Nonlinear Fiber Optics - 4 ed. Agrawal

214 Chapter 6. Polarization Effects
6.6.1 Polarization-Mode Dispersion
It is intuitively clear that the polarization state of CW light propagating in fibers with
randomly varying birefringence will generally be elliptical and would change randomly
along the fiber during propagation. In the case of optical pulses, the polarization state
can also be different for different parts of the pulse unless the pulse propagates as a
soliton. Such random polarization changes typically are not of concern for lightwave
systems because photodetectors used inside optical receivers are insensitive to the state
of polarization of the incident light (unless a coherent-detection scheme is employed).
What affects such systems is not the random polarization state but pulse broadening
induced by random changes in the birefringence. This is referred to as PMD-induced
pulse broadening.
The analytical treatment of PMD is quite complex in general because of its statistical
nature. A simple model, first introduced in 1986 [125], divides the fiber into a large
number of segments. Both the degree of birefringence and the orientation of the principal
axes remain constant in each section but change randomly from section to section.
In effect, each fiber section can be treated as a phase plate and a Jones matrix can be
used for it [130]. Propagation of each frequency component associated with an optical
pulse through the entire fiber length is then governed by a composite Jones matrix obtained
by multiplying individual Jones matrices for each fiber section. The composite
Jones matrix shows that two principal states of polarization exist for any fiber such that,
when a pulse is polarized along them, the polarization state at fiber output is frequency
independent to first order, in spite of random changes in fiber birefringence. These
states are analogs of the slow and fast axes associated with polarization-maintaining
fibers. Indeed, the differential group delay ΔT (relative time delay in the arrival time
of the pulse) is largest for the principal states of polarization [127].
The principal states of polarization provide a convenient basis for calculating the
moments of ΔT [132]. The PMD-induced pulse broadening is characterized by the
root-mean-square (RMS) value of ΔT , obtained after averaging over random birefringence
changes. Several approaches have been used to calculate this average using
different models [133]–[136]. The variance σ 2 T ≡〈(ΔT )2 〉 turns out to be the same in
all cases and is given by [137]
σ 2 T (z)=2(Δβ 1 ) 2 l 2 c [exp(−z/l c )+z/l c − 1], (6.6.1)
where the intrinsic modal dispersion Δβ 1 = d(Δβ)/dω is related to the difference in
group velocities along the two principal states of polarization. The parameter l c is
the correlation length, defined as the length over which two polarization components
remain correlated; its typical values are ∼10 m.
For short distances such that z ≪ l c , σ T =(Δβ 1 )z from Eq. (6.6.1), as expected
for a polarization-maintaining fiber. For distances z > 1 km, a good estimate of pulse
broadening is obtained using z ≫ l c . For a fiber of length L, σ T in this approximation
becomes
σ T ≈ Δβ 1
√
2lc L ≡ D p
√
L, (6.6.2)
where D p is the PMD parameter. Measured values of D p vary from fiber to fiber, typically
in the range D p = 0.1 to 2 ps/ √ km [138]. Modern fibers are designed to have low