Page 97 - Fiber Optic Communications Fund

78 Fiber Optic Communications

with L = 80 km and using Eqs. (2.219) and (2.220), we obtain

2
|S| ≤ 0.917 ps/nm ∕km. (2.221)

2.7.5 Polarization Mode Dispersion
Weakly guiding approximation implies that the electromagnetic field propagates almost in the z-direction
and, therefore, the electric and magnetic field components are nearly transverse (see Section 2.4). The scalar
field of Section 2.4 could either represent the electric field intensity E or E . Therefore, a single-mode
x y
fiber supports two independent propagation modes [30–32]. For one of the modes, the electric field com-
ponent is along the ̂x-direction, the magnetic field component is along the ̂y-direction, and the propagation
constant is (x-polarization). For the other mode, the electric field component is along the ̂y-direction, the
x
magnetic field component is along the ̂x-direction, and the propagation constant is (y-polarization). If the
y
fiber cross-section is perfectly circular, these two modes are degenerate, i.e., = . However, it is hard to
y
x
fabricate a fiber whose cross-section is perfectly circular. Because of the asymmetry introduced during the
fiber manufacturing process, and external factors such as bending or twisting, the propagation constants x
and differ. The inverse group speeds 1x and 1y corresponding to the x- and y-polarization components
y
are also different. As a result, the x- and y-polarization components of the input signal arrive at the fiber out-
put at different times, leading to pulse broadening if a direct detection receiver is used. This phenomenon is
known as polarization mode dispersion (PMD) [33–37]. Owing to random fluctuations in the fiber refractive
index along the fiber axis, there is an exchange of power between these polarization components that occurs
randomly along the fiber. Therefore, the pulse broadening due to PMD is stochastic in nature.
Using Eq. (2.110), the x- and y-components of the electric field intensity can be written as

E (x, y, z, t)= s (t, z)Φ(x, y) exp [−i(t − z)],
x x x
E (x, y, z, t)= s (t, z)Φ(x, y) exp [−i(t − z)], (2.222)
y y y
where s and s are electrical field envelopes, and are propagation constants for the x- and y-polarization
y
x
y
x
components, respectively. The transverse field distributions are nearly the same for x- and y-polarization. As
in Section 2.5, input and output electrical field envelopes are related by
̃ s (Ω) = ̃s (Ω)H (Ω, L),
x,out x,in x
̃ s (Ω) = ̃s (Ω)H (Ω, L), (2.223)
y,out y,in y
where
[ ( 2 )]
Ω z
2
H (Ω, L)= exp −z∕2 + i Ωz + , a = x, y. (2.224)
a
1a
2
Eqs. (2.223) and (2.224) can be written in matrix form using Jones’ vector notation (see Section 1.11):
[ ] [ ]
̃ s x,out (Ω) ̃ s x,in (Ω)
̃ s (Ω) = , ̃ s (Ω) = , (2.225)
in
out
̃ s
y,out (Ω) ̃ s y,in (Ω)
̃ s (Ω) = H(Ω, L)̃ s (Ω), (2.226)
out
in
[ ]
H (Ω, L) 0
x
H(Ω, L)= , (2.227)
0 H (Ω.L)
y