Polarization mode dispersion (PMD) is a source of pulse broadening which results from fiber birefringence and it can become a limiting factor for optical fiber communications at high transmission rates. It is a random effect due to both intrinsic (caused by non- circular fiber core geometry and residual stresses in the glass material near the core region) and extrinsic (caused by stress from mechanical loading, bending or twisting of the fiber) factors which in actual manufactured fibers result in group velocity variation with polarization state.
When considering a short section of single-mode fiber within a long fiber span, as shown in the time domain illustration of Figure 3.29, it can be assumed that any perturba- tions acting on it are constant over its entire length rather than varying along it. In this case the fiber becomes bimodal due to a loss of degeneracy for the two HE11modes. As these
two modes have different phase propagation constants βx and βy they exhibit different
specific group delays (see Section 2.5.4.). In the time domain for a short section of fiber, the differential group delay (DGD), Δτ = δτgL, is defined as the group delay difference
between the slow and the fast modes over the fiber lengths as indicated in Figure 3.29. The DGD can be obtained from the frequency derivative of the difference in the phase propa- gation constants from Eq. (2.107) as:
δτg= = (βx−βy) = −
= Δneff= − Δneff (3.64)
where δτg, the differential group delay per unit length, is referred to as the polarization
mode dispersion (PMD) of the fiber and is usually expressed in units of picoseconds per kilometer of fiber such that the DGD is the time domain manifestation of PMD as shown in Figure 3.29. This linear relationship to fiber length, however, applies only to short fiber lengths or intrinsic PMD in which the birefringence can be assumed to be uniform. Hence for polarization-maintaining fibers δτgcan be quite large at around 1 ns km−
1
when the two components are equally exited at the fiber input.
For conventional single-mode fibers, however, the axis of the birefringence (and its magnitude) varies randomly along the fiber. This phenomenon causes polarization mode coupling such that the fast and slow polarization modes of one segment of a long fiber migrate into both fast and slow modes in the next span. The polarization mode coupling which results from localized stress during cabling, from splices and connectors, and from variations in the fiber drawing process therefore tends to reduce the overall dispersion because the PMD effects do not accumulate linearly in very long fiber spans. In this con- text low-birefringence fibers which exhibit low PMD can also be fabricated by spinning the fiber in the drawing process (see Section 3.3.2.). Hence, as a result of mode coupling in long fiber lengths, the birefringence of each segment adds to, or subtracts from, the total
d dω ω c Δneff c ω c d dω D F ωny c ωnx c A C d dω d dω Δτ L Polarization 145
Figure 3.29 Time domain effect of polarization mode dispersion in a short fiber length with a pulse being launched with equal power on the two birefringent axes,
xand y, becoming two pulses at the output separated by the differential group delay
birefringence so that the DGD does not accumulate linearly. Indeed, in long fiber spans it has been shown that the PMD increases on average with the square root of the length [Ref. 85]. To determine a more precise value for the PMD of a specific fiber link, however, requires a statistical approach [Ref. 86]. Using this method it is possible to categorize fiber into a short- or a long-length regime on the basis of a parameter called the correlation length, defined as the length over which the two polarization modes remain correlated.
The statistical approach to the theory of PMD [Refs 87, 88] provides an expression for the mean square DGD in terms of the fiber polarization beat length LBand the correlation
length Lcdefined as the length over which the two polarization modes remain correlated
such that:
<Δτ2> =
2 ΔτB 2
exp − + −1 (3.65)
where ΔτBis the DGD corresponding to the beat length and L is the fiber length. The cor-
relation length can vary over quite a wide range, from 1 m to 1 km, depending on the specific fiber type, with typical values around 10 m. Moreover, the correlation length can be seen to define two distinct PMD regimes as follows. For LLc, Eq. (3.65) simplifies to:
(<δτ2>
)1---
2=δτ
rms=δτB (3.66)
indicating the linear relationship. When LLc, however, then Eq. (3.65) becomes:
δτrms= (2 LLc)
1 ---
2 (3.67)
demonstrating the dependence on the square root of the length L. Since optical fiber transmission systems usually operate in the long-length regime, then fiber PMD is often specified using a PMD coefficient having units of ps km−1
[Ref. 86]. Although legacy fibers from the 1980s can exhibit mean PMD coefficients greater than 0.8 ps km−1
, recently manufactured fibers usually have mean PMD coefficients lower than 0.1 ps km−1
. Hence, as a result of the L dependence, PMD-induced pulse broadening is often small in comparison with the combined material and waveguide dispersion effects, with Δτrmsonly
around 1 ps for 100 km fiber lengths. Nevertheless, PMD can become a limiting factor in optical communication systems operating over long distances at high transmission rates [Ref. 89] and hence PMD compensation techniques and devices have been developed for these situations [Refs 85, 90, 91]. In addition, it should be noted that the relation- ship in Eq. (3.65) takes no account of other elements in the fiber link which may exhibit polarization-dependent gain or loss, the latter of which can cause additional broadening. The effects of second- and higher order PMD are also important at higher transmission rates of 40 Gbit s−1
and above, particularly for a system where the first-order effects have been removed using a compensator device [Ref. 92].
Although certain single-mode fibers can be fabricated to propagate only one polariza- tion mode (see Section 3.13.3), fibers which transmit two orthogonally polarized funda- mental modes can exhibit interference between the modes which may cause polarization
1 --- 2 δτB LB Lc LB J L L Lc D F L Lc A C G I D F Lc LB A C
146 Transmission characteristics of optical fibers Chapter 3 OPTF_C03.qxd 11/6/08 10:53 Page 146
modal noise. This phenomenon occurs when the fiber is slightly birefringent and there is a component with polarization-dependent loss. Hence, when the fiber link contains an ele- ment whose insertion loss is dependent on the state of polarization, then the transmitted optical power will depend on the phase difference between the normal modes and it will fluctuate if the transmitted wavelength or the birefringence alters. Any polarization- sensitive loss will therefore result in modal noise within single-mode fiber [Ref. 93].
Polarization modal noise is generally of larger amplitude than modal noise obtained within multimode fibers (see Section 3.10.3). It can therefore significantly degrade the performance of a communication system such that high-quality analog transmission may prove impossible [Ref. 47]. Moreover, with digital transmission it is usually necessary to increase the system channel loss margin (see Section 12.6.4). It is therefore important to minimize the use of elements with polarization-dependent insertion losses (e.g. beam splitters, polarization-selective power dividers, couplers to single-polarization optical components, bends in high-birefringence fibers) on single-mode optical fiber links. However, other types of fiber perturbation such as bends in low-birefringence fibers, splices and directional couplers do not appear to introduce significant polarization sensi- tive losses [Ref. 80].
Techniques have been developed to produce both high- and low-birefringence fibers, initially to facilitate coherent optical communication systems. Birefringence occurs when the circular symmetry in single-mode fibers is broken, which can result from the effect of geometrical shape or stress. Alternatively, to design low-birefringence fibers it is neces- sary to reduce the possible perturbations within the fiber manufacture. These fiber types are discussed in the following section.