by using the condition 4Bσ < 1 and is given by [44]
B(|β 3 |L) 1/3 < 0.324. (3.4.6)
The dispersive effects are most forgiving in this case. For a typical value β 3 = 0.1
ps 3 /km, the bit rate can be as large as 150 Gb/s for L = 100 km. It decreases to only
70 Gb/s even when L increases by a factor of 10 because of the L −1/3 dependence of
the bit rate on the fiber length. The dashed line in Figure 3.10 shows this dependence
using Eq. (3.4.6) with β 3 = 0.1 ps 3 /km. Clearly, the performance of a lightwave system
can be considerably improved by operating it close to the zero-dispersion wavelength
of the fiber.