Page 95 - Fiber Optic Communications Fund

76 Fiber Optic Communications

2.7.4 Dispersion Slope
The dispersion parameter depends on the wavelength. For a single-channel low-bit-rate optical communica-
tion system, the spectral width Δf (and therefore the wavelength spread |Δ|) is quite small and the dispersion
parameter D() can be considered as a constant over a small Δ. However, for a high-bit-rate wide-band
communication system, the dependence of D on the wavelength can not be ignored. For example, the spectral
width of an optical non-return-to-zero (NRZ) signal at B = 160 Gb/s is roughly (see Chapter 4)

Δf ≃ 2B = 320 GHz. (2.206)

Since c = f,
Δ Δf
=− . (2.207)
f
At the carrier wavelength = 1550 nm, Δ =−2.56 nm. The change in dispersion over such a large wave-
c
length spread around 1550 nm can not be ignored. It is useful to define the dispersion slope as
dD
S = . (2.208)
d
Using Eq. (2.204), the dispersion slope for a standard SMF can be calculated as
[ 4 ]
S 0 3 0
S = 1 + . (2.209)
4 4

If the dispersion parameter D and the dispersion slope S at the carrier wavelength are known, the disper-
c
c
c
sion parameter in the vicinity of the carrier wavelength can be obtained by the linear approximation [29]
D()= D + S ( − ). (2.210)
c c 0
2
In Eq. (2.102), we have retained the Taylor expansion terms up to Ω ∕2. Under this approximation, or
2
2
D is constant over the spectral width of the signal. To include the impact of the dispersion slope, we need to
include a higher-order term in the Taylor expansion of Eq. (2.102),
3
2
(Ω) = + Ω+ Ω ∕2 + Ω ∕6, (2.211)
1
2
3
0
where
3 |
2
d | d |
= | = | . (2.212)
3
3
|
d | d |= 0
|= 0
Substituting Eq. (2.211) in Eq. (2.98), we obtain
∞
[ ( )]
̃
F(t, z)= exp −i t − z ∫ B(Ω)H (Ω, z) exp (−iΩt) dΩ, (2.213)
0
f
0
−∞
where the fiber transfer function is modified as
[ 2 3 ]
H (Ω, z)= exp −z + i Ωz + i Ω z∕2 + i Ω z∕6 . (2.214)
1
f
2
3
The field envelope at the output is given by Eq. (2.113) as before,
]
]
s (t)=  −1 [ ̃ s (Ω) =  −1 [ H (Ω, z)̃s (Ω) . (2.215)
f
o
i
o